Assessing the danger of selfsustained HIV epidemics in heterosexuals by population based phylogenetic cluster analysis
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 phylogenybased 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 ${R}_{0}$, we extend the branching process model to infer transmission parameters. Overall, the ${R}_{0}$ is estimated to be $0.44$ ($95\%$confidence interval $0.42$—$0.46$) 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 selfsustained epidemics from any viral sequence data.
https://doi.org/10.7554/eLife.28721.001eLife 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 selfsustained 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 selfsustained 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.28721.002Introduction
Epidemics of HIV and other bloodborne 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 highrisk 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 selfsustained, the HIV epidemic remains concentrated; otherwise the virus is spreading rapidly in the broad population leading to a generalized HIV epidemic.
In most resourcerich 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; RagonnetCronin 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 ${R}_{0}$ among heterosexuals is below $1$. However, it is not clear how far away from selfsustained 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 sourcesink dynamics, i.e., a flux of infections from a subpopulation in which the disease is selfsustained 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 LloydSmith, 2013b; Blumberg and LloydSmith, 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 humantohuman 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 animalhuman transmission the delayed introduction of the index case of an STI or bloodborne 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 ${R}_{0}$ from the size distribution of transmission chains. We further extend this approach to determine the impact of calendar time and other potential determinants on ${R}_{0}$; especially in order to assess whether ${R}_{0}$ 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 ${R}_{0}<1$ is from the epidemic threshold, that is, how far it is from being selfsustained in these populations (see Materials and methods). A classical application of this question/method is HIV1 transmission in heterosexuals in settings with a concentrated epidemic. Heterosexual HIV1 transmission in Switzerland is a case in point for such a nonselfsustained HIV epidemic. We identified $3,\phantom{\rule{negativethinmathspace}{0ex}}100$ 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 nonSwiss origin. For these latter transmission chains, we assumed that the ${R}_{0}$ of the index case was reduced by a factor of ${\rho}_{\text{index}}=0.35$ (see Materials and methods). To take into account the imperfect sampling density we fixed the subtypedepending 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).
${R}_{0}$ of the HIV transmission in Swiss heterosexuals
To obtain an overall estimate for the ${R}_{0}$ 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 ${R}_{0}$ was estimated to be $0.44$ ($95\%$confidence interval (CI) $0.42$—$0.46$). The fact that ${R}_{0}$ was clearly below $1$ ($p$value $<0.001$ from onesided Wald hypothesis testing ${H}_{0}:{R}_{0}=1$ against the alternative ${H}_{A}:{R}_{0}<1$) indicated that HIV transmission is far away from a selfsustained epidemic.
Although the overall ${R}_{0}$ estimate was clearly below $1$, individual subtypes represent different epidemiological settings and hence individual subtypes may have ${R}_{0}$ closer to the epidemic threshold. The subtypestratified analyses indeed yielded lower ${R}_{0}$ of $0.35$ ($95\%$CI $0.33$—$0.39$) for subtype B as compared to the nonB subtypes (Figure 1). The recombinant form CRF02_AG had the highest estimated ${R}_{0}$ of $0.62$ ($95\%$CI $0.56$—$0.69$). Despite these differences among the ${R}_{0}$ estimates for different subtypes they were all significantly below $1$ (with all $p$values from the onesided test smaller than $0.001$). Therefore, we concluded that there is no danger of a selfsustained HIV epidemic in Swiss heterosexuals of any HIV subtype.
Time trend of the ${R}_{0}$
Despite consistently low ${R}_{0}$ estimates, an increasing time trend for ${R}_{0}$ 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 $\mathrm{log}\left({R}_{0}\right)$ as a linear function of the establishment date of the transmission chain. We found that overall the ${R}_{0}$ is decreasing at a factor $0.89$ per $10$ years ($95\%$CI $0.83$—$0.96$). The per subtypestratified 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 Bsubtype.
To better capture the changes of ${R}_{0}$ over time we included higherorder 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 ${R}_{0}$ from the mid 1980’s to the mid 1990’s, with a peak${R}_{0}$ of $0.49$ ($95\%$CI $0.46$—$0.53$) reached in 1996 and followed by a steep and monotonic decrease. It is noteworthy that the time of peak${R}_{0}$ coincided with the introduction of highly active antiretroviral therapy. Shortly after the ${R}_{0}$ 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).
Determinants of the HIVtransmission
Finally, we identified the characteristics associated with higher ${R}_{0}$ and therefore potential focal subpopulations, in which the basic reproductive number ${R}_{0}$ could be above $1$. The simplest model containing only the linear terms of risk factors showed that the ${R}_{0}$ is decreasing with the establishment date of the transmission chain and that all nonB subtypes have higher ${R}_{0}$ compared to subtype B, which was consistent with the findings from the univariate model and persubtype 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 ${R}_{0}$, whereas the earliest CD4 cell count and the age do not have significant effects (Figure 3).
These trends remained robust (Figure 4) when allowing the covariables to enter the model nonlinearly (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 ${R}_{0}$ (see Selection of the predictive models). Allowing nonlinear terms for the time to diagnosis provided better goodnessoffit than the linear model. The steep increase of ${R}_{0}$ 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.
Discussion
Our approach demonstrates that viral sequences combined with basic demographic information can be successfully used not only to estimate the basic reproductive number ${R}_{0}$ 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 ${R}_{0}$ 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 nonselfsustained epidemics since viral sequences from genotypic resistance testing are nowadays routinely produced in most resourcerich 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 nonB subtypes (Esbjörnsson et al., 2016; Chaillon et al., 2017; RagonnetCronin 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 birthdeath 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 ${R}_{0}$ 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 nonB 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 selfsustained epidemics.
Epidemiological differences between the HIV1 subtypes, especially between B and nonB 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 nonB 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 nonB 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 ${R}_{0}$ 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 ${\rho}_{\text{index}}$, 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 ${\rho}_{\text{index}}=1$, leads to largely underestimated ${R}_{0}$ (overall ${R}_{0}$ of $0.33$, $95\%$CI $0.31$—$0.35$) 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 ${\rho}_{\text{index}}$ 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 ${\rho}_{\text{index}}$ for the transmission chains of nonSwiss origin to $0.25$ to obtain a more conservative estimate of ${R}_{0}$, which was, nevertheless, still safely below $1$ ($0.47$, $95\%$CI $0.44$—$0.49$). Furthermore, even though theoretically the transmission potential of some index cases could also be enhanced (i.e., ${\rho}_{\text{index}}>1$), 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 ${\rho}_{\text{index}}>1$ would lead to even lower ${R}_{0}$, our main conclusions would not change (in fact, the assumption of ${\rho}_{\text{index}}<1$ is conservative with respect to our conclusion of ${R}_{0}<1$).
The presented model is based on sourcesink 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 resourcerich settings similar sourcesink 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 ${R}_{0}$ is below $1$. If ${R}_{0}$ 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., 2010; von 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 ${R}_{0}$ 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 noninfectious of approximately $2$—$2.5$ 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 LloydSmith, 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 ${R}_{0}$ estimate. However, this means that our method will tend to overestimate ${R}_{0}$ and is hence conservative with respect to its main aim of assessing the danger of selfsustained epidemics; thus, if the method predicts an ${R}_{0}$ strongly below $1$, the corresponding epidemic will indeed be far away from being selfsustained. Moreover, our method neglects the transmission chain structure and consequently uses only the aggregated number of infections, and assumes the same ${R}_{0}$ 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 LloydSmith, 2013b) we did not observe any significant differences in the ${R}_{0}$ 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 Poissonbased 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 selfsustained 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 LloydSmith (2013a) to infer the basic reproductive number ${R}_{0}$ (Figure 5). In particular, we accounted for both imperfect detection (included in Blumberg and LloydSmith, 2013a) and modified transmissibility of the index case (not included in Blumberg and LloydSmith, 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 HIV1 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 nonselfsustained 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.
SHCS and viral sequences
Request a detailed protocolThe 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/206ethiccommitteeapprovalandinformedconsent), and written informed consent was obtained from all participants. Up to December 2016 over $19,500$ patients have been enrolled. The SHCS is highlyrepresentative as it covers more than $75\%$ HIVpositive 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 followup (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
Request a detailed protocolThe phylogenetic tree was constructed from the Swiss HIV sequences of the SHCS patients and nonSwiss background sequences exported from the Los Alamos National Laboratory, 2016 database ($241,783$ HIV1 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 $8$ HIV1 subtypes and circulating recombinant forms (B, C, CRF01_AE, CRF02_AG, A(12)), G, D and F(12)) 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 timereversible model of nucleotide evolution (Price et al., 2009) including $10,840$ SHCS and $90,933$ background sequences.
Transmission chains
Request a detailed protocolThe 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 nonSwiss or nonheterosexual 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 followup 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 discretetime branching process. The basic reproductive number ${R}_{0}$ 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
Request a detailed protocolWe 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 LloydSmith, 2013b) in order to infer ${R}_{0}$; 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 ${R}_{k,n}$ denotes the number of secondary infections of transmission degree $n$ caused by the $k$th individual from the preceding generation (i.e., infected individuals with transmission degree $n1$), 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
which coincides with the definition of the basic reproductive number ${R}_{0}=\mathbb{E}\left[{R}_{k,n}\right]$. 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
where ${\rho}_{\text{index}}$ denotes the index case relative transmission potential.
To assess the trends and determinants of ${R}_{0}$, we further extended the offspring distribution based on the baseline characteristics $\mathbf{\mathbf{x}}$ of the transmission chain. More precisely, we assumed that the logarithm of ${R}_{0}$ can be linearly described by the chain characteristics which resulted in the offspring distributions
for the secondary and the index cases, respectively. Hence, the ${R}_{0}$ can be predicted from the effect sizes $\mathit{\beta}$ of factors $\mathbf{\mathbf{x}}$ as
Note that since each transmission chain $i$ has its specific baseline characteristics ${\mathbf{\mathbf{x}}}_{i}$ (perhaps even sampling density ${p}_{i}$ and index case relative transmission potential ${\rho}_{\text{index},i}$) the notation above represents a simplification. More precisely, the ${R}_{0}$ of the $i$th transmission chain equals ${R}_{0,i}=\mathrm{exp}\left({\mathit{\beta}}^{\mathsf{T}}{\mathbf{x}}_{i}\right)$.
Likelihood function
Request a detailed protocolThe likelihood function was expressed in terms of the probability generating function (PGF) of the transmission chain size distribution assuming independent and stuttering (i.e., ${R}_{0}<1$ 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 $\stackrel{~}{T}$ the size of the corresponding observed transmission chain. The above two assumptions can be summarized as
and the PGF $\stackrel{~}{\mathcal{T}}$ of the observed transmission chain size hence equals
in terms of the PGF $\mathcal{T}$ of $T$. The probability that a transmission chain has observed size of $\stackrel{~}{t}\ge 0$ (where $\stackrel{~}{t}=0$ means that none of the cases of the transmission chain is detected) is given by
In particular, the probability that a transmission chain is observed (i.e., the observed size is strictly positive) can be calculated as
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 $\stackrel{~}{t}>0$ is
Finally, for a set of independent observed transmission chain sizes ${\left\{{\stackrel{~}{t}}_{i}\right\}}_{i=1}^{I}$ the likelihood function equals
if the same ${R}_{0}$, ${\rho}_{\text{index}}$ and $p$ are assumed for all transmission chains. For transmission chains with different baseline characteristics and different parameters, the generalized likelihood function is
Model fit
Request a detailed protocolThe maximum likelihood (ML) estimator for $\mathit{\bm{\beta}}$, the predictor for ${R}_{0}$ 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/elifesciencespublications/PoisTransCh). The provided confidence intervals are the Waldtype $95\%$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 HIV1 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
Request a detailed protocolPreviously 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 HIV1 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: GU344102GU344671, 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, JF769777JF769851
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 ${\rho}_{\text{index}}$. Firstly, we varied the ${\rho}_{\text{index}}$ for the transmission chains of nonSwiss origin from $0.05$ to $1.5$. Secondly, we assumed the same ${\rho}_{\text{index}}$ for all transmission chains regardless of their origin and fit the models over a range of ${\rho}_{\text{index}}$ values. Finally, we restricted the analysis only to the transmission chains of nonSwiss origin and varied ${\rho}_{\text{index}}$.
These sensitivity analyses (see Appendix 1—figure 1) implied that the conclusion of no danger for a selfsustained epidemic is stable with respect to ${\rho}_{\text{index}}$ even in the case when some of the Swissoriginated transmission chains are misclassified. In addition, while slightly higher ${R}_{0}$ estimates in the nonSwiss transmission chain subanalysis were mostly driven by the nonB subtypes, the results were safely below $1$ indicating the nonsensitivity of the main conclusion also when some nonSwiss transmission chains would be falsely identified as such.
Sampling density
To study the impact of the sampling densities we performed subtypestratified 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 ${\rho}_{\text{index}}$ remained the same as in the main analyses.
The sensitivity analyses (see Appendix 1—figure 2) showed that neither the ${R}_{0}$ from the baseline model nor the time trend are sensitive to the sampling density, namely the conclusions of ${R}_{0}$ 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 ${R}_{0}$ 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 $d\left(\tau \right)$ define the infectious period duration of an individual who became infected at time $\tau $. The worstcase 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 $\tau $ of a transmission chain $i$ that started at ${\tau}_{0}$ therefore equals
where ${\tau}_{k}$ denotes the latest possible time at which the transmission of the $k$th generation was complete and is calculated iteratively as ${\tau}_{k+1}:={\tau}_{k}+d\left({\tau}_{k}\right)$ for $k\in {\mathbb{N}}_{0}$ (Appendix 1—figure 3). If its index case is still infectious at time $\tau $, it can still produce new infections (which would have a transmission degree $1$) and hence ${N}_{\text{max},i}=0$.
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 ($\widehat{\omega}$) and based on the transmission chains that had been completed (with respect to the last sampling date) by the same date ($\omega $). The relative bias arising from neglecting the ongoing transmission hence equals
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 ${R}_{0}$ 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 LloydSmith (Blumberg and LloydSmith, 2013b; Blumberg and LloydSmith, 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 ${R}_{0}$ (including the estimated ${R}_{0}$) 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’ ${R}_{0}$ 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.
Finally, we compared the $1000$quantiles (permilles) of the transmission clusters extracted from the phylogeny against simulated transmission chains (Appendix 1—figure 5). The QQ 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 ${R}_{0}$ were above $1$ (or even close to $1$). Moreover, the size distribution of the transmission chains simulated for the estimated ${R}_{0}$ showed a good concordance with the observed transmission chains (upper left QQ 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 $\ge 2$. 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 ${R}_{0}$ (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).
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 ${\rho}_{\text{index}}=1$.
While the ${R}_{0}$ estimates for the majority of the nonB 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 ${R}_{0}$ 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 ${\rho}_{\text{index}}=1$) 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 $\xi $ 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.
With the relaxed threshold, we identified $3,039$ transmission chains and repeated the main analyses (Appendix 1—figure 8). As expected the ${R}_{0}$ slightly increased, but stayed below $1$. Overall, we did not observe any noteworthy deviations.
Missing followup data for reported sex with occasional partner
In the main analysis of the possible determinants of HIV transmission we imputed missing followup 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).
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 $b\in \{1,\mathrm{\dots},B\}$ 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 Waldtype $95\%$CIs. The overview of the parameters and the models is provided in Appendix 1—table 1.
For a single parameter $\beta $ (under the assumption that the true value equals the estimated value $\widehat{\beta}$) we therefore obtained a sample of ML estimators ${\widehat{\beta}}^{\left(1\right)},\mathrm{\dots},{\widehat{\beta}}^{\left(B\right)}$, 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 Waldtype $95\%$CIs we calculated the coverage rate as the proportion of these CIs that contained the true value $\widehat{\beta}$.
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
where ${q}^{*}$ denotes the corresponding percentile of the bootstrap sample ${\widehat{\beta}}^{\left(1\right)},\mathrm{\dots},{\widehat{\beta}}^{\left(B\right)}$. Finally, we constructed the profile likelihood based CIs (Held and Bové, 2013) and compared different types of CIs against the Waldtype CIs (Appendix 1—figure 11).
These simulations indicated no significant difference between the widths of Waldtype and profile likelihood based CIs. Besides, the Waldtype CIs did not appear to be systematically wider or narrower compared to the bootstrap CIs.
To summarize, these simulations imply that the normal approximation Waldtype CIs used in our study provide a reliable alternative to other more timecomplex types of CIs.
Appendix 2
Selection of the predictive models
Single determinant models
To construct a multivariate predictive model for ${R}_{0}$ 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.
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 goodnessoffit (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 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 ${R}_{k,n}$ denote the number of secondary infections with transmission degree $n$ produced by the $k$th individual from the preceding generation, ${S}_{n}$ the total number of new infections of transmission degree $n$ and ${Q}_{N}$ the cumulative number of cases in the transmission chain with the transmission degree at most $N$, that is,
The index case establishes the transmission chain and corresponds to the generation $0$, therefore ${S}_{0}={Q}_{0}=1$.
Assuming that the numbers of secondary infections are independent and identically distributed for all patients of the same transmission degree, let ${\mathcal{R}}_{n}$ denote the probability generating function (PGF) of ${R}_{k,n}$, namely
for each $k\in \{1,2,\mathrm{\dots},{S}_{n1}\}$. The expected number of secondary infections of degree $n$ is therefore given by
Furthermore, assume that the numbers of secondary infections caused by different individuals are independent between each other regardless of the transmission degree. The PGF ${\mathcal{Q}}_{N}$ of ${Q}_{N}$ is
because (a) of the tower property of the conditional expectation, (b) ${Q}_{N1}={\sum}_{k=0}^{N1}{S}_{n}$ is ${\left\{{S}_{n}\right\}}_{n=0}^{N1}$measurable, (c) ${\left\{{R}_{k,N}\right\}}_{k=1}^{{S}_{N1}}$ are independent, and (d) ${R}_{k,N}$ are independent from ${\left\{{S}_{n}\right\}}_{n=0}^{N1}$ for all $k=1,2,\mathrm{\dots},{S}_{N1}$. Repeating similar steps iteratively yields
The total size of the transmission chain is denoted by $T$ and equals
From the definition of $T$ it follows that its PGF $\mathcal{T}$ equals
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 $\mathcal{G}$ for all infected persons, namely
for every $n$ (i.e., the transmission is uniform across different transmission degrees). The PGF ${\mathcal{Q}}_{N}$ (Equation 1) then simplifies to
Using Equation 2, the PGF $\mathcal{T}$ for each $z$ solves the equation
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 followup 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 ${\rho}_{\text{index}}$ denote the index case relative transmission potential (ICRTP). In terms of the model the above assumptions can be summarized as
where $\mathcal{F}$ and $\mathcal{G}$ denote the PGF of two distributions, such that
namely $\mathbb{E}\left[{R}_{1,1}\right]={\rho}_{\text{index}}\mathbb{E}\left[{R}_{k,n}\right]$ for all $k\in \{1,\mathrm{\dots},{S}_{n1}\}$ 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 $\mathcal{K}$, which controls the regular part/tail of the transmission chain. Let $\mathcal{K}$ be the pointwise limit $\mathcal{K}\left(z\right):={lim}_{N\to \mathrm{\infty}}{\mathcal{K}}_{N}\left(z\right)$ of the iteratively defined functions
The skeleton therefore solves the equation
Note that in the absence of the modified transmissibility of the index case, the skeleton function $\mathcal{K}$ coincides with the PGF of the transmission chain size. Having introduced this notation one can rewrite the PGF ${\mathcal{Q}}_{N}$ (Equation 1) as
As $N\to \mathrm{\infty}$, this implies
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 ( $\stackrel{~}{}$ ) denote the observed cases. Since each case is detected at random with probability $p$ the following applies to the observed transmission chains.
If ${R}_{k,n}$ is defined as above then ${\stackrel{~}{R}}_{k,n}$ denotes the number of secondary infections with transmission degree $n$ caused by patient $k$ which are actually observed. It follows
${\stackrel{~}{R}}_{k,n}{R}_{k,n}\sim Bin({R}_{k,n},p).$Given the numbers of secondary infections with transmission degree $n$ of all the patients the observed number of infections of transmission degree $n$ equals
${\stackrel{~}{S}}_{n}=\sum _{k=1}^{{S}_{n1}}{\stackrel{~}{R}}_{k,n}$and follows a binomial distribution, namely
${\stackrel{~}{S}}_{n}{\left\{{R}_{k,n}\right\}}_{k=1}^{{S}_{n1}}\sim \sum _{k=1}^{{S}_{n1}}Bin({R}_{k,n},p)=Bin(\sum _{k=1}^{{S}_{n1}}{R}_{k,n},p)=Bin({S}_{n},p).$The observed cumulative number of infected individuals with the transmission degree at most $N$ equals
${\stackrel{~}{Q}}_{N}=\sum _{n=0}^{N}{\stackrel{~}{S}}_{n}={\stackrel{~}{Q}}_{N1}+{\stackrel{~}{S}}_{N}={\stackrel{~}{Q}}_{N1}+\sum _{k=1}^{{S}_{N1}}{\stackrel{~}{R}}_{k,N}.$By conditioning on the cumulative number of infections of transmission degree up to $N1$ and on the numbers of secondary infections of transmission degree $N$, ${\stackrel{~}{Q}}_{N}$ therefore follows a binomial distribution, that is,
${\stackrel{~}{Q}}_{N}{Q}_{N1},{\left\{{R}_{k,N}\right\}}_{k=1}^{{S}_{N1}}\sim Bin({Q}_{N1}+\sum _{k=1}^{{S}_{N1}}{R}_{k,N},p)=Bin({Q}_{N1}+{S}_{N},p)=Bin({Q}_{N},p).$Since $\mathcal{B}\left(z\right)={\left(\left(1p\right)+pz\right)}^{n}$ is the PGF of a $Bin(n,p)$distributed random variable, the PGF of ${\stackrel{~}{Q}}_{N}$ can be expressed as
$\begin{array}{rlr}{\displaystyle {\stackrel{~}{\mathcal{\mathcal{Q}}}}_{N}\left(z\right)}& {\displaystyle =\mathbb{E}\left[{z}^{{\stackrel{~}{Q}}_{N}}\right]{\displaystyle \stackrel{(\mathrm{a})}{=}\mathbb{E}\left[\mathbb{E}\left[{z}^{{\stackrel{~}{Q}}_{N}}{Q}_{N}\right]\right]}}& \\ & {\displaystyle \stackrel{(\mathrm{b})}{=}\mathbb{E}\left[\mathbb{E}\left[\mathbb{E}\left[{z}^{{\stackrel{~}{Q}}_{N}}{Q}_{N1},{\left\{{R}_{k,N}\right\}}_{k=1}^{{S}_{N1}}\right]{Q}_{N}\right]\right]}& \\ & {\displaystyle \stackrel{(\mathrm{c})}{=}\mathbb{E}\left[\mathbb{E}\left[{\left(\left(1p\right)+pz\right)}^{{Q}_{N1}+\sum _{k=1}^{{S}_{N1}}{R}_{k,N}}{Q}_{N}\right]\right]=\mathbb{E}\left[\mathbb{E}\left[{\left(\left(1p\right)+pz\right)}^{{Q}_{N}}{Q}_{N}\right]\right]}& \\ & {\displaystyle \stackrel{(\mathrm{a})}{=}\mathbb{E}\left[{\left(\left(1p\right)+pz\right)}^{{Q}_{N}}\right]}& \\ & {\displaystyle ={\mathcal{\mathcal{Q}}}_{N}\left(\left(1p\right)+pz\right)}& \end{array}$in terms of the PGF ${\mathcal{Q}}_{N}$, because (a) of the tower property, (b) of the tower property for $\sigma $algebras $\sigma \left({Q}_{N}\right)\subseteq \sigma ({Q}_{N1},{\left\{{R}_{k,N}\right\}}_{k=1}^{{S}_{N1}})$ due to the relation ${Q}_{N}={Q}_{N1}+{\sum}_{k=1}^{{S}_{N1}}{R}_{k,N}$, and (c) ${Q}_{N}$ given ${Q}_{N1}$ and ${\left\{{R}_{k,N}\right\}}_{k=1}^{{S}_{N1}}$ is binomially distributed.
This allows us to obtain the PGF $\stackrel{~}{\mathcal{T}}$ of the observed transmission chain length $\stackrel{~}{T}=\underset{N\to \mathrm{\infty}}{lim}{\stackrel{~}{Q}}_{N}$, namely
where $\mathcal{T}$ 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
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
where ${}^{\left(j\right)}$ denotes the $j$th 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
So far, we have not included the basic reproductive number or any other transmissionrelated parameters in the PGF of transmission chain size $\stackrel{~}{\mathcal{T}}$. 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 $\mathit{\bm{\omega}}$ denote a vector of transmission parameters, for instance $\mathit{\bm{\omega}}={R}_{0}$ 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 $\stackrel{~}{\mathbf{\mathbf{t}}}:={\left\{{\stackrel{~}{t}}_{i}\right\}}_{i=1}^{I}$ is defined by
where $\stackrel{~}{\mathcal{T}}(z;\mathit{\bm{\omega}})$ denotes the PGF of transmission chain size with transmission parameters $\mathit{\bm{\omega}}$. The corresponding loglikelihood function is
Since the transmission parameters are often required to be positive the logparameterization is more appropriate. Let $\mathit{\bm{\theta}}$ denote the transmission parameters on the logarithmic scale, namely $\mathit{\bm{\theta}}:=\mathrm{log}\left(\mathit{\bm{\omega}}\right)$. The logparameterized loglikelihood function is therefore $\mathrm{\ell}\left(\mathit{\bm{\theta}}\right\stackrel{~}{\mathbf{\mathbf{t}}}):={\mathrm{\ell}}_{\mathit{\bm{\omega}}}\left(\mathrm{log}\left(\mathit{\bm{\omega}}\right)\right\stackrel{~}{\mathbf{\mathbf{t}}})$. The Jacobian matrix corresponding to the logparameterization equals
where $diag\left(x\right)$ denotes a diagonal matrix with vector $x$ representing its diagonal elements.
The score function and the Fisher information matrix
The maximum likelihood (ML) estimator $\widehat{\mathit{\bm{\omega}}}$ maximizes the loglikelihood function and is a root of the score function
or equivalently, the ML estimator $\widehat{\mathit{\bm{\theta}}}$ solves
where $\mathbf{\mathbf{u}}$ denotes the score function corresponding to the logparameterized loglikelihood $\mathrm{\ell}$.
The Fisher information matrix ${\mathcal{I}}_{\mathit{\bm{\omega}}}\left(\mathit{\bm{\omega}}\right\stackrel{~}{\mathbf{\mathbf{t}}}):=\frac{{\partial}^{2}}{\partial {\mathit{\bm{\omega}}}^{2}}{\mathrm{\ell}}_{\mathit{\bm{\omega}}}\left(\mathit{\bm{\omega}}\right\stackrel{~}{\mathbf{\mathbf{t}}})$ is given by
and equals
under the logparameterization due to the chain rule in higher dimensions and the special form of the transformation corresponding to the logparameterization. The PGF function $\stackrel{~}{\mathcal{T}}$ and its derivatives are thus crucial (and sufficient) for the statistical inference, since the loglikelihood function $\mathrm{\ell}$, the score function $\mathbf{\mathbf{u}}$ and the Fisher information matrix $\mathcal{I}$ can be expressed in terms of $\stackrel{~}{\mathcal{T}}$ 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 $\theta \in \mathit{\bm{\theta}}$ we can construct the Wald $\alpha \%$confidence interval as
where ${z}_{\frac{1+\alpha}{2}}$ denotes the $\frac{1+\alpha}{2}$quantile of the standard normal distribution, and the standard error $se\left(\widehat{\theta}\right)$ is defined as
and ${}_{\theta \theta}$ denotes the diagonal element of the inversed observed Fisher information matrix $\mathcal{I}\left(\mathit{\bm{\theta}}\right\stackrel{~}{\mathbf{\mathbf{t}}})$ corresponding to parameter $\theta $. The approximate $\alpha \%$confidence interval for the original parameter $\omega $ is obtained by the reverse transformation
Similarly, to test the hypothesis ${H}_{0}:\theta ={\theta}_{0}$ against the alternative ${H}_{A}$, the Wald test statistic
can be used. Assuming the standard normal distribution of the test statistic under null hypothesis, the $p$value equals
$2\cdot \left(1\mathrm{\Phi}\left(\left{\tau}_{\theta}\left({\theta}_{0}\right)\right\right)\right)$ for the alternative hypothesis ${H}_{A}:\theta \ne {\theta}_{0}$,
$\mathrm{\Phi}\left({\tau}_{\theta}\left({\theta}_{0}\right)\right)$ for the alternative ${H}_{A}:\theta <{\theta}_{0}$, and
$1\mathrm{\Phi}\left({\tau}_{\theta}\left({\theta}_{0}\right)\right)$ for the alternative ${H}_{A}:\theta >{\theta}_{0}$;
where $\mathrm{\Phi}$ 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
are the transmission parameters of the $i$th chain with characteristics ${\mathbf{\mathbf{x}}}_{i}$, where $\mathit{\bm{\eta}}$ denotes the remaining parameters from $\mathit{\bm{\theta}}$ which are not modeled as a linear combination. Furthermore, it is plausible to assume that while the transmission chains share all the transmission parameters $(\mathit{\bm{\beta}},\mathit{\bm{\eta}})$, 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 ${\rho}_{\text{index}}=1$). Let ${\stackrel{~}{\mathcal{T}}}_{i}$ be the PGF corresponding to the transmission chain $i$ and let $\mathbf{\mathbf{X}}:={\left\{{\mathbf{\mathbf{x}}}_{i}\right\}}_{i=1}^{I}$. The generalized loglikelihood function is hence given by
where
The corresponding Jacobian matrix equals
In the generalized model, the score function is
$\mathbf{\mathbf{u}}(\mathit{\bm{\beta}},\mathit{\bm{\eta}}\stackrel{~}{\mathbf{\mathbf{t}}},\mathbf{\mathbf{X}}):={\displaystyle \frac{\partial}{\partial (\mathit{\bm{\beta}},\mathit{\bm{\eta}})}}\mathrm{\ell}(\mathit{\bm{\beta}},\mathit{\bm{\eta}}\stackrel{~}{\mathbf{\mathbf{t}}},\mathbf{\mathbf{X}})={\displaystyle \sum _{i=1}^{I}}{\mathbf{\mathbf{J}}}_{{\mathit{\bm{\omega}}}_{i}}{(\mathit{\bm{\beta}},\mathit{\bm{\eta}})}^{\U0001d5b3}({\displaystyle \frac{\frac{\partial}{\partial \mathit{\bm{\omega}}}{\stackrel{~}{\mathcal{T}}}_{i}^{\left({\stackrel{~}{t}}_{i}\right)}(0;{\mathit{\bm{\omega}}}_{i}(\mathit{\bm{\beta}},\mathit{\bm{\eta}}))}{{\stackrel{~}{\mathcal{T}}}_{i}^{\left({\stackrel{~}{t}}_{i}\right)}(0;{\mathit{\bm{\omega}}}_{i}(\mathit{\bm{\beta}},\mathit{\bm{\eta}}))}}+{\displaystyle \frac{\frac{\partial}{\partial \mathit{\bm{\omega}}}{\stackrel{~}{\mathcal{T}}}_{i}(0;{\mathit{\bm{\omega}}}_{i}(\mathit{\bm{\beta}},\mathit{\bm{\eta}}))}{1{\stackrel{~}{\mathcal{T}}}_{i}(0;{\mathit{\bm{\omega}}}_{i}(\mathit{\bm{\beta}},\mathit{\bm{\eta}}))}})$ and the Fisher information matrix as
Prediction intervals
It is tempting to construct an approximate confidence interval for the parameter ${\theta}_{i}:={\mathit{\bm{\beta}}}^{\U0001d5b3}{\mathbf{\mathbf{x}}}_{i}$. Since the parameter ${\theta}_{i}$ is a prediction rather than an estimate, the element of interest is the prediction interval, which takes into account both the characteristics ${\mathbf{\mathbf{x}}}_{i}$ and the uncertainty of all parameter estimates $\widehat{\mathit{\bm{\beta}}}$.
Assuming that the ML estimator $(\widehat{\mathit{\bm{\beta}}},\widehat{\mathit{\bm{\eta}}})$ is asymptotically normally distributed, it follows that the linear combination ${\widehat{\mathit{\bm{\beta}}}}^{\U0001d5b3}{\mathbf{\mathbf{x}}}_{i}$ is also asymptotically Gaussian, specifically
The variance $Var\left(\widehat{\mathit{\bm{\beta}}}\right)$ can be approximated by the inverse of the observed Fisher information matrix as
Finally, an approximate $\alpha \%$prediction interval for ${\theta}_{i}$ is constructed as
with
Example: Poisson model
Suppose that the number of secondary infections follows the Poisson distribution with parameter ${R}_{0}$. Taking into account the modified transmissibility of the index case ${\rho}_{\text{index}}$ (wherever applicable), the PGFs $\mathcal{F}$ and $\mathcal{G}$ for the index case and the tail, respectively, are
The skeleton function $\mathcal{K}$ thus solves
Consider an imperfectly sampled transmission chain with probability of detection $p$ and with ICRTP ${\rho}_{\text{index}}$. The aim is to obtain the Taylor coefficients of $\stackrel{~}{\mathcal{T}}$ around $z=0$ to be able to estimate the transmission parameter ${R}_{0}$ with the maximum likelihood approach since they are needed to calculate the loglikelihood (Equation 6).
Let
and
such that $\mathcal{Y}(\left(1p\right)+pz;{R}_{0})=\stackrel{~}{\mathcal{T}}(z;{R}_{0})$ and that the Equation 5 of the PGF of observed transmission chain size $\stackrel{~}{\mathcal{T}}$ simplifies to
Taking into account the PGF $\mathcal{F}$ of the index case implies
Solving for $\mathcal{K}(w;{R}_{0})$ yields
Plugging this into Equation 7 gives
With $\mathcal{Z}(w;{R}_{0}):={\left(\frac{\mathcal{Y}(w;{R}_{0})}{w}\right)}^{\frac{1}{{\rho}_{\text{index}}}}$, the last equation is equivalent to
which is an equation of the form $f\left(w\right){e}^{f\left(w\right)}=g\left(w\right)$. The latter admits a solution $f\left(w\right)={W}_{0}\left(g\left(w\right)\right)$, where ${W}_{0}$ is the principal branch of the Lambert $W$ function (Corless et al., 1996). Thus
and finally,
Using the relation $\frac{{W}_{0}\left(x\right)}{x}={e}^{{W}_{0}\left(x\right)}$ (which follows from the definition of ${W}_{0}\left(x\right)$), we have
From the Taylor expansion of ${e}^{\gamma {W}_{0}\left(x\right)}={\sum}_{m=0}^{\mathrm{\infty}}\gamma {\left(\gamma +m\right)}^{m1}\frac{{x}^{m}}{m!}$ around $x=0$ (equality (2.36) in Corless et al., 1996), we obtain
In terms of $\stackrel{~}{\mathcal{T}}$ this yields
Unfortunately, we need Taylor expansion around $z=0$ to derive the state probabilities (and consequently the loglikelihood function). By applying the binomial theorem, $\stackrel{~}{\mathcal{T}}$ can be rewritten as
with $m=k\vee 1$ denoting $m=\mathrm{max}\{k,1\}$.
Initial estimate for ${R}_{0}$
Since the optimization problem of maximizing the likelihood does not admit a closedform solution, the ML estimator is obtained with numerical techniques for which a suited initial estimate for ${R}_{0}$ 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
be the observed average chain size (based on a sample of $I$ observed chains $\stackrel{~}{\mathbf{\mathbf{t}}}$ like proposed in Blumberg and LloydSmith, 2013b). $\overline{\mu}$ represents a reasonable estimate for
The definition of the skeleton function $\mathcal{K}$ for transmission parameters $\mathit{\bm{\omega}}$ implies $\mathcal{K}(1;\mathit{\bm{\omega}})=1$. Implicitly deriving Equation 3 with respect to $z$ implies
since $\mathcal{G}(1;\mathit{\bm{\omega}})=1$ (just like for any PGF). Moreover, implicitly deriving Equation 5 with respect to $z$ yields
Under the Poisson model, the latter equals to
Next, we can use the first Taylor coefficient of $\stackrel{~}{\mathcal{T}}(z;{R}_{0})$ from Equation 8, namely $\stackrel{~}{\mathcal{T}}(0;{R}_{0})\approx {e}^{{\rho}_{\text{index}}{R}_{0}}\left(1p\right)$. In order to obtain a quadratic equation with respect to ${R}_{0}$, we further use the approximation ${e}^{{\rho}_{\text{index}}{R}_{0}}\approx 1{\rho}_{\text{index}}{R}_{0}$, such that $1\stackrel{~}{\mathcal{T}}(0;{R}_{0})\approx 1\left(1{\rho}_{\text{index}}{R}_{0}\right)\left(1p\right)$. This yields the quadratic equation
with the roots
where
Should none of the roots lie within $(0,1)$, we could use the following feature. If the average size of the observed chain equals $\overline{\mu}$, the average size of the complete transmission chains would be roughly $\frac{\overline{\mu}}{p}$ (since the mean value of the binomial distribution $Bin(n,p)$ is $np$). Hence,
Equation 4 then implies
In case of the Poisson model, the initial estimate for ${R}_{0}$ can be therefore obtained by solving the equation
which has the solution
Generalized Poisson model
Let $\stackrel{~}{\mathbf{\mathbf{T}}}:={\left\{{\stackrel{~}{\mathbf{\mathbf{t}}}}_{i}\right\}}_{i=1}^{I}$ be a sample of $I$ observed transmission chains where each observed transmission chain ${\stackrel{~}{\mathbf{\mathbf{t}}}}_{i}$ carries the following information
namely the observed chain size ${\stackrel{~}{t}}_{i}$, the chain characteristics ${\mathbf{\mathbf{x}}}_{i}$, the probability ${p}_{i}$ at which each infection in the chain is observed, and the index case relative transmission potential ${\rho}_{\text{index},i}$. In the generalized Poisson transmission chain size distribution model we assume that the heterogeneity of the basic reproductive number ${R}_{0}$ can be explained by the variability of the demographic characteristics of the transmission chains, namely
The vector $\mathit{\bm{\beta}}$ describes the effect of the chain characteristics on the basic reproductive number ${R}_{0}$ and it is the same for all transmission chains.
To obtain the maximum likelihood estimates for $\mathit{\bm{\beta}}$, 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 ${R}_{0}$ for each transmission chain. More precisely, imagine that each transmission chain ${\stackrel{~}{\mathbf{\mathbf{t}}}}_{i}$ is a sample of transmission chains itself and therefore we can obtain the initial ${r}_{0,i}$ estimates as described above. In the next step, we fit the linear regression model
and use ${\widehat{\mathit{\bm{\beta}}}}_{0}$ 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 ${R}_{0}$ and dispersion parameter $\xi $. Its PGF equals
For the simplicity assume that index case has the same transmission potential as the remaining part of the transmission chain, namely $\mathcal{F}\equiv \mathcal{G}$ (which coincides with ${\rho}_{\text{index}}=1$). The skeleton function $\mathcal{K}(z;{R}_{0},\xi )$ is therefore a solution of the equation
which does not admit a closedform solution. However, as a consequence of the Lagrange inversion theorem its Taylor coefficients around $z=0$ can be explicitly calculated (Blumberg and LloydSmith, 2013b) as
Since we assumed $\mathcal{F}\equiv \mathcal{G}$, it follows $\mathcal{T}(z;{R}_{0},\xi )=\mathcal{K}(z;{R}_{0},\xi )$ 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
By applying the binomial theorem to the Taylor expansion around $z=0$ of $\mathcal{K}(z;{R}_{0},\xi )$ the higherorder derivatives
are obtained (which coincides with the result from Blumberg and LloydSmith, 2013a).
In similar manner as in the case of Poisson model, the generalized negative binomial model can be derived by introducing
The sampling density $p$ can vary between the transmission chains (or their characteristics, for instance between the subtypes), while the dispersion parameter $\xi $ is kept constant among all the transmission chains.
References

Inference of R(0) and transmission heterogeneity from the size distribution of stuttering chainsPLoS Computational Biology 9:e1002993.https://doi.org/10.1371/journal.pcbi.1002993

Effectiveness of highly active antiretroviral therapy in reducing heterosexual transmission of HIVJAIDS Journal of Acquired Immune Deficiency Syndromes 40:96–101.https://doi.org/10.1097/01.qai.0000157389.78374.45

Antiretroviral Therapy for the Prevention of HIV1 TransmissionNew England Journal of Medicine 375:830–839.https://doi.org/10.1056/NEJMoa1600693

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Acute HIV1 InfectionNew England Journal of Medicine 364:1943–1954.https://doi.org/10.1056/NEJMra1011874

Advances in Computational Mathematics329–359, On the Lambert W Function, Advances in Computational Mathematics, Vol. 5.

MUSCLE: multiple sequence alignment with high accuracy and high throughputNucleic Acids Research 32:1792–1797.https://doi.org/10.1093/nar/gkh340

HIV1 transmission, by stage of infectionThe Journal of Infectious Diseases 198:687–693.https://doi.org/10.1086/590501

Effect of early versus deferred antiretroviral therapy for HIV on survivalNew England Journal of Medicine 360:1815–1826.https://doi.org/10.1056/NEJMoa0807252

Increases in Condomless Sex in the Swiss HIV Cohort StudyOpen Forum Infectious Diseases 2:ofv077.https://doi.org/10.1093/ofid/ofv077

Molecular epidemiology reveals longterm changes in HIV type 1 subtype B transmission in SwitzerlandThe Journal of Infectious Diseases 201:1488–1497.https://doi.org/10.1086/651951

Initiation of Antiretroviral Therapy in Early Asymptomatic HIV InfectionThe New England Journal of Medicine 373:795–807.https://doi.org/10.1056/NEJMoa1506816

Phylodynamics of the major HIV1 CRF02_AG African lineages and its global disseminationInfection, Genetics and Evolution 46:190–199.https://doi.org/10.1016/j.meegid.2016.05.017

FastTree: computing large minimum evolution trees with profiles instead of a distance matrixMolecular Biology and Evolution 26:1641.https://doi.org/10.1093/molbev/msp077

Transmission of NonB HIV Subtypes in the United Kingdom Is Increasingly Driven by Large NonHeterosexual Transmission ClustersJournal of Infectious Diseases 213:1410–1418.https://doi.org/10.1093/infdis/jiv758

Molecular epidemiology of HIV1 in Iceland: Early introductions, transmission dynamics and recent outbreaks among injection drug usersInfection, Genetics and Evolution 49:157–163.https://doi.org/10.1016/j.meegid.2017.01.004

Cohort profile: the Swiss HIV Cohort studyInternational Journal of Epidemiology 39:1179–1189.https://doi.org/10.1093/ije/dyp321

Estimating the basic reproductive number from viral sequence dataMolecular Biology and Evolution 29:347.https://doi.org/10.1093/molbev/msr217

A joint back calculation model for the imputation of the date of HIV infection in a prevalent cohortStatistics in Medicine 27:4835–4853.https://doi.org/10.1002/sim.3294

The role of migration and domestic transmission in the spread of HIV1 nonB subtypes in SwitzerlandThe Journal of Infectious Diseases 204:1095–1103.https://doi.org/10.1093/infdis/jir491
Decision letter

Ryosuke OmoriReviewing 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 selfsustained 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 noninfectious after 22.5 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), ρ_{index} 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 heterosexual 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.28721.028Author 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 noninfectious after 22.5 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 ρ_{index} (which was already assessed in a sensitivity analysis by setting ρ_{index}=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 R_{0}). 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 reevaluated 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 ρ_{index} (Appendix 1—figure 1), where the lower ρ_{index} values implicitly correspond to later immigration to Switzerland and/or the importance of acute phase on transmission, and vice versa – higher ρ_{index} indicates either immigration early during infection or high infectivity long into the chronic phase. We added a more explicit relation between the ρ_{index} 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 ρ_{index} 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), ρ_{index} 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 ρ_{index} exceeds 1, this would imply even lower R_{0}. We added this comment to the Discussion: "Furthermore, even though theoretically the transmission potential of some index cases could also be enhanced (i.e., ρ_{index}>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 ρ_{index}>1 would lead to even lower R_{0}, our main conclusions would not change (in fact, the assumption of ρ_{index}<1 is conservative with respect to our conclusion of R_{0}<1)." Furthermore, we replaced the "reduced" with "modified" transmission potential/transmissibility throughout the manuscript and extended the range of the sensitivity analyses regarding ρ_{index} 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 $1\mathbb{P}\left[\stackrel{~}{T}=0\right]$). 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 LloydSmith, 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 $\stackrel{~}{t}\ge 0$ (where $\stackrel{~}{t}=0$ means that none of the cases of the transmission chain is detected) is given by
In particular, the probability that a transmission chain is observed (i.e., the observed size is strictly positive) can be calculated as
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 $\stackrel{~}{t}>0$ is
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 heterosexual 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 tradeoff 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 followup 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 Waldtype 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 Waldtype confidence intervals are systematically wider/narrower than the bootstrap based confidence intervals (Appendix 1—figure 11).
We therefore concluded that the normal approximation Waldtype 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 R_{0} 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 R_{0} 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 R_{0} 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.28721.029Article and author information
Author details
Funding
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (33CS30148522)
 Huldrych F Günthard
YvonneJacob 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 (PZ00P3142411)
 Roger D Kouyos
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung (BSSGI0155851)
 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 highquality 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 TicinoComitato Ethico Cantonale, Commission Cantonale d'Étique de la Recherche sur l'Être Humain, Ethikkommission beiderBasel; all approvals are available on http://www.shcs.ch/206ethiccommitteeapprovalandinformedconsent), and written informed consent was obtained from all participants.
Reviewing Editor
 Ryosuke Omori, Hokkaido University, Japan
Version history
 Received: May 17, 2017
 Accepted: August 28, 2017
 Accepted Manuscript published: September 12, 2017 (version 1)
 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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Further reading

 Epidemiology and Global Health
Background:
The aim of our study was to test the hypothesis that the community contact tracing strategy of testing contacts in households immediately instead of at the end of quarantine had an impact on the transmission of SARSCoV2 in schools in Reggio Emilia Province.
Methods:
We analysed surveillance data on notification of COVID19 cases in schools between 1 September 2020 and 4 April 2021. We have applied a mediation analysis that allows for interaction between the intervention (before/after period) and the mediator.
Results:
Median tracing delay decreased from 7 to 3.1 days and the percentage of the known infection source increased from 34–54.8% (incident rate ratioIRR 1.61 1.40–1.86). Implementation of prompt contact tracing was associated with a 10% decrease in the number of secondary cases (excess relative risk –0.1 95% CI –0.35–0.15). Knowing the source of infection of the index case led to a decrease in secondary transmission (IRR 0.75 95% CI 0.63–0.91) while the decrease in tracing delay was associated with decreased risk of secondary cases (1/IRR 0.97 95% CI 0.94–1.01 per one day of delay). The direct effect of the intervention accounted for the 29% decrease in the number of secondary cases (excess relative risk –0.29 95%–0.61 to 0.03).
Conclusions:
Prompt contact testing in the community reduces the time of contact tracing and increases the ability to identify the source of infection in school outbreaks. Although there are strong reasons for thinking it is a causal link, observed differences can be also due to differences in the force of infection and to other control measures put in place.
Funding:
This project was carried out with the technical and financial support of the Italian Ministry of Health – CCM 2020 and Ricerca Corrente Annual Program 2023.

 Epidemiology and Global Health
In biomedical science, it is a reality that many published results do not withstand deeper investigation, and there is growing concern over a replicability crisis in science. Recently, Ellipse of Insignificance (EOI) analysis was introduced as a tool to allow researchers to gauge the robustness of reported results in dichotomous outcome design trials, giving precise deterministic values for the degree of miscoding between events and nonevents tolerable simultaneously in both control and experimental arms (Grimes, 2022). While this is useful for situations where potential miscoding might transpire, it does not account for situations where apparently significant findings might result from accidental or deliberate data redaction in either the control or experimental arms of an experiment, or from missing data or systematic redaction. To address these scenarios, we introduce Region of Attainable Redaction (ROAR), a tool that extends EOI analysis to account for situations of potential data redaction. This produces a bounded cubic curve rather than an ellipse, and we outline how this can be used to identify potential redaction through an approach analogous to EOI. Applications are illustrated, and source code, including a webbased implementation that performs EOI and ROAR analysis in tandem for dichotomous outcome trials is provided.