David J. Gerberry

Quantification of the role of discordant couples in driving incidence of HIV in sub-Saharan Africa

Data presented in the meta-analysis by Oghenowede Eyawo and colleagues sheds new light on the role of serodiscordant couples in driving the incidence of HIV in Africa. The HIV epidemic in Africa is driven by heterosexual transmission. Studies have shown transmission in stable discordant couples can be as low as 1·9 per 100 person-years to as high as 19·0 per 100 person-years; these couples are typically in stable relationships lasting at least a year. However, the extent to which they drive countrylevel incidence in sub-Saharan Africa percentage of the population in stable relationships (regardless of whether they are in discordant or concordant couples), the more discordant couples drive incidence. Notably, their role in driving incidence is country-specifi c. For example, if 40% of the population in Ghana is in stable relationships, stable discordant couples could account for 34% of the country-level incidence. However, if the same percentage in Rwanda is in stable relationships, they could only account for 18% of the country-level incidence. Results from a previous study suggest transmission in stable discordant couples could account for about 50% of the countrylevel incidence. Our results show that such transmission is possible, but is more likely in some countries than in others; for example, it seems possible in Kenya but unlikely in Guinea (fi gure). We agree that public health programmes aimed at preventing HIV infection in stable discordant couples could be very useful in reducing transmission. However, determination of the degree to which these couples drive incidence of HIV in specifi c countries is crucial, because this will determine the necessity and intensity of other interventions. Figure: The proportion of country-level incidence of HIV due to transmission by stable discordant couples versus the proportion of the population in stable relationships lasting a year or more. Data includes concordant and discordant couples. SDC=stable discordant couple. Burkina Faso Ethiopia Guinea Kenya Malawi Rwanda Tanzania

Using geospatial modelling to optimize the rollout of antiretroviral-based pre-exposure HIV interventions in Sub-Saharan Africa

Antiretroviral-based pre-exposure HIV interventions may soon be rolled out in resource-constrained Sub-Saharan African countries, but rollout plans have yet to be designed. Here we use geospatial modeling and optimization techniques to compare two rollout plans for ARV-based microbicides in South Africa: a utilitarian plan that minimizes incidence by using geographic targeting, and an egalitarian plan that maximizes geographic equity in access to interventions. We find significant geographic variation in the efficiency of interventions in reducing HIV transmission, and that efficiency increases disproportionately with increasing incidence. The utilitarian plan would result in considerable geographic inequity in access to interventions, but (by exploiting geographic variation in incidence) could prevent ~40% more infections than the egalitarian plan. Our results show that the geographic resource allocation decisions made at the beginning of a rollout, and the location where the rollout is initiated, will be crucial in determining the success of interventions in reducing HIV epidemics.

An SEIQR model for childhood diseases

It has been shown that the inclusion of an isolated class in the classical SIR model for childhood diseases can be responsible for self-sustained oscillations. Hence, the recurrent outbreaks of such diseases can be caused by autonomous, deterministic factors. We extend the model to include a latent class (i.e. individuals who are infected with the disease, but are not yet able to pass the disease to others) and study the resulting dynamics. The existence of Hopf bifurcations is shown for the model, as well as a homoclinic bifurcation for a perturbation to the model. For historical data on scarlet fever in England, our model agrees with the epidemiological data much more closely than the model without the latent class. For other childhood diseases, our model suggests that isolation is unlikely to be a major factor in sustained oscillations.

Practical aspects of backward bifurcation in a mathematical model for tuberculosis.

In this work, we examine practical aspects of backward bifurcation for a data-based model of tuberculosis that incorporates multiple features which have previously been shown to produce backward bifurcation (e.g. exogenous reinfection and imperfect vaccination) and new considerations such as the treatment of latent TB infection (LTBI) and the BCG vaccines interference with detecting LTBI. Understanding the interplay between these multiple factors and backward bifurcation is particularly timely given that new diagnostic tests for LTBI detection could dramatically increase rates of both LTBI detection and vaccination in the coming decades. By establishing analytic thresholds for the existence of backward bifurcation, we identify those aspects of TBs complicated pathology that make backward bifurcation more or less likely to occur. We also examine the magnitude of the backward bifurcation produced by the model and its sensitivity to various model parameters. We find that backward bifurcation is unlikely to occur. While increased vaccine coverage and/or increased detection and treatment of LTBI can push the threshold for backward bifurcation into the region of biological plausibility, the resulting bifurcations may still be too small to have any noticeable epidemiological impact.

Trade-off between BCG vaccination and the ability to detect and treat latent tuberculosis

Policies regarding the use of the Bacille Calmette-Guérin (BCG) vaccine for tuberculosis vary greatly throughout the international community. In several countries, consideration of discontinuing universal vaccination programs is currently under way. The arguments against mass vaccination are that the effectiveness of BCG in preventing tuberculosis is uncertain and that BCG vaccination can interfere with the detection and treatment of latent tuberculosis. In this work, we pose a dynamical systems model for the population-level dynamics of tuberculosis in order to study the trade-off which occurs between vaccination and detection/treatment of latent tuberculosis. We assume that latent infection in vaccinated individuals is completely undetectable. For the case of a country with very low levels of tuberculosis, we establish analytic thresholds, via stability analysis and the basic reproductive number, which determine the optimal vaccination policy, given the effectiveness of the vaccine and the detection/treatment rate of latent tuberculosis. The results of this work suggest that it is unlikely that a country detects and treats latent tuberculosis at a high enough rate to justify the discontinuation of mass vaccination from this perspective.

Could we - should we - calculate a risk exposure score for an HIV-negative individual in a serodiscordant couple?

The Bernoulli risk equation has long been used within HIV transmission models to calculate population-level incidence rates [1–3]. In an article in this issue, Fox et al. [4] use the Bernoulli risk equation to quantify the degree to which an uninfected individual in a discordant couple has been exposed to HIV, based on their past sexual practices. Consequently, Fox et al. describe the Bernoulli risk equation as a risk exposure score; notably, it is a measure of past and not future risk. They calculate risk exposure scores for heterosexual women, heterosexual men, and MSM based on four risk factors. Two risk factors are based on characteristics of the HIV-positive partner: viral load and HIV stage (primary, chronic, or late). The other two risk factors, genital ulcer disease (GUD) and genital herpes (herpes simplex virus-2; HSV2), either increase susceptibility of the HIV-negative partner and/or increase infectivity of the HIV-positive partner. Notably, the presence of bacterial sexually transmitted infections is not included as a risk factor in the proposed risk exposure score. However, pregnancy is included as an additional risk factor for women and it is assumed to increase risk by 116% [confidence interval (CI) 39–237%] based on results from one study [5]. Circumcision is included, for heterosexual men, as a factor that reduces the risk of acquiring HIV. The HIV risk exposure score is determined for any individual by calculating their cumulative exposure risk from their specific risk factors and is based on the number of unprotected acts they engaged in during the sexual relationship. To simplify the calculations, Fox et al. assume that condoms are 100% effective and that the HIVpositive partner is not on treatment. In addition, they assume that all of the risk factors are independent. However, clearly, the viral load of the HIV-infected partner and their stage of infection are not independent and neither are GUD and HSV-2. By assuming independence, the authors are overestimating the risk of exposure to HIV.

Exogenous re-infection does not always cause backward bifurcation in TB transmission dynamics

Exogenous re-infection does not always cause backward bifurcation in TB transmission dynamics.Backward bifurcation in TB disease is more likely to occur if (a) the rates of re-infection and transmissibility of re-infected individuals are sufficiently high (b) the fraction of slow progressors is increased or if the rates of treatment and disease-induced mortality are increased.Backward bifurcation in TB disease is less likely to occur for increasing rate of endogenous re-activation of latent TB cases. Models for the transmission dynamics of mycobacterium tuberculosis (TB) that incorporate exogenous re-infection are known to induce the phenomenon of backward bifurcation, a dynamic phenomenon associated with the existence of two stable attractors when the reproduction number of the model is less than unity. This study shows, by way of a counter example, that exogenous re-infection does not always cause backward bifurcation in TB transmission dynamics. In particular, it is shown that it is the transmission ability of the re-infected individuals, and not just the re-infection process, that causes the backward bifurcation phenomenon. When re-infected individuals do not transmit infection, the disease-free equilibrium of the model is shown to be globally-asymptotically stable (GAS) when the associated reproduction number is less than unity. The model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity. It is shown, using a Lyapunov function, that the unique endemic equilibrium is GAS for the special case with no disease-induced mortality and no transmission by re-infected individuals. It is further shown that even if re-infected individuals do transmit infection, backward bifurcation only occurs if their transmissibility exceeds a certain threshold. Sensitivity analyses, with respect to the derived backward bifurcation threshold, show that the phenomenon of backward bifurcation is more likely to occur if the rates of re-infection and transmissibility of re-infected individuals are sufficiently high. Furthermore, it is likely to occur if the fraction of slow progressors (to active TB) is increased or if the rates of treatment (of symptomatic cases) and disease-induced mortality are increased. On the other hand, backward bifurcation is less likely to occur for increasing rates of endogenous re-activation of latent TB cases.

The effect of the incidence function on the existence of backward bifurcation

In modelling, the dynamics of infectious disease, the choice of the specific mathematical formulation of disease transmission (i.e. the incidence function) is one of the initial assumptions to be made. While inconsequential in many situations, we show that the incidence function can have an effect on the existence of backward bifurcation (the phenomenon where a disease can persist even when the basic reproductive number is less than 1). More specifically, we compare mass action (MA) and standard incidence (SI) (the most common incidence functions) versions of two hallmark models in the backward bifurcation literature and an original combination model. Our findings indicate that the SI formation of disease transmission is more conducive to backward bifurcation than MA, a trend seen in all the models analysed.

Predicting the level of vaccine-induced cross-immunity necessary to eliminate HIV epidemics composed of multiple subtypes.

In a very interesting study, Kiwanuka et al. [1] report significant differences in the rates of transmission associated with different HIV-1 subtypes in Rakai, Uganda. Controlling for other factors, they find the transmission rate of subtype A to be nearly double that of subtype D. The authors suggest that differential transmission rates among subtypes are important for: (i) HIV vaccine development and testing, (ii) understanding the dynamics of HIV-1 epidemics in different geographical regions and (iii) projections of the pandemic. Over a decade ago, we constructed a mathematical model of an HIV epidemic composed of multiple co-circulating subtypes that differed on the basis of transmissibility [2]. We constructed this subtype model because preliminary data, collected in the mid 1990’s, indicated that HIV subtypes might exhibit differences in transmission efficiency [3,4]. We used our model to predict temporal trends in prevalence of co-circulating subtypes and the potential impact of prophylactic vaccines. Specifically, we investigated (theoretical) vaccines that would provide a degree of protection against infection by one subtype and induce cross-immunity against infection by another subtype. As the focus of our modeling work is so closely in line with the implications of [1], we now reexamine our subtype model using the remarkable data of Kiwanuka et al. in order to gain insights into current epidemiological patterns and predict the long-term outcome of mass vaccination against HIV in Uganda. In [2], we used the model to calculate basic reproductive numbers (R0) for each of the co-circulating subtypes. R0 represents the expected number of secondary infections caused by the introduction of one infectious individual into a completely susceptible population, and hence is a measure of fitness. Our modeling showed that, in the absence of vaccination, the subtype with the largest R0 will eventually outcompete and eliminate the other subtype. However, we found elimination would take over 100 years to occur and that the prevalence of the less-fit subtype could remain high for several decades. If subtype D emerged before subtype A in Uganda, our results could explain why the less-fit subtype D is currently more prevalent than subtype A and is slowly decreasing (71 to 63% from 1994 to 2002), whereas the prevalence of subtype A is slowly increasing (15 to 20% from 1994 to 2002) [5]. As Kiwanuka et al. point out, it is important to develop vaccines that are effective against several subtypes in order to control HIV epidemics. Our previous modeling has shown that mass vaccination could result in several long-term outcomes: (i) elimination of both subtypes, (ii) elimination of only one subtype, and (iii) persistence of both subtypes [2]. We showed that which outcome would occur depends on the R0’s, the vaccine coverage level, and three characteristics of the vaccine: take (i.e. the fraction of individuals for which the vaccine produces a protective immune response), degree of protection against one subtype, and the level of cross-immunity (i.e. degree of protection against the second subtype) [2]. The model was originally formulated to reflect transmission of HIV in a community of men who have sex with men. However, the model can be parameterized to reflect heterosexual transmission in Uganda because no significant differences between male-to-female and female-to-male transmission were observed in [1]. Doing so, we assume that the transmissibility of subtype A is approximately double that of subtype D [1] and that the average time to develop AIDS is 8 and 6.5 years, for subtypes A and D, respectively [6]. Figure 1 shows the long-term outcomes of mass vaccination given different levels of vaccine-induced cross-immunity, vaccine take and coverage. Calculations were made assuming the vaccine would provide an 80% degree of protection against infection by subtype A. It may be seen that even using this highly effective vaccine, HIV elimination will only be possible if a high fraction of the population is effectively vaccinated (Figure 1). For example, even if the vaccine take is 75%, it would be necessary to vaccinate over 90% of the population and for the vaccine to induce more than 30% cross-immunity against subtype D. If the level of cross-immunity is too low, elimination would not be possible and the less-fit subtype D could outcompete subtype A (Figure 1). This occurs when the vaccine’s protection renders subtype A less fit than subtype D. Fig. 1 Predicted long-term outcomes of a vaccination policy in Uganda using a vaccine with an 80% degree of protection against infection from subtype A. The fraction effectively vaccinated is calculated as the product of take and population coverage (e.g. if ... Our findings indicate that the presence of multiple subtypes will make HIV elimination more difficult and that mass vaccination campaigns may produce surprising results. Sincerely Dr. David J. Gerberry and Professor Sally Blower

An exact approach to calibrating infectious disease models to surveillance data: the case of HIV and HSV-2

When mathematical models of infectious diseases are used to inform health policy, an important first step is often to calibrate a model to disease surveillance data for a specific setting (or multiple settings). It is increasingly common to also perform sensitivity analyses to demonstrate the robustness, or lack thereof, of the modeling results. Doing so requires the modeler to find multiple parameter sets for which the model produces behavior that is consistent with the surveillance data. While frequently overlooked, the calibration process is nontrivial at best and can be inefficient, poorly communicated and a major hurdle to the overall reproducibility of modeling results. In this work, we describe a general approach to calibrating infectious disease models to surveillance data. The technique is able to match surveillance data to high accuracy in a very efficient manner as it is based on the Newton-Raphson method for solving nonlinear systems. To demonstrate its robustness, we use the calibration technique on multiple models for the interacting dynamics of HIV and HSV-2.