1. Epidemiology and Global Health
  2. Microbiology and Infectious Disease
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Assessing the danger of self-sustained HIV epidemics in heterosexuals by population based phylogenetic cluster analysis

  1. Teja Turk
  2. Nadine Bachmann
  3. Claus Kadelka
  4. Jürg Böni
  5. Sabine Yerly
  6. Vincent Aubert
  7. Thomas Klimkait
  8. Manuel Battegay
  9. Enos Bernasconi
  10. Alexandra Calmy
  11. Matthias Cavassini
  12. Hansjakob Furrer
  13. Matthias Hoffmann
  14. Huldrych F Günthard
  15. Roger D Kouyos Is a corresponding author
  16. Swiss HIV Cohort Study
  1. University Hospital Zurich, Switzerland
  2. University of Zurich, Switzerland
  3. Geneva University Hospitals, Switzerland
  4. University Hospital Lausanne, Switzerland
  5. University of Basel, Switzerland
  6. University Hospital Basel, Switzerland
  7. Regional Hospital Lugano, Switzerland
  8. Lausanne University Hospital, Switzerland
  9. Bern University Hospital, University of Bern, Switzerland
  10. Cantonal Hospital St. Gallen, Switzerland
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Cite as: eLife 2017;6:e28721 doi: 10.7554/eLife.28721

Abstract

Assessing the danger of transition of HIV transmission from a concentrated to a generalized epidemic is of major importance for public health. In this study, we develop a phylogeny-based statistical approach to address this question. As a case study, we use this to investigate the trends and determinants of HIV transmission among Swiss heterosexuals. We extract the corresponding transmission clusters from a phylogenetic tree. To capture the incomplete sampling, the delayed introduction of imported infections to Switzerland, and potential factors associated with basic reproductive number R0, we extend the branching process model to infer transmission parameters. Overall, the R0 is estimated to be 0.44 (95%-confidence interval 0.420.46) and it is decreasing by 11% per 10 years (4%17%). Our findings indicate rather diminishing HIV transmission among Swiss heterosexuals far below the epidemic threshold. Generally, our approach allows to assess the danger of self-sustained epidemics from any viral sequence data.

https://doi.org/10.7554/eLife.28721.001

eLife digest

In epidemiology, the “basic reproductive number” describes how efficiently a disease is transmitted, and represents the average number of new infections that an infected individual causes. If this number is less than one, many people do not infect anybody and hence the transmission chains die out. On the other hand, if the basic reproductive number is larger than one, an infected person infects on average more than one new individual, which leads to the virus or bacteria spreading in a self-sustained way.

Turk et al. have now developed a method to estimate the basic reproductive number using the genetic sequences of the virus or bacteria, and have used it to investigate how efficiently HIV spreads among Swiss heterosexuals. The results show that the basic reproductive number of HIV in this group is far below the critical value of one and that over the last years this number has been decreasing. Furthermore, the basic reproductive number differs for different subtypes of the HIV virus, indicating that the geographical region where the infection was acquired may play a role in transmission. Turk et al. also found that people who are diagnosed later or who often have sex with occasional partners spread the virus more efficiently.

These findings might be helpful for policy makers as they indicate that the risk of self-sustained transmission in this group in Switzerland is small. Furthermore the method allows HIV epidemics to be monitored at high resolution using sequence data, assesses the success of currently implemented preventive measures, and helps to target subgroups who are at higher risk of an infection – for instance, by supporting frequent HIV testing of these people. The method developed by Turk et al. could also prove useful for assessing the danger of other epidemics.

https://doi.org/10.7554/eLife.28721.002

Introduction

Epidemics of HIV and other blood-borne and sexually transmitted diseases (for instance syphilis, HBV and HCV) can be subdivided into concentrated and generalized epidemics. While for the former, the rapid infectious agent transmission is restricted to core transmission groups involved in high-risk behaviors (such as men who have sex with men and injecting drug users), the generalized epidemic refers to fast pathogen spreading in the heterosexual (general) population resulting in higher overall disease prevalence. Mechanistically, the key factor explaining whether the HIV transmission is concentrated or generalized, is the ability of HIV to spread among heterosexuals. If the epidemic in this population is not self-sustained, the HIV epidemic remains concentrated; otherwise the virus is spreading rapidly in the broad population leading to a generalized HIV epidemic.

In most resource-rich settings HIV transmission is concentrated, that is, driven mostly by transmission among men who have sex with men (MSM) and injecting drug users (IDU), whereas the limited transmission among heterosexuals is maintained by either imported infections or spillovers from other transmission groups (Kouyos et al., 2010; von Wyl et al., 2011; Ragonnet-Cronin et al., 2016; Xiridou et al., 2010; Esbjörnsson et al., 2016; Sallam et al., 2017). This suggests that in most Western European countries and similar epidemiological settings the basic reproductive number R0 among heterosexuals is below 1. However, it is not clear how far away from self-sustained the epidemic is in heterosexuals. Moreover, the change in HIV transmission among heterosexuals over time is another important, yet unknown, factor, especially with evidenced increasing risky sexual behavior (Kouyos et al., 2015). It is therefore crucial to assess both the transmission and its time trend in order to obtain meaningful insights into the epidemic.

Assessing the subcritical transmission of HIV in the general population shares some methodological similarities with the analysis of stage III zoonoses, for instance, monkeypox (Wolfe et al., 2007), which also exhibit stuttering transmission chains. Both cases follow a source-sink dynamics, i.e., a flux of infections from a subpopulation in which the disease is self-sustained to a population where it is not. For the case of stage III zoonoses and tuberculosis, it has been shown that the distribution of outbreak sizes can be used to quantify the pathogen spread (Blumberg and Lloyd-Smith, 2013b; Blumberg and Lloyd-Smith, 2013a; Borgdorff et al., 1998). The fundamental approach of our study is to apply this concept to transmission of HIV in the general population. However, there are two key differences between emerging zoonotic pathogens and human-to-human infectious agents. Firstly, while the contact tracing data are not available for many sexually transmitted infections (STI), the viral sequences carry valuable information about the transmission chain size distribution. Thus, the approach of quantifying transmissibility from chain size distributions needs to be combined with a tool to derive clusters from viral sequences. Compared to the animal-human transmission the delayed introduction of the index case of an STI or blood-borne virus to the subpopulation of interest plays an important role, especially in viruses like HIV with long infectious periods in the absence of treatment and higher transmissibility during the acute phase (Marzel et al., 2016; Powers et al., 2011; Rieder et al., 2010; Rodger et al., 2016; Hollingsworth et al., 2008; Cohen et al., 2011b; Cohen et al., 2011a; Cohen et al., 2016). This is especially important because a considerable fraction of HIV cases in heterosexuals is found in migrants (Del Amo et al., 2004; von Wyl et al., 2011; European Centre for Disease Prevention and Control/WHO Regional Office for Europe, 2016). If, for example, a migrant infected with HIV abroad moves to Switzerland in the chronic stage of the infection, he/she has (from the perspective of the Swiss population) lost some transmission potential upon entering Swiss heterosexual transmission network.

In order to quantify the subcritical transmission we combine phylogenetic cluster analysis with an adapted version of a branching process model based estimator that derives the basic reproductive number R0 from the size distribution of transmission chains. We further extend this approach to determine the impact of calendar time and other potential determinants on R0; especially in order to assess whether R0 exhibits an increasing time trend or is high in particular subgroups. Applying this method to the phylogenetic transmission clusters among heterosexuals in the Swiss HIV Cohort Study (SHCS), we can assess transmission of HIV in this population and in particular the risk of a generalized HIV epidemic together with the main determinants of transmission.

Results

We developed a method to assess how far HIV transmission in populations with basic reproductive number R0<1 is from the epidemic threshold, that is, how far it is from being self-sustained in these populations (see Materials and methods). A classical application of this question/method is HIV-1 transmission in heterosexuals in settings with a concentrated epidemic. Heterosexual HIV-1 transmission in Switzerland is a case in point for such a non-self-sustained HIV epidemic. We identified 3,100 transmission clusters among heterosexuals in the SHCS. These clusters were small in size (Table 1) and comprised individuals of broad demographic background (see Table 2). Based on the most likely geographic origin of the transmission clusters, we classified 1,133 transmission chains as being of Swiss origin, that is, to represent introductions from other transmission groups in Switzerland into the heterosexual population, and 1,967 to be of non-Swiss origin. For these latter transmission chains, we assumed that the R0 of the index case was reduced by a factor of ρindex=0.35 (see Materials and methods). To take into account the imperfect sampling density we fixed the subtype-depending sampling probabilities based on the results from the study by Shilaih et al. (2016), corrected by the proportion of the HIV infected individuals linked to care (80% based on Kohler et al., 2015) and the fraction of heterosexuals from the SHCS with an HIV sequence in the phylogenetic tree (57.22%). The model parameters used in this study are summarized in Table 1 (see Sensitivity analyses, Appendix 1—figure 1 and Appendix 1—figure 2 for the corresponding sensitivity analyses).

Table 1
Transmission chain size distribution and model parameters.
https://doi.org/10.7554/eLife.28721.003
SubtypeOverall
BC01_AE02_AGAOther
Total number of chains, n (%)1643
(53%)
322
(10%)
239
(7.7%)
331
(11%)
327
(11%)
238
(7.7%)
3100
(100%)
Chain size, n (%)
 11437
(87%)
280
(87%)
206
(86%)
272
(82%)
269
(82%)
195
(82%)
2659
(86%)
 2158
(9.6%)
34
(11%)
31
(13%)
40
(12%)
44
(13%)
36
(15%)
343
(11%)
 330
(1.8%)
7
(2.2%)
1
(0.42%)
10
(3.0%)
10
(3.1%)
6
(2.5%)
64
(2.1%)
 412
(0.73%)
-1
(0.42%)
6
(1.8%)
3
(0.92%)
1
(0.42%)
23
(0.74%)
 51
(0.06%)
1
(0.31%)
-2
(0.6%)
1
(0.31%)
-5
(0.16%)
 61
(0.06%)
--1
(0.3%)
--2
(0.06%)
 71
(0.06%)
-----1
(0.03%)
 82
(0.12%)
-----2
(0.06%)
 91
(0.06%)
-----1
(0.03%)
Sampling probability, p (SD)0.390.290.340.260.330.290.35
(0.05)
Chain origin, n (%)
 Swiss (ρindex=1)948
(58%)
36
(11%)
36
(15%)
36
(11%)
47
(14%)
30
(13%)
1133
(37%)
 non-Swiss (ρindex=0.35)695
(42%)
286
(89%)
203
(85%)
295
(89%)
280
(86%)
208
(87%)
1967
(63%)

R0 of the HIV transmission in Swiss heterosexuals

To obtain an overall estimate for the R0 of HIV transmission in Swiss heterosexuals, the baseline model was fitted to all of the previously described transmission chain data. In this baseline model the R0 was estimated to be 0.44 (95%-confidence interval (CI) 0.420.46). The fact that R0 was clearly below 1 (p-value <0.001 from one-sided Wald hypothesis testing H0:R0=1 against the alternative HA:R0<1) indicated that HIV transmission is far away from a self-sustained epidemic.

Although the overall R0 estimate was clearly below 1, individual subtypes represent different epidemiological settings and hence individual subtypes may have R0 closer to the epidemic threshold. The subtype-stratified analyses indeed yielded lower R0 of 0.35 (95%-CI 0.330.39) for subtype B as compared to the non-B subtypes (Figure 1). The recombinant form CRF02_AG had the highest estimated R0 of 0.62 (95%-CI 0.560.69). Despite these differences among the R0 estimates for different subtypes they were all significantly below 1 (with all p-values from the one-sided test smaller than 0.001). Therefore, we concluded that there is no danger of a self-sustained HIV epidemic in Swiss heterosexuals of any HIV subtype.

Overall basic reproductive number R0 and R0 per subtype from stratified analysis.

The dark gray point indicates the overall basic reproductive number R0 estimate (by neglecting the transmission chain subtypes) and the corresponding 95%-confidence interval is shown with the dark gray line and the gray-shaded band. The analogous results from the per-subtype stratified analysis are represented by colored points and lines, each color corresponding to one of the subtypes (B, C, CRF01_AE, CRF02_AG or A) or the group of subtypes (other).

https://doi.org/10.7554/eLife.28721.004

Time trend of the R0

Despite consistently low R0 estimates, an increasing time trend for R0 would impose a potential concern, especially if the time trend would predict a crossing of the epidemic threshold in the near future. To investigate this, we fitted a univariate model with log(R0) as a linear function of the establishment date of the transmission chain. We found that overall the R0 is decreasing at a factor 0.89 per 10 years (95%-CI 0.830.96). The per subtype-stratified analyses showed the consistently decreasing time trend among the subtypes ranging from factor 0.65 per 10 years for subtype A to 0.89 for B-subtype.

To better capture the changes of R0 over time we included higher-order polynomials of the establishment date to our model (Figure 2). With the reference date on the 1st of January 1996 (which corresponds to the median estimated date of infection - see Table 2) a cubic spline (without the linear term) was identified as the optimal model according to the Bayesian information criterion (BIC). This model exhibits a mild increase of the R0 from the mid 1980’s to the mid 1990’s, with a peak-R0 of 0.49 (95%-CI 0.460.53) reached in 1996 and followed by a steep and monotonic decrease. It is noteworthy that the time of peak-R0 coincided with the introduction of highly active antiretroviral therapy. Shortly after the R0 started to rapidly decrease and has never rebounded. This extrapolation should be, however, taken with a grain of salt and seen more as a trend rather than a prognosis, since only a few transmission chains have been observed for the recent years (which is reflected by wide confidence intervals).

Time trends for R0.

The upper smaller panels show the time trends for R0 from the subtype-stratified analyses, in which the log(R0)’s were modeled as linear functions of establishment date (i.e., for each subtype the time trend rate was assumed to be constant). The colored shaded-bands correspond to the 95%-prediction bands. The (best-fitting) nonlinear time trend for R0 from the overall analysis is displayed in the lower panel (dark gray curve) together with the 95%-prediction band (gray-shaded area). The black points represent the R0 estimates from the per establishment year stratified analyses and the gray vertical lines the corresponding 95%-confidence intervals.

https://doi.org/10.7554/eLife.28721.005
Table 2
Patients’ demographic characteristics.
https://doi.org/10.7554/eLife.28721.006
PatientsTransmission chains
Index case
Total number, n36983100
Age at estimated date of infection [in years], median (IQR)29.2 (23.1—37.8)28.8 (22.8—37.4)
Estimated date of infection, median (IQR)Jun 1996 (Sep 1990—Nov 2001)Nov 1995 (Sep 1989—May 2001)
Time to diagnosis [in years], median (IQR)3.40 (1.66—5.24)3.54 (1.78—5.43)
Reported sex with occasional partner [as fraction of FUPs*], median (IQR)0.53 (0.09—0.89)0.50 (0.07—0.88)
 No available FUP, n (%)250 (6.8%)226 (7.3%)
Earliest CD4 count [per μL], median (IQR)310 (143—510)300 (134—507)
  1. *Follow-up visit (FUP).

    Patients without FUP questionnaire regarding the sexual risk behavior. See Sensitivity analyses.

  2. One patient did not have any available CD4 cell count. The missing value was imputed with the mean CD4 cell count.

Determinants of the HIV-transmission

Finally, we identified the characteristics associated with higher R0 and therefore potential focal subpopulations, in which the basic reproductive number R0 could be above 1. The simplest model containing only the linear terms of risk factors showed that the R0 is decreasing with the establishment date of the transmission chain and that all non-B subtypes have higher R0 compared to subtype B, which was consistent with the findings from the univariate model and per-subtype stratified analyses. Moreover, we found that reporting sex with occasional partners and longer time to HIV diagnosis of the index case are associated with higher R0, whereas the earliest CD4 cell count and the age do not have significant effects (Figure 3).

Effect of different factors on the basic reproductive number R0 from the multivariate model with only linear factor terms.

The black square and the black line show the reference basic reproductive number R0 and its 95%-confidence interval (for a transmission chain of subtype B which started on 1.1.1996, and in which the index case was diagnosed 3 years after the infection, was 32 years old upon infection, never reported on having sex with occasional partner and had the earliest CD4 cell count of 350 cells per μL). The vertical gray line separates the factors associated with lower R0 (left; effect factor <1) and from the factors contributing to higher R0 (right; effect factor >1). The black points on this line refer to the reference transmission chain. The colored and dark gray lines represent the effect sizes from multivariate model (black circles depicting the estimates) for different factors and their 95%-confidence intervals. The corresponding p-values are shown in the rightmost column. FUP, follow-up visit.

https://doi.org/10.7554/eLife.28721.007

These trends remained robust (Figure 4) when allowing the covariables to enter the model non-linearly (for instance as polynomials like in the case of the time trend above). The final multivariate model identified subtype, establishment date of the transmission chain, frequency of reporting sex with occasional partner and time to diagnosis of the index case as the significant risk factors associated with R0 (see Selection of the predictive models). Allowing nonlinear terms for the time to diagnosis provided better goodness-of-fit than the linear model. The steep increase of R0 in the early/acute phase (see Figure 4) of the infection indicates the importance of early diagnosis (which is nowadays closely related to early treatment initiation) while the time becomes less relevant in the cases diagnosed late in the chronic phase.

Final multivariate model’s profile plots of factors associated with the basic reproductive number R0.

The vertical dotted lines depict the reference transmission chain (of subtype B, started on 1.1.1996, in which the observed index case did not report having sex with occasional partner and was diagnosed after 3 years after the infection). The left y-axis represents the basic reproductive number whereas the right y-axis corresponds to the relative values of R0 as compared to the baseline R0. The R0 as the function of specific factor (with the other factors held fixed at the reference value) is displayed by the colored (for HIV-1 subtype) and the dark gray (establishment date, sexual risk behavior and time to diagnosis) lines. The vertical bars and the shaded bands, respectively, correspond to the 95%-confidence intervals.

https://doi.org/10.7554/eLife.28721.008

Discussion

Our approach demonstrates that viral sequences combined with basic demographic information can be successfully used not only to estimate the basic reproductive number R0 of HIV in a subcritical setting and thereby assess the danger of a generalized HIV epidemic but also to shed light on the trends and other determinants of viral transmission. As a proof of concept, this approach was applied to HIV transmission in Swiss heterosexuals, for which we found an R0 far below the epidemic threshold with a decreasing time trend - indicating a low and decreasing danger of a generalized epidemic. Even though the Swiss HIV epidemic is captured in outstanding detail and representativeness by the SHCS, our approach can be easily used in other non-self-sustained epidemics since viral sequences from genotypic resistance testing are nowadays routinely produced in most resource-rich settings. Moreover, the generalizability of our approach might be broadened to other settings and viruses due to the increased availability of viral sequences boosted by decreasing sequencing costs and the ability of the method to adjust for imperfect sampling.

To our knowledge our study represents the first systematic assessment of the basic reproductive number for subcritical HIV transmission among heterosexuals, which makes it difficult to compare our results to other estimates. In addition, it was conducted in one of the most densely sampled settings. Most of the studies investigated the transmission route composition of larger transmission clusters across different B and non-B subtypes (Esbjörnsson et al., 2016; Chaillon et al., 2017Ragonnet-Cronin et al., 2016; Sallam et al., 2017; Kouyos et al., 2010; von Wyl et al., 2011), or focused on homosexual men or injecting drug users as the main drivers of HIV transmission (Amundsen et al., 2004). Stadler et al. (2012) previously presented a birth-death process based analysis of HIV transmission in Switzerland. However, since this approach is restricted to sufficiently large clusters, it is not suitable for subcritical settings and might potentially overestimate R0 due to selection bias. Hence, our approach, which is tailored to subcritical viral transmission, is complementary to theirs. Among other studies specific for heterosexual populations, Hughes et al. (2009) focused on the clusters of size at least 2 across non-B subtypes, and Xiridou et al. (2010) studied the impact of sexual behavior of migrants on the HIV prevalence, while none of them directly assessed the danger of self-sustained epidemics.

Epidemiological differences between the HIV-1 subtypes, especially between B and non-B subtypes, have been pointed out previously (Kouyos et al., 2010; von Wyl et al., 2011). Yet the exact factors contributing to the differences are difficult to identify. On the one hand, the non-B subtypes are often seen in relation to the infections imported from abroad, which could be introduced either by immigrants or by residents who got infected while temporarily abroad. A proportion of these introductions could be attributed to the sex tourism (Rogstad, 2004). However, even the differences between the various non-B subtypes could be substantial, as they represent different epidemiological settings. For instance, the CRF01_AE is often found in Asians and it also most likely originates from Southeastern Asia (Angelis et al., 2015), while subtypes originating from Africa, such as CRF02_AG (Mir et al., 2016), are frequently found in people of black ethnicity. Additionally, poverty and different policies regulating prostitution worldwide also have an impact on the transmission patterns, like on rate of condom use, access to HIV testing and treatment (Shannon et al., 2015). On the other hand, disentangling the effect of different epidemiological characteristics and even of the strains remains challenging, as R0 was significantly affected by the HIV subtype even in the multivariate model (Figure 3).

One of the key components of our model is the index case relative transmission potential ρindex, which is also associated with some degree of uncertainty. To illustrate its role and influence on the transmission parameters we performed a range of sensitivity analyses (Appendix 1—figure 1). On the one hand, omitting the reduced transmissibility of the index case, that is, assuming ρindex=1, leads to largely underestimated R0 (overall R0 of 0.33, 95%-CI 0.310.35) affirming the importance of this extension. Then again, the concrete value chosen may be debatable, especially due to arguable infectivity in chronic phase (studied by Bellan et al., 2015); thus a small ρindex can be caused both by immigration later during chronic infection and by elevated infectivity in the acute phase. To address this issue we lowered the ρindex for the transmission chains of non-Swiss origin to 0.25 to obtain a more conservative estimate of R0, which was, nevertheless, still safely below 1 (0.47, 95%-CI 0.440.49). Furthermore, even though theoretically the transmission potential of some index cases could also be enhanced (i.e., ρindex>1), for instance for sex workers, we do not expect that this is the case for many transmission chains and would therefore have only marginal effect on our estimates. Besides, since a ρindex>1 would lead to even lower R0, our main conclusions would not change (in fact, the assumption of ρindex<1 is conservative with respect to our conclusion of R0<1).

The presented model is based on source-sink dynamics, which is reflected in the importance of the index case and its immigration background, while the role of emigration is neglected. However, in many resource-rich settings similar source-sink patterns can be observed, both in the migration related influxes and the new virus introductions in the heterosexual population from other risk groups. Namely, the immigration from a setting with a generalized epidemic to a setting with a concentrated epidemic is by far more likely than the emigration. Similarly, occasional spillovers from other risk groups, such as MSM and IDU, to the generalized population are more probable than the reverse. Therefore, the assumption of absence of such outflow from the epidemiological setting under consideration is not problematic when considering a country like Switzerland, but might present a potential limitation if the unit of interest is smaller, like a region or a city.

Our approach has theoretically several limitations, which we, however, expect to have only moderate impact. First, we assumed stuttering transmission chains, or in other words, that the basic reproductive number R0 is below 1. If R0 was larger than 1 the observed transmission chains would have been much longer (see Sensitivity analyses and Appendix 1—figure 5) which is inconsistent with rather small clusters observed in HIV transmission among Swiss heterosexuals (Kouyos et al., 2010von Wyl et al., 2011 and Shilaih et al., 2016). Second, some transmission chains might still be active, meaning that some patients from the chain could be still infectious and therefore able to further spread the virus. The consequence of this would be an underestimation of R0 for recent years. However, given much higher transmissibility of HIV in the acute and recent infection (Marzel et al., 2016) and estimated mean time to being non-infectious of approximately 22.5 years in recent years (Stadler et al., 2012; Hughes et al., 2009) the majority of the observed transmission chains had most likely been stopped by the time of sampling and hence we do expect that this issue will not lead to a major bias of our estimates (see Sensitivity analyses and Appendix 1—figure 4). Third, since our method is based on transmission clusters their misidentification and negligence of their structure could be another constraint. Possible overlapping transmission chains (as it was also noted in Blumberg and Lloyd-Smith, 2013b), that is, misidentifying two transmission chains resulting from two separate introductions of closely related viruses as one single chain, represent the biggest concern in this regard. Failing to identify separate clusters would lead to a higher R0 estimate. However, this means that our method will tend to overestimate R0 and is hence conservative with respect to its main aim of assessing the danger of self-sustained epidemics; thus, if the method predicts an R0 strongly below 1, the corresponding epidemic will indeed be far away from being self-sustained. Moreover, our method neglects the transmission chain structure and consequently uses only the aggregated number of infections, and assumes the same R0 for the entire chain except for the index case. Yet, this issue is likely to have a weak impact, since we focus on subcritical transmission; the transmission chains are hence short (see Table 1), and their structure conveys only limited information. Indeed, although a huge variation in sexual behavior has been shown previously (Liljeros et al., 2001), our sensitivity analyses exhibited no major impact of varying sexual risk behavior on risk determinants (Sensitivity analyses and Appendix 1—figure 6). Finally, even though the negative binomial model was proposed as the favorable choice for the offspring distribution compared to the Poisson distribution (Blumberg and Lloyd-Smith, 2013b) we did not observe any significant differences in the R0 estimates (see Sensitivity analyses and Appendix 1—figure 7). On the contrary, due to the simplicity of the Poisson distribution we managed to integrate the index case transmission potential reduction and the heterogeneity between the transmission chains into our Poisson-based model in a more systematic manner through the observed variability of the demographic characteristics.

Conclusion

Generally, our approach allows the assessment of the danger of a concentrated epidemic to become generalized based on the viral sequence data. We demonstrated this approach for the case of heterosexual HIV transmission in Switzerland. In particular, even though the study highlighted some heterogeneity between the HIV subtypes, our findings indicate that there is no imminent danger of a self-sustained epidemic among Swiss heterosexuals, but rather diminishing HIV transmission far below the epidemic threshold. Hence, the HIV epidemic in Switzerland is and most likely will remain restricted to high risk core groups, especially MSM. Moreover, the results suggest that integrated prevention measures in Switzerland taken over time were successful within the heterosexual population.

Materials and methods

We combined a phylogenetic cluster detection approach to identify transmission chains in the population under consideration with an adapted version of the model developed in Blumberg and Lloyd-Smith (2013a) to infer the basic reproductive number R0 (Figure 5). In particular, we accounted for both imperfect detection (included in Blumberg and Lloyd-Smith, 2013a) and modified transmissibility of the index case (not included in Blumberg and Lloyd-Smith, 2013a) from the perspective of the setting under consideration because it enters the population only (late) in chronic infection – e.g., via immigration. Moreover, we included the baseline transmission chain characteristics (such as HIV-1 subtype, date of infection, time to diagnosis, risky sexual behavior, etc.) to explain the heterogeneity among transmission chains. Note that our approach in principle estimates the effective reproductive number defined as the number of secondary infections for the current state of population; however, in case of a non-self-sustained epidemic with low prevalence, the vast majority of the population is susceptible and hence the effective reproductive number is a very good approximation for the basic reproductive number.

Graphical representation of our phylogeny-based statistical approach.

(i): HIV transmission among heterosexuals in Switzerland (white arrow) has never led to a self-sustained epidemic. However, the unknown potential of imported infections (black arrows) either from abroad or from other transmission groups in Switzerland remains a large concern. (ii): The HIV transmission chains corresponding to Swiss heterosexuals (depicted in red) were identified from the phylogenetic tree containing the SHCS and background viral sequences. (iii): Our mathematical model is based on the discrete-time branching process with nodes of three different types: sampled Swiss infection (red), unsampled Swiss infection (light red) and foreign infection infected by a Swiss index case before moving to Switzerland (green). (iv): Our method for inferring R0 accounts for both imperfect sampling and modified transmission potential of the index case. (v): Moreover, it includes the baseline transmission chain characteristics to assess the determinants of R0.

https://doi.org/10.7554/eLife.28721.009

SHCS and viral sequences

The SHCS is a multicenter, nationwide, prospective observational study of HIV infected individuals in Switzerland, established in 1988 (Swiss HIV Cohort Study et al., 2010). The SHCS was approved by the ethics committees of the participating institutions (Kantonale Ethikkommission Bern, Ethikkommission des Kantons St. Gallen, Comite Departemental d’Ethique des Specialites Medicales et de Medicine Communataire et de Premier Recours, Kantonale Ethikkommission Zürich, Repubblica e Cantone Ticino–Comitato Ethico Cantonale, Commission Cantonale d’Étique de la Recherche sur l’Être Humain, Ethikkommission beiderBasel; all approvals are available on http://www.shcs.ch/206-ethic-committee-approval-and-informed-consent), and written informed consent was obtained from all participants. Up to December 2016 over 19,500 patients have been enrolled. The SHCS is highly-representative as it covers more than 75% HIV-positive individuals on antiretroviral therapy (ART) in Switzerland (Swiss HIV Cohort Study et al., 2010). In addition to the extensive demographic and clinical data collected at biannual/quarterly follow-up (FUP) visits, for approximately 60% of the patients at least one partial pol sequence from the genotypic resistance testing is available (in total 22,036 sequences from the SHCS resistance database until August 2015). The patients with heterosexual contact as the most likely transmission route comprise about one third of all SHCS participants.

Phylogenetic tree

The phylogenetic tree was constructed from the Swiss HIV sequences of the SHCS patients and non-Swiss background sequences exported from the Los Alamos National Laboratory, 2016 database (241,783 HIV-1 viral sequences of any subtype and including the circulating recombinant forms 01–74 retrieved on February 23rd, 2016 spanning over the protease and RT regions with fragments of at least 250 nucleotides; the HXB2 sequence and sequences from Switzerland were removed afterwards). The sequences of 8 HIV-1 subtypes and circulating recombinant forms (B, C, CRF01_AE, CRF02_AG, A(1-2)), G, D and F(1-2)) were pairwise aligned to the reference genome HXB2 (accession number K03455) using Muscle v3.8.31 (Edgar, 2004). Sequences with insufficient sequencing quality of the protease region (coverage of less than 200 nucleotides between the positions 2253 and 2549 of HXB2) or reverse transcriptase region (less than 500 nucleotides between positions 2550 and 3869) were excluded. Using the earliest available of the remaining sequences for each patient, the phylogenetic tree was built with the FastTree algorithm under the generalized time-reversible model of nucleotide evolution (Price et al., 2009) including 10,840 SHCS and 90,933 background sequences.

Transmission chains

The Swiss heterosexual transmission chains were defined as clusters in the phylogenetic tree containing exclusively Swiss HIV sequences from individuals with heterosexual contact as the most likely route of the transmission, regardless of the respective genetic distances and local support values (see Sensitivity analyses and Appendix 1—figure 8 for alternative definition). The transmission chains and the patients enrolled in the SHCS forming them were identified with custom written functions in R (version 3.3.2).

For each transmission chain we determined if it was introduced to the Swiss HIV heterosexuals either as an imported infection from abroad or from other HIV transmission groups within Switzerland. The geographic origin for a given chain was obtained as the country of the closest sequence, which did not belong to Swiss heterosexuals. Specifically, we considered the smallest clade that contained both the transmission chain and either a non-Swiss or non-heterosexual sequence, and chose the sequence with the smallest pairwise genetic distance to the transmission chain (with respect to the Jukes and Cantor (JC69) model).

Additionally, in each extracted transmission chain the observed index case was identified as the patient with the earliest estimated date of infection in the chain. The date of HIV infection for each single individual was imputed with the model described by Taffé et al. (2008) if the patient had enough CD4 cell count measurements before the ART initiation and the estimated date of infection fell within the seroconversion window; otherwise the midpoint of the seroconversion window was used. The demographic characteristics (Table 2) of the index case were extracted from the SHCS, including age at infection, time to diagnosis, first available CD4 cell count and sexual risk behavior. The latter was quantified as the fraction of semiannual follow-up visits at which the patient reported sex with occasional partners. The patients with no available questionnaire regarding the sexual risk behavior were assumed to have never reported on having sex with occasional partner (see Sensitivity analyses and Appendix 1—figure 9 for the corresponding sensitivity analysis). The characteristics of the index case were then used to define the features of each corresponding transmission chain.

Estimating the basic reproductive number from a model

Our model is based on the basic discrete-time branching process. The basic reproductive number R0 was inferred from the model as the expected number of offsprings, therefore the offspring distribution represents the crucial component of the chain size distribution model. In the following sections we describe the main extensions of the basic branching process theory, which were implemented in our model. The detailed derivations can be found in Appendix 3.

Offspring distribution

We modeled the offspring distribution in a transmission chain using a Poisson distribution, which is a special case of the negative binomial distribution. The latter has been suggested in the literature (Blumberg and Lloyd-Smith, 2013b) in order to infer R0; however since we did not observe any large differences between the two distributions (see Sensitivity analyses and Appendix 1—figure 7), we decided to use the simpler Poisson model.

Suppose that Rk,n denotes the number of secondary infections of transmission degree n caused by the kth individual from the preceding generation (i.e., infected individuals with transmission degree n-1), where the transmission degree refers to the number of transmissions needed to transfer the pathogen from the index case (see Appendix 3 for detailed model description). Under the Poisson offspring distribution the number of secondary infections is modeled by

Rk,nPois(R0),

which coincides with the definition of the basic reproductive number R0=𝔼[Rk,n]. Some index cases may have lower transmission potential, e.g., immigrants that arrive during their chronic infection phase, while other index cases may exhibit enhanced transmissibility, for example, sex workers or foreigners living in Switzerland without a partner. To capture a potentially modified transmissibility of the index case we assumed a different offspring distribution of the root, namely

R1,0Pois(ρindexR0),

where ρindex denotes the index case relative transmission potential.

To assess the trends and determinants of R0, we further extended the offspring distribution based on the baseline characteristics 𝐱 of the transmission chain. More precisely, we assumed that the logarithm of R0 can be linearly described by the chain characteristics which resulted in the offspring distributions

Rk,nPois(exp(βTx))andR1,0Pois(ρindexexp(βTx))

for the secondary and the index cases, respectively. Hence, the R0 can be predicted from the effect sizes β of factors 𝐱 as

R0=exp(𝜷𝖳𝐱).

Note that since each transmission chain i has its specific baseline characteristics 𝐱i (perhaps even sampling density pi and index case relative transmission potential ρindex,i) the notation above represents a simplification. More precisely, the R0 of the ith transmission chain equals R0,i=exp(βTxi).

Likelihood function

The likelihood function was expressed in terms of the probability generating function (PGF) of the transmission chain size distribution assuming independent and stuttering (i.e., R0<1 assures that each transmission chain goes extinct almost surely) transmission chains. The following assumptions were made when incorporating the incomplete sampling of the sequences:

  • For each transmission chain at most one observed transmission chain can be extracted from the phylogeny. In other words, all observed cases belonging to the same transmission chain can be identified as the cases forming the corresponding observed transmission chain, although some intermediate transmitters might not have been sampled. For a phylogeny, this represents by a definition a weak assumption; in contrast, for contact tracing approaches missing one ancestor can lead to misidentifying one transmission chain as two or more.

  • The sampling density is independent of the transmission chain size or the transmission degree of the individual, namely each case of the transmission chain can be observed independently from the rest of the chain with probability p.

Let T denote the true size of a transmission chain and T~ the size of the corresponding observed transmission chain. The above two assumptions can be summarized as

T~TBin(T,p),

and the PGF 𝒯~ of the observed transmission chain size hence equals

𝒯~(z;R0,ρindex,p)=𝒯((1-p)+pz;R0,ρindex)

in terms of the PGF 𝒯 of T. The probability that a transmission chain has observed size of t~0 (where t~=0 means that none of the cases of the transmission chain is detected) is given by

[T~=t~]=1t~!𝒯~(t~)(0;R0,ρindex,p).

In particular, the probability that a transmission chain is observed (i.e., the observed size is strictly positive) can be calculated as

[T~>0]=1-[T~=0]=1-𝒯~(0;R0,ρindex,p).

However, since only the transmission chains with at least one detected case can be extracted from the phylogeny (and therefore to account for the unobserved transmission chains) we are interested in the probability that an observed transmission chain has a specific size. The probability of observing a transmission chain of size t~>0 is

[T~=t~|T~>0]=[T~=t~][T~>0]=1t~!𝒯~(t~)(0;R0,ρindex,p)1-𝒯~(0;R0,ρindex,p).

Finally, for a set of independent observed transmission chain sizes {t~i}i=1I the likelihood function equals

L(R0|{t~i}i=1I,ρindex,p)=i=1I1t~i!𝒯~(t~i)(0;R0,ρindex,p)1-𝒯~(0;R0,ρindex,p)

if the same R0, ρindex and p are assumed for all transmission chains. For transmission chains with different baseline characteristics and different parameters, the generalized likelihood function is

L(𝜷|{t~i,𝐱i,ρindex,i,pi}i=1I)=i=1I1t~i!𝒯~(t~i)(0;exp(𝜷𝖳𝐱i),ρindex,i,pi)1-𝒯~(0;exp(𝜷𝖳𝐱i),ρindex,i,pi).

Model fit

The maximum likelihood (ML) estimator for 𝜷, the predictor for R0 and the corresponding statistics (confidence intervals, p-values, etc.) were implemented in the R package PoisTransCh (Turk, 2017, https://github.com/tejaturk/PoisTransCh; copy archived at https://github.com/elifesciences-publications/PoisTransCh). The provided confidence intervals are the Wald-type 95%-confidence intervals (see Sensitivity analyses for the comparison against different types) and the p-values are based on the Wald statistic. Initially, we assessed the impact of covariables potentially associated with HIV transmission. Specifically, we considered HIV-1 subtype, establishment date of the transmission chain (i.e., the earliest estimated date of infection in the transmission chain), reported sex with occasional partner, age at infection, first measured CD4 cell count and time to diagnosis of the index case. Final model selection was carried out by the forward selection and backward elimination algorithms based on the Akaike and Bayesian information criterion (AIC and BIC, respectively). The detailed steps are provided in Selection of the predictive models.

Datasets

Previously published datasets from ​Kouyos et al. (2010) and ​von Wyl et al. (2011) were used in this study. As previously discussed in these publications, due to the large sampling density this data would, in principle, allow for the reconstruction of entire transmission networks and could thereby endanger the privacy of the patients. This is especially problematic because HIV-1 sequences frequently have been used in court cases. Therefore, a random subset of 10% of the sequences are accessible via GenBank. These accession numbers are as follows: GU344102-GU344671, EF449787, EF449788, EF449796, EF449798, EF449828, EF449829, EF449838, EF449844, EF449852, EF449853, EF449854, EF449860, EF449880, EF449883, EF449889, EF449895, EF449901, EF449904, EF449905, EF449917, EF449921, EF449928, EF449930, EF449943, EF449950, EF449960, EF449971, EF449980, EF449987, EF450004, EF450005, EF450011, EF450024, EF450026, GQ848113, GQ848120, GQ848140, GQ848145, GQ848149, JF769777-JF769851

Appendix 1

Sensitivity analyses

Relative transmission potential of the index case

To assess the role of the index case relative transmission potential we carried out three different sensitivity analyses regarding parameter ρindex. Firstly, we varied the ρindex for the transmission chains of non-Swiss origin from 0.05 to 1.5. Secondly, we assumed the same ρindex for all transmission chains regardless of their origin and fit the models over a range of ρindex values. Finally, we restricted the analysis only to the transmission chains of non-Swiss origin and varied ρindex.

Appendix 1—figure 1
Sensitivity analysis regarding the index case relative transmission potential.

Panel (i) shows the sensitivity of the R0 estimates from baseline model and panel (ii) the sensitivity of the time trend factor. The colored lines represent the subtype-stratified analyses, while the results from the overall models are shown in gray. In the first sensitivity analysis, the ρindex of Swiss-originating transmission chains was held at 1 and the ρindex of non-Swiss origin varied (solid lines). In the second analysis, the ρindex of Swiss and non-Swiss origin was the same (dashed lines). The dotted lines show the results from the sensitivity subanalysis including only the transmission chains of non-Swiss origin. The vertical and horizontal lines depict the parameters and estimates from the main analysis, respectively.

https://doi.org/10.7554/eLife.28721.012

These sensitivity analyses (see Appendix 1—figure 1) implied that the conclusion of no danger for a self-sustained epidemic is stable with respect to ρindex even in the case when some of the Swiss-originated transmission chains are misclassified. In addition, while slightly higher R0 estimates in the non-Swiss transmission chain subanalysis were mostly driven by the non-B subtypes, the results were safely below 1 indicating the non-sensitivity of the main conclusion also when some non-Swiss transmission chains would be falsely identified as such.

Sampling density

To study the impact of the sampling densities we performed subtype-stratified sensitivity analyses as well as the overall sensitivity analysis by keeping the sampling density constant among the transmission chains. In all scenarios, we varied the sampling density between 0.02 and 1, while ρindex remained the same as in the main analyses.

Appendix 1—figure 2
Sensitivity analysis regarding the sampling density.

The index case relative transmission potential parameter ρindex was the same as used in the main analyses, while the sampling densities varied (x-axis). In the pooled analysis (larger plots) the sampling density was the same for all transmission chains. Panel (i) shows the corresponding estimates of the basic reproductive number R0 and the time trend factor estimates are displayed in panel (ii). The dotted vertical lines depict the sampling densities used for each subtype in our study (subtype-stratified plots) and the mean sampling density over all transmission chains (overall plots). The horizontal dotted lines represent the estimates from the main analysis.

https://doi.org/10.7554/eLife.28721.013

The sensitivity analyses (see Appendix 1—figure 2) showed that neither the R0 from the baseline model nor the time trend are sensitive to the sampling density, namely the conclusions of R0 being significantly below 1 and decreasing time trend could be made even for slightly lower or higher sampling densities.

Ongoing transmission and stuttering transmission chains assumption

Duration of infectious period in relation to ongoing transmission

Some of the observed transmission chains may still experience ongoing transmission due to either not yet diagnosed cases or unsuppressed patients within the transmission chain who still have the ability to spread the virus. The transmission chain sizes might thus be too small and R0 underestimated. However, the gradually increasing treatment success (Castilla et al., 2005; Kohler et al., 2015), benefits of earlier ART initiation (Kitahata et al., 2009; INSIGHT START Study Group et al., 2015) and consequently updated treatment guidelines (Günthard et al., 2016) resulted in a shorter duration of infectious period. Transmission chains which started earlier are thus more strongly affected by ongoing transmission than recent transmission clusters.

One possibility to assess this issue is to investigate the highest possible transmission degree that has completed a transmission at a given time point; that is the maximum number of generations which are not infectious anymore and therefore have used their transmission potential. We assumed that the length of the infectious period is changing linearly with calendar year and fitted a linear regression model to the duration of infectious period of the index cases (measured by time to suppression or treatment start). To ensure a more conservative approach we truncated the fitted infectious period durations from below, such that the minimum was 3 years. Let d(τ) define the infectious period duration of an individual who became infected at time τ. The worst-case scenario in the context of ongoing transmission and related potential bias is represented by a transmission chain, in which each infected individual transmits the virus just at the end of his/her infectious period. The (conservative) maximum number of completed transmission degrees at time τ of a transmission chain i that started at τ0 therefore equals

Nmax,i(τ)=max{k|τkτ},

where τk denotes the latest possible time at which the transmission of the kth generation was complete and is calculated iteratively as τk+1:=τk+d(τk) for k0 (Appendix 1—figure 3). If its index case is still infectious at time τ, it can still produce new infections (which would have a transmission degree 1) and hence Nmax,i=0.

Appendix 1—figure 3
Conservative (with respect to ongoing transmission) maximum number of completed transmission degrees by a given date.

The red lines show the date (y-axis) by which at least a certain number (red numbers) of transmission degrees have been completed for a transmission chain with a specific establishment date (x-axis). The diagonal dotted gray lines depict the number of years since the establishment date, and the horizontal blue line represents the last sampling date.

https://doi.org/10.7554/eLife.28721.014

Ongoing transmission

To assess the potential bias due to ongoing transmission we compared the estimates based on the transmission chains formed by the cases with the estimated date of infection before a specific date (ω^) and based on the transmission chains that had been completed (with respect to the last sampling date) by the same date (ω). The relative bias arising from neglecting the ongoing transmission hence equals

δrel=ω^-ωω.
Appendix 1—figure 4
Relative bias due to ongoing transmission.

The upper panel shows the relative bias of the basic reproductive number R0 from the baseline model and the lower panel the relative bias of the linear time trend factor from the corresponding generalized linear model. The proportion of active transmission chains over time is represented by the black line. The relative bias associated with overestimation and underestimation is displayed with green and red bars-points, respectively. Absence of bias is depicted by the horizontal gray lines.

https://doi.org/10.7554/eLife.28721.015

The proportion of ongoing transmission chains is decreasing with time, which is in line with a decreasing duration of infectious period, hence indicating that the ongoing transmission is less of an issue for recent years than for older transmission chains. Our sensitivity analyses revealed that the expected bias stemming from neglecting the ongoing transmission is less than 5% since the early 2000’s for both key questions (Appendix 1—figure 4): the basic reproductive number R0 and its linear time trend factor. Moreover, the relative bias is positive for most of the recent dates, implying that the negligence of ongoing transmission results in rather conservative estimates with respect to our conclusions.

Subcritical transmission assumption

Like the models described by Blumberg and Lloyd-Smith (Blumberg and Lloyd-Smith, 2013bBlumberg and Lloyd-Smith, 2013a), our model also implicitly assumes subcritical transmission. To justify that the extracted HIV transmission chain sizes of the Swiss heterosexuals did not violate this assumption, we simulated transmission chains for various R0 (including the estimated R0) and compared the empirical quantiles between the simulated transmission chain sizes and the transmission chain sizes extracted from the phylogenetic tree. Since some transmission chains (observed or simulated) might still exhibit ongoing transmission at the time of the sampling, we restricted the maximal number of generations (i.e., transmission degrees), which were simulated according to the duration of infectious periods (Appendix 1—figure 3).

More precisely, from each observed Swiss heterosexual transmission chain we kept sampling transmission chains (for different ‘known true’ R0 scenarios) with the maximal number of simulated generations until a simulated transmission chain was observed (i.e., at least one case was observed) to reflect the more realistic observed transmission chain size distribution. We repeated these steps for each extracted Swiss heterosexual transmission chain.

Appendix 1—figure 5
Sensitivity analysis regarding the stuttering transmission chains assumption.

The Q-Q plots compare the hypothetical transmission chain size distributions (y-axis showing their empirical permilles) with the transmission chain size distribution (empirical permilles on the x-axis) inferred from the phylogeny. The upper left plot compares the distribution of the simulated transmission chain sizes based on the estimated R0 with the (from the phylogeny) observed transmission chain sizes and thus verifies the R0 estimate. The remaining plots compare the simulated transmission chain size distributions against the extracted transmission chain sizes for R0 closer to 1 to justify the subcritical transmission assumption. Each point represents a permille, hence the darker points indicate more overlapping permilles.

https://doi.org/10.7554/eLife.28721.016

Finally, we compared the 1000-quantiles (permilles) of the transmission clusters extracted from the phylogeny against simulated transmission chains (Appendix 1—figure 5). The Q-Q plots clearly show that the extracted transmission chains would be indeed much longer (the largest observed transmission chain would be of size greater than 30) if the true R0 were above 1 (or even close to 1). Moreover, the size distribution of the transmission chains simulated for the estimated R0 showed a good concordance with the observed transmission chains (upper left Q-Q plot of Appendix 1—figure 5).

Variation in sexual behavior along transmission chains

Our model assumes constant sexual risk behavior along transmission chains. In this sensitivity analysis we assessed how a changing sexual risk behavior would affect our conclusions. We approached this question by slightly changing the definition of the sexual risk behavior of each transmission chain, while the other characteristics stayed the same.

  • Instead of the index case determining the risk behavior for each transmission chain a randomly sampled infected individual from the transmission chain was chosen to determine the sexual risk behavior of the transmission chain. Noteworthy, this only affects the minority of the transmission chains, namely those with the observed length 2. The multivariate model including only the linear terms was then fitted to the transmission chains with slightly modified sexual risk behaviors. We repeated this 1000 times to get the empirical distribution of the effect sizes on R0 (Appendix 1—figure 6).

  • We considered the reported sex with an occasional partner on the level of a transmission chain as a proxy for its sexual risk behavior. More precisely, we used the fraction of FUPs of all infected individuals in a transmission chain in which any of these patients reported sex with occasional partner. We then fitted the same multivariate model with only linear terms as in the main analysis and compared the effect sizes and directions (Appendix 1—figure 6).

Appendix 1—figure 6
Comparison of effect sizes in the multivariate model with linear terms only for different sexual risk behavior definitions of a transmission chain.

The thick lines with black circles show the original effect sizes (where the index case determined the sexual risk behavior of the transmission chain) and their 95%-confidence intervals. The empirical distribution of the effect sizes where a random individual in a transmission chain determines its sexual risk behavior is displayed by the shaded areas. The thinner horizontal double sided arrows with the filled circles correspond to the effect sizes and their 95%-confidence intervals for the transmission chain level fraction of follow-up visits (FUPs) with reported sex with occasional partner by any of the infected individuals from the transmission chain. The vertical dotted gray line depicts the reference R0 from the original model, i.e., using the index case to define the sexual risk behavior.

https://doi.org/10.7554/eLife.28721.017

Our transmission chains are short in size; therefore we did not expect to see a huge impact of the variations in sexual behavior on the effects. Indeed, the analyses revealed that even with the modified definitions of the risky sexual behavior (and therefore addressing its variation) the effect directions did not change, while the effects sizes did not exhibit a huge difference. In particular, the significance of all risk determinants at the 5% level remained the same.

These findings indicate that the simplification of the equal distribution for the number of secondary infections does not exhibit a dramatic impact on the outcomes in the case of short transmission chains, which dominate in subcritical settings.

Comparison between Poisson and negative binomial offspring distribution based models

To evaluate the rationale of using the simpler Poisson model we compared the estimates from the baseline models over a range of sampling densities for both Poisson and negative binomial offspring distribution. Since an implementation with modified transmission potential of the index case is not available for the negative binomial model, we conducted the sensitivity analyses with a fixed ρindex=1.

Appendix 1—figure 7
Comparison between the Poisson and the negative binomial offspring distribution baseline model R0 estimates.

The dark gray and colored lines show the estimates from the model with Poisson offspring distribution, while the black lines correspond to the negative binomial distribution. The index case relative transmission potential parameter ρindex was fixed to 1 and the sampling density (x-axis) varied. In the overall analysis the sampling density was the same for all transmission chains regardless of their subtype. The vertical gray lines depict the sampling densities used for each subtype in our study (above panels) and the mean sampling density in the overall analysis (bottom panel).

https://doi.org/10.7554/eLife.28721.018

While the R0 estimates for the majority of the non-B subtypes were practically equal between the two models (see Appendix 1—figure 7), the observed differences in the overall analysis and in the case of B and 02_AG subtypes were mostly larger for low sampling densities. However, we also found that the Poisson model provided rather conservative R0 estimates and therefore this should not affect our main conclusions.

In addition, we performed a likelihood ratio test to evaluate if the multivariate linear negative binomial model (with ρindex=1) is significantly better than the corresponding Poisson model (from Figure 3). The p-value of 0.74 indicated no strong preference of the negative binomial over the Poisson model. Noteworthy, this implies that modelling the variability among the transmission chains in terms of their characteristics sufficiently explains the heterogeneity (dispersion parameter ξ of the negative binomial distribution) between the infected heterosexuals forming these transmission chains.

Relaxed transmission cluster definition

We defined the Swiss heterosexual transmission chains as clusters on the phylogeny containing 100% viral sequences belonging to Swiss heterosexuals. To assess the impact of this definition we relaxed the 100% threshold to 75%. All the sequences belonging to the Swiss heterosexuals from these clusters formed more liberally defined transmission chains.

Appendix 1—figure 8
Sensitivity analysis regarding the transmission cluster definition.

The upper panel (i) compares the estimated R0 with the original cluster definition (brighter lines) with the R0 estimated based on the relaxed cluster definition (darker lines) from the overall analysis (in gray) and subtype-stratified analyses (in colors). Similarly, the bottom panel (ii) shows the comparison between the estimated time trend factors obtained from the transmission chain sizes based on different cluster definition thresholds.

https://doi.org/10.7554/eLife.28721.019

With the relaxed threshold, we identified 3,039 transmission chains and repeated the main analyses (Appendix 1—figure 8). As expected the R0 slightly increased, but stayed below 1. Overall, we did not observe any noteworthy deviations.

Missing follow-up data for reported sex with occasional partner

In the main analysis of the possible determinants of HIV transmission we imputed missing follow-up information regarding sex with occasional partner with never reporting it (which is equivalent to 0 reporting rate). To evaluate this imputation, we fitted the same multivariate model with linear terms to the subset of the transmission chains in which the data about the sex with occasional partner of the index case was available. However, the effect sizes did not change dramatically; in particular, the effect directions did not change and the same set of determinants was found to be significant (Appendix 1—figure 9).

Appendix 1—figure 9
Subanalysis for the transmission chains with available follow-up information about sex with occasional partner of the index case compared to the main analysis with imputed data.

The effect sizes from the subanalysis are shown in brighter colors and those from the main analysis in dark. In the main analysis, the missing data were replaced by never reporting sex with an occasional partner.

https://doi.org/10.7554/eLife.28721.020

Confidence intervals

In our study we used the normal approximation of the ML estimator to construct the 95%-CIs and the prediction intervals. To verify the reliability of this assumption we considered bootstrap and profile likelihood based CIs for each of the models.

For the parametric bootstrap, we sampled B=1000 new datasets of transmission chains from the estimated transmission parameters (i.e., under the assumption that our estimated parameters are the true parameters) for each model. To ensure that the newly sampled datasets had the same sample size, in each repetition b{1,,B} we kept simulating from each transmission chain until the new transmission chain had at least one observed infection (i.e., such that the observed length was positive). Finally, for each sampled dataset we fitted the same model, extracted the estimated transmission parameters and the corresponding Wald-type 95%-CIs. The overview of the parameters and the models is provided in Appendix 1—table 1.

Appendix 1—table 1
Overview of all the parameters, their estimates and the 95%-confidence intervals fitted in all the models presented in this study.
https://doi.org/10.7554/eLife.28721.021
SubtypesParameter 
number
Parameter 
name
Parameter 
estimate
Wald-type
95%-CI
Profile likelihood
95%-CI
Overall1log(R0)-0.823(-0.876,-0.770)(-0.878,-0.772)
B2log(R0)-1.037(-1.121,-0.952)(-1.124,-0.955)
C3log(R0)-0.719(-0.879,-0.559)(-0.892,-0.571)
01_AE4log(R0)-0.826(-1.036,-0.615)(-1.057,-0.632)
02_AG5log(R0)-0.483(-0.587,-0.378)(-0.594,-0.384)
A6log(R0)-0.618(-0.751,-0.485)(-0.760,-0.492)
other7log(R0)-0.605(-0.758,-0.451)(-0.771,-0.461)
Overall8log(R0,𝑟𝑒𝑓)-0.839(-0.894,-0.784)(-0.895,-0.785)
9

𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510

-0.112(-0.187,-0.037)(-0.188,-0.037)
B10log(R0,𝑟𝑒𝑓)-1.070(-1.165,-0.975)(-1.169,-0.979)
11

𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510

-0.112(-0.234,0.010)(-0.236,0.008)
C12log(R0,𝑟𝑒𝑓)-0.692(-0.851,-0.533)(-0.864,-0.544)
13

𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510

-0.209(-0.466,0.049)(-0.473,0.046)
01_AE14log(R0,𝑟𝑒𝑓)-0.781(-0.991,-0.570)(-1.013,-0.588)
15

𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510

-0.255(-0.616,0.106)(-0.629,0.101)
02_AG16log(R0,𝑟𝑒𝑓)-0.434(-0.539,-0.329)(-0.545,-0.333)
17

𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510

-0.415(-0.609,-0.222)(-0.615,-0.226)
A18log(R0,𝑟𝑒𝑓)-0.725(-0.892,-0.558)(-0.907,-0.571)
19

𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510

-0.430(-0.660,-0.199)(-0.672,-0.209)
other20log(R0,𝑟𝑒𝑓)-0.600(-0.754,-0.446)(-0.767,-0.456)
21

𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510

-0.162(-0.397,0.073)(-0.403,0.072)
Overall22log(R0,𝑟𝑒𝑓)-0.710(-0.780,-0.640)(-0.782,-0.641)
23

(𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510)2

-0.313(-0.451,-0.176)(-0.457,-0.182)
24

(𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510)3

-0.184(-0.283,-0.086)(-0.288,-0.091)
Overall25log(R0,𝑟𝑒𝑓)-1.252(-1.366,-1.137)(-1.369,-1.140)
26𝑆𝑢𝑏𝑡𝑦𝑝𝑒C0.352(0.167,0.538)(0.158,0.531)
27𝑆𝑢𝑏𝑡𝑦𝑝𝑒01_𝐴𝐸0.274(0.046,0.502)(0.029,0.490)
28𝑆𝑢𝑏𝑡𝑦𝑝𝑒02_𝐴𝐺0.575(0.428,0.721)(0.426,0.720)
29𝑆𝑢𝑏𝑡𝑦𝑝𝑒A0.430(0.271,0.588)(0.266,0.584)
30𝑆𝑢𝑏𝑡𝑦𝑝𝑒𝑜𝑡ℎ𝑒𝑟0.426(0.247,0.606)(0.238,0.600)
31

𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510

-0.214(-0.301,-0.127)(-0.301,-0.128)
32𝐴𝑔𝑒-32100.007(-0.045,0.058)(-0.046,0.057)
33

CD4-350100

0.000(-0.018,0.019)(-0.019,0.018)
34𝑅𝑎𝑡𝑒𝑟𝑖𝑠𝑘0.230(0.095,0.364)(0.096,0.365)
35

𝑌𝑒𝑎𝑟𝑠𝑑𝑖𝑎𝑔𝑛𝑜𝑠𝑖𝑠-310

0.351(0.210,0.492)(0.207,0.490)
Overall36

log(R0,𝑟𝑒𝑓)

-1.173(-1.301,-1.045)(-1.304,-1.048)
37

110log(𝑌𝑒𝑎𝑟𝑠𝑑𝑖𝑎𝑔𝑛𝑜𝑠𝑖𝑠3)

1.727(1.049,2.405)(1.064,2.420)
38𝑆𝑢𝑏𝑡𝑦𝑝𝑒C0.322(0.140,0.505)(0.131,0.498)
39𝑆𝑢𝑏𝑡𝑦𝑝𝑒01_𝐴𝐸0.246(0.020,0.472)(0.004,0.460)
40𝑆𝑢𝑏𝑡𝑦𝑝𝑒02_𝐴𝐺0.516(0.374,0.659)(0.372,0.658)
41𝑆𝑢𝑏𝑡𝑦𝑝𝑒A0.404(0.246,0.562)(0.241,0.558)
42𝑆𝑢𝑏𝑡𝑦𝑝𝑒𝑜𝑡ℎ𝑒𝑟0.401(0.223,0.580)(0.214,0.574)
43

(𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510)3

-0.231(-0.337,-0.124)(-0.345,-0.131)
44

𝑅𝑎𝑡𝑒𝑟𝑖𝑠𝑘

0.230(0.094,0.366)(0.096,0.368)
45

(𝐷𝑎𝑡𝑒𝑖𝑛𝑓𝑒𝑐𝑡𝑖𝑜𝑛-1.1.199636510)4

-0.129(-0.227,-0.031)(-0.235,-0.038)

For a single parameter β (under the assumption that the true value equals the estimated value β^) we therefore obtained a sample of ML estimators β^(1),,β^(B), from which we estimated the kernel densities and compared them to the normal approximation densities used in the Wald CIs construction (Appendix 1—figure 10). Moreover, from the sample of Wald-type 95%-CIs we calculated the coverage rate as the proportion of these CIs that contained the true value β^.

Appendix 1—figure 10
Empirical distribution of maximum likelihood (ML) estimator and the Wald-type confidence intervals (CI) coverage rates.

Each plot represents a single parameter from a single model (see Appendix 1—table 1 for the parameters overview including their values), where the number in the lower left corner denotes the parameter’s consecutive parameter number. The light gray-shaded area represents the proportion of the Wald-type 95%-CIs from the parametric bootstrap simulations which contained the true value (depicted by the vertical orange line), while the green-shaded area corresponds to those CIs from the simulations that missed the true value. The numbers in the upper left corners are the coverage rates from the parametric bootstrap. The original Wald 95%-CIs used in our study are displayed with the light orange-area. The dark blue and gray lines show the empirical distribution of ML estimators from the parametric bootstrap samples and the normal approximation based probability density function, respectively. The horizontal red lines depict the target coverage rate of 95%.

https://doi.org/10.7554/eLife.28721.022

Comparing the empirical distribution of the ML estimator from these simulations (Appendix 1—figure 10) with the normal approximation from the Wald test, we concluded that the latter represents a valid approximation. In addition, the coverage rates were all very close to the target 95% or above.

Next, in addition to the parametric bootstrap as described above, we also performed a nonparametric bootstrap. New datasets were generated by randomly sampling with replacement from the existing dataset. To each newly sampled dataset all the models were fitted to obtain nonparametric bootstrap samples of ML estimators for each individual transmission parameter from Appendix 1—table 1. We then constructed the basic bootstrap 95%-CIs (Davison and Hinkley, 1997) as

(2β^-q97.5%*,2β^-q2.5%*),

where q* denotes the corresponding percentile of the bootstrap sample β^(1),,β^(B). Finally, we constructed the profile likelihood based CIs (Held and Bové, 2013) and compared different types of CIs against the Wald-type CIs (Appendix 1—figure 11).

Appendix 1—figure 11
Comparison of different types of 95%-confidence intervals (CI) with the normal approximation based Wald-type 95%-CIs.

Each column corresponds to a different type of CIs, namely the profile likelihood based CIs, the basic nonparametric bootstrap CIs and the basic parametric bootstrap CIs. Each row represents a single parameter (the overview of the parameters is provided in Appendix 1—table 1). The colorful lines show the specific CIs compared to the corresponding Wald-type CIs, namely their relative widths and positions. The gray-shaded areas represent the Wald-type 95%-CIs.

https://doi.org/10.7554/eLife.28721.023

These simulations indicated no significant difference between the widths of Wald-type and profile likelihood based CIs. Besides, the Wald-type CIs did not appear to be systematically wider or narrower compared to the bootstrap CIs.

To summarize, these simulations imply that the normal approximation Wald-type CIs used in our study provide a reliable alternative to other more time-complex types of CIs.

Appendix 2

Selection of the predictive models

Single determinant models

To construct a multivariate predictive model for R0 we first focused on each single determinant. More precisely, to find a best predictive model for a single factor we performed both forward selection and backward elimination based on the AIC and BIC criteria (see Appendix 2—table 1 for the case of establishment date). All terms which appeared in at least one of the single determinant models were later used in the multivariate model.

Appendix 2—table 1
Establishment date models obtained with the AIC/BIC forward selection and backward elimination and their respective AIC and BIC values as well as the p-values from the likelihood ratio test compared to the null model without any covariates.

Terms that were part of the respective final model are marked by ×.

https://doi.org/10.7554/eLife.28721.025
AICBIC
ForwardBackwardForwardBackward

Dateinfection1.1.199636510

(Dateinfection1.1.199636510)2

××

(Dateinfection1.1.199636510)3

××××

(Dateinfection1.1.199636510)4

××
AIC3364.33364.23364.33364.2
BIC3382.43382.33382.43382.3
p-value from LR test<0.0001<0.0001<0.0001<0.0001

We chose the model obtained with the backward elimination procedure as the predictive model based solely on the establishment date (Figure 2). It provided both the lowest BIC and AIC value, therefore indicating the best goodness-of-fit (Appendix 2—table 1).

Multiple determinants model

Using the terms obtained in the single determinant predictive models (establishment date, age at infection, earliest CD4 cell count, frequency of reporting sex with occasional partner and time to diagnosis) and a viral subtype indicator, we constructed the final multiple determinants model for the prediction as follows. Like before, we carried out both forward and backward selection algorithms for both criteria. Among the resulting algorithms we picked the one minimizing the BIC, since the BIC penalizes the model complexity stronger than the AIC (Appendix 2—table 2).

Appendix 2—table 2
Multivariate models obtained with the AIC/BIC forward selection and backward elimination algorithms.

The terms listed in the table are the terms identified from the single determinant model selections and the crosses indicate the terms entering the multivariate models. The null model from the likelihood ratio test refers to the baseline model without any covariates (not even the subtype).

https://doi.org/10.7554/eLife.28721.026
AICBIC
ForwardBackwardForwardBackward

Subtype

××××

(Dateinfection1.1.199636510)2

(Dateinfection1.1.199636510)3

××××

(Dateinfection1.1.199636510)4

×××

𝑅𝑎𝑡𝑒𝑟𝑖𝑠𝑘

××××

Raterisk

×××

110log(Yearsdiagnosis3)

××

Yearsdiagnosis310

××

Yearsdiagnosis310

×××

(Yearsdiagnosis310)3

××

CD435010

(Age3210)2
AIC3254325232543262
BIC3314333133143316
p-value from LR test<0.0001<0.0001<0.0001<0.0001

Appendix 3

Detailed derivation of the transmission chain size model and statistical inference

Transmission chain size model

Transmission chains can naturally be modeled as branching processes. The index case corresponds to the root of the process; each new infection represents a new offspring. The generation of an individual in a transmission chain can therefore be interpreted as the transmission degree relative to the index case - the first generation individuals got infected directly from the index case, the second generation indirectly through one mediator, etc. In other words, the transmission degree of a patient is the number of transmission events needed to transfer the virus to this patient from the index case.

Towards probability generating function of the transmission chain size

Let Rk,n denote the number of secondary infections with transmission degree n produced by the kth individual from the preceding generation, Sn the total number of new infections of transmission degree n and QN the cumulative number of cases in the transmission chain with the transmission degree at most N, that is,

Sn=k=1Sn-1Rk,n,QN=n=0NSn=QN-1+SN.

The index case establishes the transmission chain and corresponds to the generation 0, therefore S0=Q0=1.

Assuming that the numbers of secondary infections are independent and identically distributed for all patients of the same transmission degree, let n denote the probability generating function (PGF) of Rk,n, namely

n(z):=𝔼[zRk,n]

for each k{1,2,,Sn-1}. The expected number of secondary infections of degree n is therefore given by

E[Rk,n]=ddzn(z)|z=1=n(1)(1).

Furthermore, assume that the numbers of secondary infections caused by different individuals are independent between each other regardless of the transmission degree. The PGF 𝒬N of QN is

𝒬N(z):=E[zQN]=E[zQN1zSN]=(a)E[E[zQN1zSN|{Sn}n=0N1]]=(b)E[zQN1E[zSN|{Sn}n=0N1]]=E[zQN1E[k=1SN1zRk,N|{Sn}n=0N1]]=(c)E[zQN1k=1SN1E[zRk,N|{Sn}n=0N1]]=(d)E[zQN1k=1SN1E[zRk,N]]=E[zQN1k=1SN1N(z)]=E[zQN1N(z)SN1],

because (a) of the tower property of the conditional expectation, (b) QN-1=k=0N-1Sn is {Sn}n=0N-1-measurable, (c) {Rk,N}k=1SN-1 are independent, and (d) Rk,N are independent from {Sn}n=0N-1 for all k=1,2,,SN-1. Repeating similar steps iteratively yields

(1)𝒬N(z)=E[zQN2zSN1N(z)SN1]=E[zQN2E[(zN(z))SN1|{Sn}n=0N2]]=E[zQN3zSN2N1(zN(z))SN2]==E[zQN4(zN2(zN1(zN(z))))SN3]==E[(z1(z2((zN(z)))))S0]=z1(z2((zN(z)))).

The total size of the transmission chain is denoted by T and equals

T:=limNQN.

From the definition of T it follows that its PGF 𝒯 equals

(2) 𝒯(z)=limN𝒬N(z)

for all z.

Probability generating function of a completely observed uniform transmission chain

Assume that the number of secondary infections follows the same distribution with PGF 𝒢 for all infected persons, namely

n𝒢

for every n (i.e., the transmission is uniform across different transmission degrees). The PGF 𝒬N (Equation 1) then simplifies to

𝒬1(z)=z𝒢(z)𝒬2(z)=z𝒢(z𝒢(z))=z𝒢(𝒬1(z))𝒬N(z)=z𝒢(𝒬N-1(z)).

Using Equation 2, the PGF 𝒯 for each z solves the equation

𝒯(z)=!z𝒢(𝒯(z)).

Probability generating function of a transmission chain with modified transmission potential of the index case

From the perspective of the Swiss HIV heterosexual population the index case might have lost some of its potential to transmit the virus prior to establishing the transmission chain in the population under consideration. The follow-up cases are infected while already in the subpopulation and can therefore fully contribute to spreading. Sex workers and lonely foreigners in Switzerland represent two examples of index cases with an enhanced transmission potential. We assume that apart from the index case the numbers of secondary infections are equally and independently distributed for all the other infected individuals. Let ρindex denote the index case relative transmission potential (ICRTP). In terms of the model the above assumptions can be summarized as

1(z)=(z),n(z)=𝒢(z),for n>1,

where and 𝒢 denote the PGF of two distributions, such that

(1)(1)=ρindex𝒢(1)(1),

namely 𝔼[R1,1]=ρindex𝔼[Rk,n] for all k{1,,Sn-1} and n>1. In other words, the ICRTP is the expected number of secondary infections of the index case relative to the expected number of secondary infections of the rest of the transmission chain.

To compute the PGF of the transmission chain with modified transmissibility of the index case we first introduce a skeleton function 𝒦, which controls the regular part/tail of the transmission chain. Let 𝒦 be the pointwise limit 𝒦(z):=limN𝒦N(z) of the iteratively defined functions

𝒦1(z):=z𝒦N(z):=z𝒢(𝒦N-1(z)).

The skeleton therefore solves the equation

(3) 𝒦(z)=!z𝒢(𝒦(z)).

Note that in the absence of the modified transmissibility of the index case, the skeleton function 𝒦 coincides with the PGF of the transmission chain size. Having introduced this notation one can rewrite the PGF 𝒬N (Equation 1) as

𝒬1(z)=z(z)=z(𝒦1(z))𝒬2(z)=z(z𝒢(z))=z(𝒦2(z))𝒬3(z)=z(z𝒢(z𝒢(z)))=z(𝒦3(z))𝒬N(z)=z(𝒦N(z)).

As N, this implies

(4) 𝒯(z)=z(𝒦(z))

for all z.

Probability generating function of an incompletely observed transmission chain

Since not every HIV infected person is included in a cohort, linked to care or even diagnosed, we only observe parts of the transmission chains. Suppose that each infection is detected with probability p, independently of the others. Furthermore, assume that despite not all cases being observed, the sampled patients belonging to the same true transmission chain could be identified as members of this transmission cluster (and not as members of two or more separate transmission clusters).

The true transmission chain can still be modeled with the branching process as above. Let tilde ( ~ ) denote the observed cases. Since each case is detected at random with probability p the following applies to the observed transmission chains.

  • If Rk,n is defined as above then R~k,n denotes the number of secondary infections with transmission degree n caused by patient k which are actually observed. It follows

    R~k,n|Rk,nBin(Rk,n,p).
  • Given the numbers of secondary infections with transmission degree n of all the patients the observed number of infections of transmission degree n equals

    S~n=k=1Sn-1R~k,n

    and follows a binomial distribution, namely

    S~n|{Rk,n}k=1Sn-1k=1Sn-1Bin(Rk,n,p)=Bin(k=1Sn-1Rk,n,p)=Bin(Sn,p).
  • The observed cumulative number of infected individuals with the transmission degree at most N equals

    Q~N=n=0NS~n=Q~N-1+S~N=Q~N-1+k=1SN-1R~k,N.

    By conditioning on the cumulative number of infections of transmission degree up to N-1 and on the numbers of secondary infections of transmission degree N, Q~N therefore follows a binomial distribution, that is,

    Q~N|QN-1,{Rk,N}k=1SN-1Bin(QN-1+k=1SN-1Rk,N,p)=Bin(QN-1+SN,p)=Bin(QN,p).

    Since (z)=((1-p)+pz)n is the PGF of a Bin(n,p)-distributed random variable, the PGF of Q~N can be expressed as

    𝒬~N(z)=E[zQ~N]=(a)E[E[zQ~N|QN]]=(b)E[E[E[zQ~N|QN1,{Rk,N}k=1SN1]|QN]]=(c)E[E[((1p)+pz)QN1+k=1SN1Rk,N|QN]]=E[E[((1p)+pz)QN|QN]]=(a)E[((1p)+pz)QN]=𝒬N((1p)+pz)

    in terms of the PGF 𝒬N, because (a) of the tower property, (b) of the tower property for σ-algebras σ(QN)σ(QN-1,{Rk,N}k=1SN-1) due to the relation QN=QN-1+k=1SN-1Rk,N, and (c) QN given QN-1 and {Rk,N}k=1SN-1 is binomially distributed.

This allows us to obtain the PGF 𝒯~ of the observed transmission chain length T~=limNQ~N, namely

𝒯~(z)=𝒯((1-p)+pz),

where 𝒯 denotes the PGF of the true underlying transmission chain.

Finally, the PGF of the observed transmission chain size with modified transmissibility of the index case equals

(5) 𝒯~(z)=𝒯((1-p)+pz)=((1-p)+pz)(𝒦((1-p)+pz)).

Inferring the transmission parameters

Probability generating functions enable us to obtain the state probabilities, namely the probability of observing a transmission chain of length j can be calculated as

[T~=j]=𝒯~(j)(0)j!,

where (j) denotes the jth derivative. The transmission chains with no observed cases are not observable, therefore we are interested in the probability that an observed chain is of length j, which equals

[T~=j|T~>0]=[T~=j]1-[T~=0]=1j!𝒯~(j)(0)1-𝒯~(0).

So far, we have not included the basic reproductive number or any other transmission-related parameters in the PGF of transmission chain size 𝒯~. In the following paragraphs we extensively present the statistical inference (following Held and Bové, 2013) of the transmission parameters based on the transmission chain size model described above.

The likelihood function

Let 𝝎 denote a vector of transmission parameters, for instance 𝝎=R0 in case of a single transmission parameter corresponding to the basic reproductive number. Assuming that the transmission chain sizes are independent, the likelihood function of the sample of I observed transmission chain sizes 𝐭~:={t~i}i=1I is defined by

L𝝎(𝝎|𝐭~):=i=1I1t~i!𝒯~(t~i)(0;𝝎)1-𝒯~(0;𝝎),

where 𝒯~(z;𝝎) denotes the PGF of transmission chain size with transmission parameters 𝝎. The corresponding log-likelihood function is

(6) 𝝎(𝝎|𝐭~)=i=1I(log(𝒯~(t~i)(0;𝝎))-log(t~i!)-log(1-𝒯~(0;𝝎))).

Since the transmission parameters are often required to be positive the log-parameterization is more appropriate. Let 𝜽 denote the transmission parameters on the logarithmic scale, namely 𝜽:=log(𝝎). The log-parameterized log-likelihood function is therefore (𝜽|𝐭~):=𝝎(log(𝝎)|𝐭~). The Jacobian matrix corresponding to the log-parameterization equals

𝐉𝝎(𝜽)=diag(e𝜽)=diag(𝝎),

where diag(x) denotes a diagonal matrix with vector x representing its diagonal elements.

The score function and the Fisher information matrix

The maximum likelihood (ML) estimator 𝝎^ maximizes the log-likelihood function and is a root of the score function

𝐮𝝎(𝝎|𝐭~):=𝝎𝝎(𝝎|𝐭~)=i=1I(𝝎𝒯~(t~i)(0;𝝎)𝒯~(t~i)(0;𝝎)+𝝎𝒯~(0;𝝎)1-𝒯~(0;𝝎)),

or equivalently, the ML estimator 𝜽^ solves

u(θ^|t~)=Jω(θ)Tuω(eθ^|t~)=!0,

where 𝐮 denotes the score function corresponding to the log-parameterized log-likelihood .

The Fisher information matrix 𝝎(𝝎|𝐭~):=-2𝝎2𝝎(𝝎|𝐭~) is given by

ω(ω|t~)=i=1I(2ω2𝒯~(t~i)(0;ω)𝒯~(t~i)(0;ω)(ω𝒯~(t~i)(0;ω)𝒯~(t~i)(0;ω))2+2ω2𝒯~(0;ω)1𝒯~(0;ω)+(ω𝒯~(0;ω)1𝒯~(0;ω))2)

and equals

(𝜽|𝐭~)=𝐉𝝎(𝜽)𝖳𝝎(e𝜽|𝐭~)𝐉𝝎(𝜽)-diag(𝐮𝝎(e𝜽|𝐭~))𝐉𝝎(𝜽)

under the log-parameterization due to the chain rule in higher dimensions and the special form of the transformation corresponding to the log-parameterization. The PGF function 𝒯~ and its derivatives are thus crucial (and sufficient) for the statistical inference, since the log-likelihood function , the score function 𝐮 and the Fisher information matrix can be expressed in terms of 𝒯~ only.

Confidence intervals and hypothesis testing

Assuming that the regularity conditions are satisfied (Held and Bové, 2013) the ML estimator is unbiased and asymptotically normally distributed with variance equal to the inverse observed Fisher information matrix. Hence, for each parameter θ𝜽 we can construct the Wald α%-confidence interval as

𝒞θ,α=(θ^z1+α2se(θ^),θ^+z1+α2se(θ^))

where z1+α2 denotes the 1+α2-quantile of the standard normal distribution, and the standard error se(θ^) is defined as

se(θ^):=-1(𝜽^|𝐭~)θθ,

and θθ denotes the diagonal element of the inversed observed Fisher information matrix (𝜽|𝐭~) corresponding to parameter θ. The approximate α%-confidence interval for the original parameter ω is obtained by the reverse transformation

𝒞ω,α=e𝒞θ,α.

Similarly, to test the hypothesis H0:θ=θ0 against the alternative HA, the Wald test statistic

τθ(θ0):=θ^-θ0se(θ^)

can be used. Assuming the standard normal distribution of the test statistic under null hypothesis, the p-value equals

  • 2(1-Φ(|τθ(θ0)|)) for the alternative hypothesis HA:θθ0,

  • Φ(τθ(θ0)) for the alternative HA:θ<θ0, and

  • 1-Φ(τθ(θ0)) for the alternative HA:θ>θ0;

where Φ is the cumulative distribution function of the standard normal distribution.

Generalized transmission chain size model

Suppose that the variability of one of the parameters can be explained through a linear combination of different covariates, namely

𝜽i:=(𝜷𝖳𝐱i,𝜼)

are the transmission parameters of the ith chain with characteristics 𝐱i, where 𝜼 denotes the remaining parameters from 𝜽 which are not modeled as a linear combination. Furthermore, it is plausible to assume that while the transmission chains share all the transmission parameters (𝜷,𝜼), their transmission chain size distribution may differ due to different sampling densities or different offspring distribution of the index case (for instance, for the transmission chains originating from other Swiss transmission groups the ICRTP is irrelevant/equals ρindex=1). Let 𝒯~i be the PGF corresponding to the transmission chain i and let 𝐗:={𝐱i}i=1I. The generalized log-likelihood function is hence given by

(𝜷,𝜼|𝐭~,𝐗)=i=1I(log(𝒯~i(t~i)(0;𝝎i(𝜷,𝜼)))-log(t~i!)-log(1-𝒯~i(0;𝝎i(𝜷,𝜼)))),

where

𝝎i(𝜷,𝜼):=(e𝜷𝖳𝐱i,e𝜼).

The corresponding Jacobian matrix equals

𝐉𝝎i(𝜷,𝜼)=[e𝜷𝖳𝐱i𝐱i𝖳𝟎𝟎diag(e𝜼)].

In the generalized model, the score function is

𝐮(𝜷,𝜼|𝐭~,𝐗):=(𝜷,𝜼)(𝜷,𝜼|𝐭~,𝐗)=i=1I𝐉𝝎i(𝜷,𝜼)𝖳(𝝎𝒯~i(t~i)(0;𝝎i(𝜷,𝜼))𝒯~i(t~i)(0;𝝎i(𝜷,𝜼))+𝝎𝒯~i(0;𝝎i(𝜷,𝜼))1-𝒯~i(0;𝝎i(𝜷,𝜼))) and the Fisher information matrix as

(β,η|t~,X):=22(β,η)(β,η|t~,X)=i=1IJωi(β,η)T(diag(ω𝒯~i(t~i)(0;ωi(β,η))𝒯~i(t~i)(0;ωi(β,η))+ω𝒯~i(0;ωi(β,η))1𝒯~i(0;ωi(β,η)))Jθi(β,η)=i=1IJωi(β,η)T(+(2ω2𝒯~i(t~i)(0;ωi(β,η))𝒯~i(t~i)(0;ω)(ω𝒯~i(t~i)(0;ωi(β,η))𝒯~i(t~i)(0;ωi(β,η)))2=i=1IJωi(β,η)T(+(+2ω2𝒯~i(0;ωi(β,η))1𝒯~i(0;ωi(β,η))+(ω𝒯~i(0;ωi(β,η))1𝒯~i(0;ωi(β,η)))2)Jωi(β,η))

Prediction intervals

It is tempting to construct an approximate confidence interval for the parameter θi:=𝜷𝖳𝐱i. Since the parameter θi is a prediction rather than an estimate, the element of interest is the prediction interval, which takes into account both the characteristics 𝐱i and the uncertainty of all parameter estimates 𝜷^.

Assuming that the ML estimator (𝜷^,𝜼^) is asymptotically normally distributed, it follows that the linear combination 𝜷^𝖳𝐱i is also asymptotically Gaussian, specifically

β^Txia.𝒩(βTxi,xiTVar(β^)xi).

The variance Var(𝜷^) can be approximated by the inverse of the observed Fisher information matrix as

Var(𝜷^)-1(𝜷^,𝜼^|𝐭~,𝐗)𝜷𝜷.

Finally, an approximate α%-prediction interval for θi is constructed as

θi,α=(β^Txiz1+α2se(β^Txi),β^Txi+z1+α2se(β^Txi)),

with

se(𝜷^𝖳𝐱i):=𝐱i𝖳Var(𝜷^)𝐱i.

Example: Poisson model

Suppose that the number of secondary infections follows the Poisson distribution with parameter R0. Taking into account the modified transmissibility of the index case ρindex (wherever applicable), the PGFs and 𝒢 for the index case and the tail, respectively, are

(z;R0)=eρindexR0(z-1),𝒢(z;R0)=eR0(z-1).

The skeleton function 𝒦 thus solves

(7) 𝒦(z;R0)=!zeR0(𝒦(z;R0)1),z.

Consider an imperfectly sampled transmission chain with probability of detection p and with ICRTP ρindex. The aim is to obtain the Taylor coefficients of 𝒯~ around z=0 to be able to estimate the transmission parameter R0 with the maximum likelihood approach since they are needed to calculate the log-likelihood (Equation 6).

Let

w:=(1-p)+pz

and

𝒴(w;R0):=𝒯~(w-(1-p)p;R0)

such that 𝒴((1-p)+pz;R0)=𝒯~(z;R0) and that the Equation 5 of the PGF of observed transmission chain size 𝒯~ simplifies to

𝒴(w;R0)=w(𝒦(w;R0)).

Taking into account the PGF of the index case implies

𝒴(w;R0)=weρindexR0(𝒦(w;R0)-1).

Solving for 𝒦(w;R0) yields

𝒦(w;R0)=1ρindexR0log(𝒴(w;R0)w)+1.

Plugging this into Equation 7 gives

1ρindexR0log(𝒴(w;R0)w)+1=we1ρindexlog(𝒴(w;R0)w)1ρindexR0log(𝒴(w;R0)w)=w(𝒴(w;R0)w)1ρindex-1.

With 𝒵(w;R0):=(𝒴(w;R0)w)1ρindex, the last equation is equivalent to

1R0log(𝒵(w;R0))=w𝒵(w;R0)-1𝒵(w;R0)=eR0(w𝒵(w;R0)-1)𝒵(w;R0)e-R0w𝒵(w;R0)=e-R0-R0w𝒵(w;R0)e-R0w𝒵(w;R0)=-R0we-R0,

which is an equation of the form f(w)ef(w)=g(w). The latter admits a solution f(w)=W0(g(w)), where W0 is the principal branch of the Lambert W function (Corless et al., 1996). Thus

𝒵(w;R0)=W0(-R0we-R0)-R0w,

and finally,

𝒴(w;R0)=w𝒵(w;R0)ρindex=w(W0(R0weR0)R0w)ρindex=w(eR0W0(R0weR0)R0weR0)ρindex=weρindexR0(W0(R0weR0)R0weR0)ρindex.

Using the relation W0(-x)-x=e-W0(-x) (which follows from the definition of W0(-x)), we have

𝒴(w;R0)=we-ρindexR0e-ρindexW0(-R0we-R0).

From the Taylor expansion of e-γW0(-x)=m=0γ(γ+m)m-1xmm! around x=0 (equality (2.36) in Corless et al., 1996), we obtain

𝒴(w;R0)=weρindexR0m=0ρindex(m+ρindex)m1m!(R0weR0)m=m=0ρindex(m+ρ index)m1R0meR0(m+ρindex)m!wm+1=m=1ρindex(m+ρ index1)m2R0m1eR0(m+ρindex1)(m1)!wm.

In terms of 𝒯~ this yields

(8) 𝒯~(z;R0)=m=1ρindex(m+ρindex-1)m-2R0m-1e-R0(m+ρindex-1)(m-1)!((1-p)+pz)m.

Unfortunately, we need Taylor expansion around z=0 to derive the state probabilities (and consequently the log-likelihood function). By applying the binomial theorem, 𝒯~ can be re-written as

𝒯~(z;R0)=m=1ρindex(m+ρ index1)m2R0m1eR0(m+ρindex1)(m1)!k=0m(mk)(1p)mkpkzk=k=0m=k1ρ index(m+ρindex1)m2R0m1eR0(m+ρindex1)(m1)!(mk)(1p)mkpkzk𝒯~(z;R0)=k=0ρindex(p1p)kk!(m=k1m(m+ρ index1)m2R0m1eR0(m+ρindex1)(1p)m(mk)!)zk,

with m=k1 denoting m=max{k,1}.

Initial estimate for R0

Since the optimization problem of maximizing the likelihood does not admit a closed-form solution, the ML estimator is obtained with numerical techniques for which a suited initial estimate for R0 is required. In the following paragraphs we present one possibility for obtaining a useful starting value (which was also implemented and used in our analyses).

Let

μ¯:=1Ii=1It~i

be the observed average chain size (based on a sample of I observed chains 𝐭~ like proposed in Blumberg and Lloyd-Smith, 2013b). μ¯ represents a reasonable estimate for

μ¯E[T~|T~>0]=E[T~]1P[T~=0]=𝒯~(1)(1;R0)1𝒯~(0;R0).

The definition of the skeleton function 𝒦 for transmission parameters 𝝎 implies 𝒦(1;𝝎)=1. Implicitly deriving Equation 3 with respect to z implies

𝒦(1)(1;𝝎)=11-𝒢(1)(1;𝝎),

since 𝒢(1;𝝎)=1 (just like for any PGF). Moreover, implicitly deriving Equation 5 with respect to z yields

𝒯~(1)(1;ω)=p(𝒦(1;ω);ω)+(1)(𝒦(1;ω);ω)𝒦(1)(1;ω)p=p(1;ω)+(1)(1;ω)11𝒢(1)(1;ω)p=p(1+(1)(1;ω)1𝒢(1)(1;ω)).

Under the Poisson model, the latter equals to

𝒯~(1)(1;R0)=p(1+ρindexR01R0).

Next, we can use the first Taylor coefficient of 𝒯~(z;R0) from Equation 8, namely 𝒯~(0;R0)e-ρindexR0(1-p). In order to obtain a quadratic equation with respect to R0, we further use the approximation e-ρindexR01-ρindexR0, such that 1-𝒯~(0;R0)1-(1-ρindexR0)(1-p). This yields the quadratic equation

μ¯=!p(1+ρ indexr01r0)p+(1p)ρindexr0

with the roots

r0=a±bc,

where

a=ρindex(μ¯(p-1)+p)+p(μ¯-1)b=4ρindex(μ¯-1)μ¯(1-p)p+(ρindexμ¯+p-(ρindex+μ¯+ρindexμ¯)p)2c=-2ρindexμ¯(1-p).

Should none of the roots lie within (0,1), we could use the following feature. If the average size of the observed chain equals μ¯, the average size of the complete transmission chains would be roughly μ¯p (since the mean value of the binomial distribution Bin(n,p) is np). Hence,

μ¯p𝔼[T]=𝒯(1)(1;𝝎).

Equation 4 then implies

𝒯(1)(1;ω)=(𝒦(1;ω);ω)+(1)(𝒦(1;ω);ω)𝒦(1)(1;ω)=1+(1)(1;ω)11𝒢(1)(1;ω).

In case of the Poisson model, the initial estimate for R0 can be therefore obtained by solving the equation

μ¯p=!1+ρindexr01r0,

which has the solution

r0=μ¯-pρindexp+μ¯-p.

Generalized Poisson model

Let 𝐓~:={𝐭~i}i=1I be a sample of I observed transmission chains where each observed transmission chain 𝐭~i carries the following information

𝐭~i:=(t~i,𝐱i,pi,ρindex,i),

namely the observed chain size t~i, the chain characteristics 𝐱i, the probability pi at which each infection in the chain is observed, and the index case relative transmission potential ρindex,i. In the generalized Poisson transmission chain size distribution model we assume that the heterogeneity of the basic reproductive number R0 can be explained by the variability of the demographic characteristics of the transmission chains, namely

log(R0,i):=𝜷𝖳𝐱i.

The vector 𝜷 describes the effect of the chain characteristics on the basic reproductive number R0 and it is the same for all transmission chains.

To obtain the maximum likelihood estimates for 𝜷, we need initial values of the estimates. One possibility is to use the coefficients from the linear regression model, in which the response values are the individual initial estimates for R0 for each transmission chain. More precisely, imagine that each transmission chain 𝐭~i is a sample of transmission chains itself and therefore we can obtain the initial r0,i estimates as described above. In the next step, we fit the linear regression model

log(r0,i):=𝜷0𝖳𝐱i+εi,εi𝒩(0,σ2),

and use 𝜷^0 as the initial values.

Example: Negative binomial model

Assume that the number of secondary infections caused by an individual is negative binomially distributed with mean R0 and dispersion parameter ξ. Its PGF equals

𝒢(z;R0,ξ)=(1+R0ξ(1-z))-ξ.

For the simplicity assume that index case has the same transmission potential as the remaining part of the transmission chain, namely 𝒢 (which coincides with ρindex=1). The skeleton function 𝒦(z;R0,ξ) is therefore a solution of the equation

𝒦(z;R0,ξ)=z(1+R0ξ(1-𝒦(z;R0,ξ)))-ξ,

which does not admit a closed-form solution. However, as a consequence of the Lagrange inversion theorem its Taylor coefficients around z=0 can be explicitly calculated (Blumberg and Lloyd-Smith, 2013b) as

𝒦(k)(0;R0,ξ)=Γ(ξk+k-1)Γ(ξk)(R0ξ)k-1(1+R0ξ)ξk+k-1.

Since we assumed 𝒢, it follows 𝒯(z;R0,ξ)=𝒦(z;R0,ξ) for all z. For a transmission chain in which each case is observed with probability p, the PGF of the observed transmission chain size equals

𝒯~(z;R0,ξ)=𝒯((1-p)+pz;R0,ξ).

By applying the binomial theorem to the Taylor expansion around z=0 of 𝒦(z;R0,ξ) the higher-order derivatives

𝒯~(k)(0;R0,ξ)=m=kΓ(ξm+m-1)Γ(ξm)Γ(m-k+1)(R0ξ)m-1(1+R0ξ)ξm+m-1pk(1-p)m-k

are obtained (which coincides with the result from Blumberg and Lloyd-Smith, 2013a).

In similar manner as in the case of Poisson model, the generalized negative binomial model can be derived by introducing

log(R0,i)=𝜷𝖳𝐱i.

The sampling density p can vary between the transmission chains (or their characteristics, for instance between the subtypes), while the dispersion parameter ξ is kept constant among all the transmission chains.

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Decision letter

  1. Ryosuke Omori
    Reviewing Editor; Hokkaido University, Japan

In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.

Thank you for submitting your article "Assessing the danger of self-sustained HIV epidemics in heterosexuals by population based phylogenetic cluster analysis" for consideration by eLife. Your article has been reviewed by three peer reviewers, one of whom, Ryosuke Omori (Reviewer #1) served as Guest Editor, and the evaluation has been overseen by Prabhat Jha as the Senior Editor. The following individuals involved in review of your submission have agreed to reveal their identity: Dimitrios Paraskevis (Reviewer #2); Nico Nagelkerke (Reviewer #3).

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

Summary:

The authors developed a new method for estimating epidemiological parameters e.g., R0, using viral sequences sampled from patients, via estimating infection tree per each cluster in phylogenetic tree. Using the model they assessed the risk of a generalized heterosexual HIV epidemic in Switzerland. They find that the basic reproduction number is well below 1 they conclude that this risk is negligible.

Essential revisions:

The robustness of the results should be assessed with relaxing the several biological assumptions assumed by the authors. 1) If the first CD4 count (must have been taken before ART initiation) is 310 (median, some even <200 and thus having AIDS) it is questionable whether the assumption of patient becoming non-infectious after 2-2.5 years is realistic. 2) The high acute phase infectivity of HIV has been challenged (e.g. Bellan et al., 2015), this also should be mentioned and considered in the analysis.

Since index cases may differ substantially from other cases (e.g. foreigners in Switzerland without their partner, moving frequently between places), ρindex may well exceed 1.

It is not entirely clear whether – when considering missing patients – the case of a missing index case without "offspring" (i.e., no patient is observed at all) is considered.

The authors assume that the number of secondary infections follows the same distribution regardless of the transmission degree (subsection “Probability generating function of a completely observed uniform transmission chain”). This assumption means that the sexual behaviors are similar among hetero-sexual population whereas the field data of sexual behavior shows huge variation (e.g., Liljeros et al., 2001).

Regarding the estimate for confidence intervals, it is not clear whether the distribution of ML estimator can be approximated by normal distribution or not due to small sample size. If the authors will not apply Wald approximation, the width of confidence interval may change.

The point estimates of R0 shown in the lower panel of Figure 2 (black dot) are likely to be increased since 2009 although the authors' model predicts monotonic decrease. Why does this discrepancy happen? The authors' model predicts HIV will go extinct around 2020, however, this prediction may be optimistic if the predictive model lacks some critical factors.

https://doi.org/10.7554/eLife.28721.028

Author response

Essential revisions:

The robustness of the results should be assessed with relaxing the several biological assumptions assumed by the authors. 1) If the first CD4 count (must have been taken before ART initiation) is 310 (median, some even <200 and thus having AIDS) it is questionable whether the assumption of patient becoming non-infectious after 2-2.5 years is realistic. 2) The high acute phase infectivity of HIV has been challenged (e.g. Bellan et al., 2015), this also should be mentioned and considered in the analysis.

We believe that this might be a misunderstanding, since none of the two assumptions was made in our analyses directly but rather used as an intuitive explanation of 1) why the ongoing transmissions are not problematic and 2) to justify the introduction of ρindex (which was already assessed in a sensitivity analysis by setting ρindex=1). When verifying the stuttering transmission chains assumption we originally used even an infectious period of 5 years because it is more conservative for this test (longer infectious periods imply more ongoing transmission chains and hence potentially stronger underestimation of R0). Despite the minor impact of these assumptions on our method, we further addressed these issues as follows:

Regarding 1) we extended and adjusted the section “Stuttering transmission chains assumption” (now named “Ongoing transmission and stuttering transmission chains assumption”) to assess the role of the duration of infectiousness in our study. Rather than assuming a constant duration, we assumed decreasing periods over time and used the information from the SHCS to estimate the decrease over time. Next, we re-evaluated the conservative maximum number of transmission degrees that had been completed by a certain date (Appendix 1—figure 3), which were then used in the simulation study to verify the subcritical transmission assumption (now Appendix 1—figure 5, previously Appendix 1—figure 3). Additionally, we explicitly assessed the relative bias stemming from ongoing transmission (Appendix 1—figure 4).

We had addressed point 2) in the original version of the manuscript by performing the sensitivity analysis over a range of ρindex (Appendix 1—figure 1), where the lower ρindex values implicitly correspond to later immigration to Switzerland and/or the importance of acute phase on transmission, and vice versa – higher ρindex indicates either immigration early during infection or high infectivity long into the chronic phase. We added a more explicit relation between the ρindex values and infectivity in the Discussion. The revised sentence in the Discussion reads:

"Then again, the concrete value chosen may be debatable, especially due to arguable infectivity in chronic phase (studied by Bellan et al., 2015; thus a small ρindex can be caused both by immigration later during chronic infection and by elevated infectivity in the acute phase."

Since index cases may differ substantially from other cases (e.g. foreigners in Switzerland without their partner, moving frequently between places), ρindex may well exceed 1.

This is an excellent point. In principle the index case could exhibit enhanced transmission potential. However, we do not expect that this is often the case and even if the ρindex exceeds 1, this would imply even lower R0. We added this comment to the Discussion: "Furthermore, even though theoretically the transmission potential of some index cases could also be enhanced (i.e., ρindex>1), for instance for sex workers, we do not expect that this is the case for many transmission chains and would therefore have only marginal effect on our estimates. Besides, since a ρindex>1 would lead to even lower R0, our main conclusions would not change (in fact, the assumption of ρindex<1 is conservative with respect to our conclusion of R0<1)." Furthermore, we replaced the "reduced" with "modified" transmission potential/transmissibility throughout the manuscript and extended the range of the sensitivity analyses regarding ρindex beyond 1 (Appendix 1—figure 1).

It is not entirely clear whether – when considering missing patients – the case of a missing index case without "offspring" (i.e., no patient is observed at all) is considered.

We are very grateful for this remark. In our model we indeed considered the transmission chains without any observed cases (i.e., all patients including the index case are missing) by normalizing the state probabilities for the observed transmission chain size distribution by the probability that a transmission chain is observed (namely 1P[T~=0]). In this sense our model takes into account that the sample of observed transmission chains from the phylogeny is biased, simply because the transmission chains with no observed cases cannot be detected. The same solution was suggested by Blumberg and Lloyd-Smith, 2013b. In the Methods and Materials we added some lines to explicitly explain how the unobserved transmission chains were handled:

"The probability that a transmission chain has observed size of t~0 (where t~=0 means that none of the cases of the transmission chain is detected) is given by

P[T~=t~]=1t~!T~(t~)(0;R0,ρindex,p).

In particular, the probability that a transmission chain is observed (i.e., the observed size is strictly positive) can be calculated as

P[T~>0]=1P[T~=0]=1T~(0;R0,ρindex,p).

However, since only the transmission chains with at least one detected case can be extracted from the phylogeny (and therefore to account for the unobserved transmission chains) we are interested in the probability that an observed transmission chain has a specific size. The probability of observing a transmission chain of size t~>0 is

P[T~=t~|T~>0]=1t~!T~(t~)(0;R0,ρindex,p)1T(0;R0,ρindex,p)."

The second source of confusion could be the transmission chains in which the index case (and/or some other intermediaries) are missing. These cases are described with the two assumptions in the Likelihood function section. For instance, if an index case with a single offspring was not sampled while its offspring was detected on the phylogeny, our model would treat this transmission pair as an observed transmission chain of observed size 1 and the sampling density p<1 would account for the 'missing' index case.

The authors assume that the number of secondary infections follows the same distribution regardless of the transmission degree (subsection “Probability generating function of a completely observed uniform transmission chain”). This assumption means that the sexual behaviors are similar among hetero-sexual population whereas the field data of sexual behavior shows huge variation (e.g., Liljeros et al., 2001).

We indeed made the above assumption as a trade-off between accuracy and simplicity, as modelling the human sexual behavior would most likely require changing patterns even within an individual over time. Yet, since our transmission chains are very short (mean 1.19, median 1), we did not expect that the heterogeneity between the individuals within a chain would have an important impact, as we had explained in the Discussion. The following three arguments affirmed this explanation:

• The significance at 5%-level and the direction of different determinants did not change by selecting a random infected individual from each chain to determine the risky sexual behavior of the chain as compared to the main analysis, in which the index case determined the sexual risk behavior of a transmission chain (newly added Appendix 1—figure 6).

• Similarly, the determinants were robust when considering all the follow-up questionnaires (about sex with an occasional partner) from all the patients from a transmission chain to determine the chain’s risky behavior (new Appendix 1—figure 6).

• The comparison between the negative binomial and Poisson based transmission chain size distribution model did not exhibit a strong preference of the former over the latter. Noteworthy, the dispersion parameter of the negative binomial distribution takes into account the heterogeneity between the individuals, hence no significant difference between the models implies that the heterogeneity between the individuals is sufficiently reflected by the variability between the transmission chains in terms of their characteristics (i.e., the model covariables), including the risky sexual behavior (see Comparison between Poisson and negative binomial offspring distribution based models in Appendix 1).

We added a section in the Sensitivity analyses appendix to explain how the effect of the variability in sexual behavior was assessed ("Variation in sexual behavior along transmission chains").

Regarding the estimate for confidence intervals, it is not clear whether the distribution of ML estimator can be approximated by normal distribution or not due to small sample size. If the authors will not apply Wald approximation, the width of confidence interval may change.

We agree with the reviewers that the Wald approximation based on the normal distribution might be debatable. To address this question we did the following:

• We performed parametric bootstrap to assess the assumption of the normal approximation and the performance of the Wald-type confidence intervals in mean of coverage rates. The obtained empirical distribution of the ML estimator could be well approximated by the normal distribution and the coverage rates were very close to or above the target 95% (newly added Appendix 1—figure 10).

• For all the transmission parameters from any of the models presented in our study (newly added Appendix 1 Table 1) we constructed the profile likelihood based confidence intervals, as well as the basic bootstrap confidence intervals (from the parametric and nonparametric bootstrapping) to compare their widths. For our sample size, the Wald confidence intervals turned out to be almost the same as the profile likelihood based confidence intervals, while we did not observe that the Wald-type confidence intervals are systematically wider/narrower than the bootstrap based confidence intervals (Appendix 1—figure 11).

We therefore concluded that the normal approximation Wald-type confidence intervals for our sample size are a reliable and computationally less expensive alternative. We added a section "Confidence intervals" to Appendix 1.

The point estimates of R0 shown in the lower panel of Figure 2 (black dot) are likely to be increased since 2009 although the authors' model predicts monotonic decrease. Why does this discrepancy happen? The authors' model predicts HIV will go extinct around 2020, however, this prediction may be optimistic if the predictive model lacks some critical factors.

This is a very valid remark. In our model, the R0 point estimates exhibit a slight increase after 2009, however the number of yet sampled transmission chains from these recent years is still small, which is also reflected by very wide confidence intervals. We admit that the extrapolation up to 2020 might be too optimistic; therefore we modified the plot (Figure 2 and the profiled multivariate plot in Figure 4) such that the R0 is now only shown up to year 2015 to avoid misinterpretations. The plot shows the general trend and should be understood as such and not as a definitive prognosis of R0 for the future years, as all the models are related to some kind of uncertainty. We also added a note on this issue at the end of the paragraph describing Figure 2 in the Results section: "This extrapolation should be, however, taken with a grain of salt and seen more as a trend rather than a prognosis, since only a few transmission chains have been observed for the recent years (which is reflected by wide confidence intervals)."

https://doi.org/10.7554/eLife.28721.029

Article and author information

Author details

  1. Teja Turk

    1. Division of Infectious Diseases and Hospital Epidemiology, University Hospital Zurich, Zurich, Switzerland
    2. Institute of Medical Virology, University of Zurich, Zurich, Switzerland
    Contribution
    Conceptualization, Software, Formal analysis, Visualization, Methodology, Writing—original draft, Writing—review and editing
    Competing interests
    No competing interests declared
    ORCID icon 0000-0003-3065-8578
  2. Nadine Bachmann

    1. Division of Infectious Diseases and Hospital Epidemiology, University Hospital Zurich, Zurich, Switzerland
    2. Institute of Medical Virology, University of Zurich, Zurich, Switzerland
    Contribution
    Software, Formal analysis, Methodology, Writing—review and editing
    Competing interests
    No competing interests declared
    ORCID icon 0000-0002-7303-9542
  3. Claus Kadelka

    1. Division of Infectious Diseases and Hospital Epidemiology, University Hospital Zurich, Zurich, Switzerland
    2. Institute of Medical Virology, University of Zurich, Zurich, Switzerland
    Contribution
    Formal analysis, Methodology, Writing—review and editing
    Competing interests
    No competing interests declared
  4. Jürg Böni

    Institute of Medical Virology, University of Zurich, Zurich, Switzerland
    Contribution
    Investigation, Writing—review and editing
    Competing interests
    No competing interests declared
  5. Sabine Yerly

    Laboratory of Virology, Geneva University Hospitals, Geneva, Switzerland
    Contribution
    Investigation, Writing—review and editing
    Competing interests
    No competing interests declared
  6. Vincent Aubert

    Division of Immunology and Allergy, University Hospital Lausanne, Lausanne, Switzerland
    Contribution
    Investigation, Writing—review and editing
    Competing interests
    No competing interests declared
  7. Thomas Klimkait

    Molecular Virology, Department of Biomedicine - Petersplatz, University of Basel, Basel, Switzerland
    Contribution
    Investigation, Writing—review and editing
    Competing interests
    No competing interests declared
  8. Manuel Battegay

    Division of Infectious Diseases and Hospital Epidemiology, University Hospital Basel, Basel, Switzerland
    Contribution
    Investigation, Writing—review and editing
    Competing interests
    No competing interests declared
  9. Enos Bernasconi

    Division of Infectious Diseases, Regional Hospital Lugano, Lugano, Switzerland
    Contribution
    Investigation, Writing—review and editing
    Competing interests
    E.B. has been a consultant for BMS, Gilead, ViiV Healthcare, Pfizer, MSD, and Janssen; has received unrestricted research grants from Gilead, Abbott, Roche, and MSD; and has received travel grants from BMS, Boehringer Ingelheim, Gilead, MSD, and Janssen.
  10. Alexandra Calmy

    Division of Infectious Diseases, Geneva University Hospitals, Geneva, Switzerland
    Contribution
    Investigation, Writing—review and editing
    Competing interests
    No competing interests declared
  11. Matthias Cavassini

    Service of Infectious Diseases, Department of Medicine, Lausanne University Hospital, Lausanne, Switzerland
    Contribution
    Investigation, Writing—review and editing
    Competing interests
    No competing interests declared
  12. Hansjakob Furrer

    Department of Infectious Diseases, Bern University Hospital, University of Bern, Bern, Switzerland
    Contribution
    Investigation, Writing—review and editing
    Competing interests
    The institution of H.F. has received unrestricted grant support from ViiV, Gilead, Abbott, Janssen, Roche, Bristol-Myers Squibb (BMS), Merck Sharp & Dohme (MSD), and Boehringer Ingelheim.
    ORCID icon 0000-0002-1375-3146
  13. Matthias Hoffmann

    Division of Infectious Diseases, Cantonal Hospital St. Gallen, St. Gallen, Switzerland
    Contribution
    Investigation, Writing—review and editing
    Competing interests
    No competing interests declared
  14. Huldrych F Günthard

    1. Division of Infectious Diseases and Hospital Epidemiology, University Hospital Zurich, Zurich, Switzerland
    2. Institute of Medical Virology, University of Zurich, Zurich, Switzerland
    Contribution
    Conceptualization, Supervision, Funding acquisition, Investigation, Writing—original draft, Writing—review and editing
    Contributed equally with
    Roger D Kouyos
    Competing interests
    H.F.G. has been an adviser and/or consultant for GlaxoSmithKline, Abbott, Gilead, Merck, Novartis, Boehringer Ingelheim, Roche, Tibotec, Pfizer, and BMS and has received unrestricted research and educational grants from Roche, Abbott, BMS, Gilead, Astra-Zeneca, GlaxoSmithKline, and MSD (all money to the institution).
  15. Roger D Kouyos

    1. Division of Infectious Diseases and Hospital Epidemiology, University Hospital Zurich, Zurich, Switzerland
    2. Institute of Medical Virology, University of Zurich, Zurich, Switzerland
    Present address
    Division of Infectious Diseases and Hospital Epidemiology, University Hospital Zurich, Zurich, Switzerland
    Contribution
    Conceptualization, Formal analysis, Supervision, Funding acquisition, Methodology, Writing—original draft, Writing—review and editing
    Contributed equally with
    Huldrych F Günthard
    For correspondence
    roger.kouyos@usz.ch
    Competing interests
    R.D.K. has received speaker honoraria and travel grants from Gilead Sciences. None if these are in relation with the submitted manuscript.
    ORCID icon 0000-0002-9220-8348
  16. Swiss HIV Cohort Study

    1. V Aubert
    2. M Battegay
    3. E Bernasconi
    4. J Böni
    5. DL Braun
    6. HC Bucher
    7. A Calmy
    8. M Cavassini
    9. A Ciuffi
    10. G Dollenmaier
    11. M Egger
    12. L Elzi
    13. J Fehr
    14. J Fellay
    15. H Furrer
    16. CA Fux
    17. HF Günthard
    18. D Haerry
    19. B Hasse
    20. HH Hirsch
    21. M Hoffmann
    22. I Hösli
    23. C Kahlert
    24. L Kaiser
    25. O Keiser
    26. T Klimkait
    27. RD Kouyos
    28. H Kovari
    29. B Ledergerber
    30. G Martinetti
    31. B Martinez de Tejada
    32. C Marzolini
    33. KJ Metzner
    34. N Müller
    35. D Nicca
    36. G Pantaleo
    37. P Paioni
    38. A Rauch
    39. C Rudin
    40. AU Scherrer
    41. P Schmid
    42. R Speck
    43. M Stöckle
    44. P Tarr
    45. A Trkola
    46. P Vernazza
    47. G Wandeler
    48. R Weber
    49. S Yerly

Funding

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (33CS30-148522)

  • Huldrych F Günthard

Yvonne-Jacob Foundation

  • Huldrych F Günthard

University of Zurich's Clinical Research Priority Program's ZPHI

  • Huldrych F Günthard

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (159868)

  • Huldrych F Günthard

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (PZ00P3-142411)

  • Roger D Kouyos

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (BSSGI0-155851)

  • Roger D Kouyos

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Acknowledgements

We thank Mohaned Shilaih for providing the original data regarding the sampling density. Furthermore, we thank Alex Marzel, Katharina Kusejko, Bruno Ledergerber and Roland R Regoes for fruitful discussions. We thank the patients who participate in the Swiss HIV Cohort Study (SHCS); the physicians and study nurses for excellent patient care; the resistance laboratories for high-quality genotypic drug resistance testing; SmartGene (Zug, Switzerland) for technical support; Johannes Abegglen, Bojana Milosevic, Alexandra U Scherrer, Anna Traytel, and Susanne Wild from the SHCS Data Center (Zurich, Switzerland) for data management; and Danièle Perraudin and Mirjam Minichiello for administrative assistance.

Ethics

Human subjects: The SHCS was approved by the ethics committees of the participating institutions (Kantonale Ethikkommission Bern, Ethikkommission des Kantons St. Gallen, Comite Departemental d'Ethique des Specialites Medicales et de Medicine Communataire et de Premier Recours, Kantonale Ethikkommission Zürich, Repubblica e Cantone Ticino-Comitato Ethico Cantonale, Commission Cantonale d'Étique de la Recherche sur l'Être Humain, Ethikkommission beiderBasel; all approvals are available on http://www.shcs.ch/206-ethic-committee-approval-and-informed-consent), and written informed consent was obtained from all participants.

Reviewing Editor

  1. Ryosuke Omori, Reviewing Editor, Hokkaido University, Japan

Publication history

  1. Received: May 17, 2017
  2. Accepted: August 28, 2017
  3. Accepted Manuscript published: September 12, 2017 (version 1)
  4. Version of Record published: October 20, 2017 (version 2)

Copyright

© 2017, Turk et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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