Incorporating spatial variation in density enhances the stability of simple population dynamics models

Simple discrete time models of population growth admit a wide variety of dynamic behaviors, including population cycles and chaos. Yet studies of natural and laboratory populations typically reveal their dynamics to be relatively stable. Many explanations for the apparent rarity of unstable or chaotic behavior in real populations have been developed, including the possible stabilizing roles of migration, refugia, abrupt density-dependence, and genetic variation in sensitivity to density. We develop a theoretical framework for incorporating random spatial variation in density into simple models of population growth, and apply this approach to two commonly used models in ecology: the Ricker and Hassell maps. We show that the incorporation of spatial density variation into both these models has a strong stabilizing influence on their dynamic behavior, and leads to their exhibiting stable point equilibria or stable limit cycles over a relatively much larger range of parameter values. We suggest that one reason why chaotic population dynamics are less common than the simple models indicate is, these models typically neglect the potentially stabilizing role of spatial variation in density.     

Dynamic population epidemic models.

Most multipopulation epidemic models are of the contact distribution type, in which the locations of successive contacts are chosen independently from appropriate contact distributions. This paper is concerned with an alternative class of models, termed dynamic population epidemic models, in which infectives move among the populations and can infect only within their current population. Both the stochastic and deterministic versions of such models are considered. Their threshold behavior is analyzed in some depth, as are their final outcomes. Velocities of spread of infection are considered when the populations have a spatial structure. A criterion for finding the equivalent contact distribution epidemic for any given dynamic population epidemic is provided, enabling comparisons to be made for the velocities and final outcomes displayed by the two classes of models. The relationship between deterministic and stochastic epidemic models is also discussed briefly.     

一类总人口在变化的HIV/AIDS传播的动力学模型

建立了HIV/AIDS传播的具有常数移民和指数出生的SI型模型,其中易感人群按照有无不良行为被分为两组.分别对具双线性传染率和具标准传染率的模型讨论了其无病平衡点和地方病平衡点的存在性,并就某些重要的特殊情况进行了平衡点和稳定性的分析.     

Dynamic versus instantaneous models of diet choice

We investigate the dynamics of a series of two-prey-one-predator models in which the predator exhibits adaptive diet choice based on the different energy contents and/or handling times of the two prey species. The predator is efficient at exploiting its prey and has a saturating functional response; these two features combine to produce sustained population cycles over a wide range of parameter values. Two types of models of behavioral change are compared. In one class of models ("instantaneous choice"), the probability of acceptance of the poorer prey by the predator instantaneously approximates the optimal choice, given current prey densities. In the second class of models ("dynamic choice"), the probability of acceptance of the poorer prey is a dynamic variable, which begins to change in an adaptive direction when prey densities change but which requires a finite amount of time to approach the new optimal behavior. The two types of models frequently predict qualitatively different population dynamics of the three-species system, with chaotic dynamics and complex cycles being a common outcome only in the dynamic choice models. In dynamic choice models, factors that reduce the rate of behavioral change when the probability of accepting the poorer prey approaches extreme values often produce complex population dynamics. Instantaneous and dynamic models often predict different average population densities and different indirect interactions between prey species. Alternative dynamic models of behavior are analyzed and suggest, first, that instantaneous choice models may be good approximations in some circumstances and, second, that different types of dynamic choice models often lead to significantly different population dynamics. The results suggest possible behavioral mechanisms leading to complex population dynamics and highlight the need for more empirical study of the dynamics of behavioral change.     

Incorporating Disease and Population Structure into Models of SIR Disease in Contact Networks

We consider the recently introduced edge-based compartmental models (EBCM) for the spread of susceptible-infected-recovered (SIR) diseases in networks. These models differ from standard infectious disease models by focusing on the status of a random partner in the population, rather than a random individual. This change in focus leads to simple analytic models for the spread of SIR diseases in random networks with heterogeneous degree. In this paper we extend this approach to handle deviations of the disease or population from the simplistic assumptions of earlier work. We allow the population to have structure due to effects such as demographic features or multiple types of risk behavior. We allow the disease to have more complicated natural history. Although we introduce these modifications in the static network context, it is straightforward to incorporate them into dynamic network models. We also consider serosorting, which requires using dynamic network models. The basic methods we use to derive these generalizations are widely applicable, and so it is straightforward to introduce many other generalizations not considered here. Our goal is twofold: to provide a number of examples generalizing the EBCM method for various different population or disease structures and to provide insight into how to derive such a model under new sets of assumptions.     

Semiparametric functional mapping of quantitative trait loci governing long-term HIV dynamics

MOTIVATION: Functional mapping has proven to be powerful for characterizing quantitative trait loci (QTL) that control complex dynamic traits. More recently, functional mapping has been extended to identify the host QTL responsible for HIV dynamics by incorporating a parametric bi-exponential function for earlier stages of viral load trajectories. However, existing functional mapping cannot be used to map long-term HIV dynamics because no mathematical functions are available for later stages of HIV dynamic changes. RESULTS: We derived a statistical model for functional mapping of dynamic QTL through characterizing HIV load trajectories during a long-term period semiparametrically. The new model was constructed within the maximum likelihood framework and implemented with the EM-simplex algorithm. It allows for the test of differences in the genetic control of short- and long-term HIV dynamics and the characterization of the effects of viral-host genome interaction. Extensive simulation studies have been performed to test the statistical behavior of this model. The new model will provide an important tool for genetic and genomic studies of human complex diseases like HIV/AIDS and their pathological progression. AVAILABILITY: Available on request from the corresponding author.     

Maximum Likelihood Estimation of Long‐Term HIV Dynamic Models and Antiviral Response

Summary HIV dynamics studies, based on differential equations, have significantly improved the knowledge on HIV infection. While first studies used simplified short‐term dynamic models, recent works considered more complex long‐term models combined with a global analysis of whole patient data based on nonlinear mixed models, increasing the accuracy of the HIV dynamic analysis. However statistical issues remain, given the complexity of the problem. We proposed to use the SAEM (stochastic approximation expectation‐maximization) algorithm, a powerful maximum likelihood estimation algorithm, to analyze simultaneously the HIV viral load decrease and the CD4 increase in patients using a long‐term HIV dynamic system. We applied the proposed methodology to the prospective COPHAR2–ANRS 111 trial. Very satisfactory results were obtained with a model with latent CD4 cells defined with five differential equations. One parameter was fixed, the 10 remaining parameters (eight with between‐patient variability) of this model were well estimated. We showed that the efficacy of nelfinavir was reduced compared to indinavir and lopinavir.     

The concurrent evolution of cooperation and the population structures that support it

The evolution of cooperation often depends upon population structure, yet nearly all models of cooperation implicitly assume that this structure remains static. This is a simplifying assumption, because most organisms possess genetic traits that affect their population structure to some degree. These traits, such as a group size preference, affect the relatedness of interacting individuals and hence the opportunity for kin or group selection. We argue that models that do not explicitly consider their evolution cannot provide a satisfactory account of the origin of cooperation, because they cannot explain how the prerequisite population structures arise. Here, we consider the concurrent evolution of genetic traits that affect population structure, with those that affect social behavior. We show that not only does population structure drive social evolution, as in previous models, but that the opportunity for cooperation can in turn drive the creation of population structures that support it. This occurs through the generation of linkage disequilibrium between socio-behavioral and population-structuring traits, such that direct kin selection on social behavior creates indirect selection pressure on population structure. We illustrate our argument with a model of the concurrent evolution of group size preference and social behavior.     

Modeling adaptive regulatory T-cell dynamics during early HIV infection

Regulatory T-cells (Tregs) are a subset of CD4(+) T-cells that have been found to suppress the immune response. During HIV viral infection, Treg activity has been observed to have both beneficial and deleterious effects on patient recovery; however, the extent to which this is regulated is poorly understood. We hypothesize that this dichotomy in behavior is attributed to Treg dynamics changing over the course of infection through the proliferation of an 'adaptive' Treg population which targets HIV-specific immune responses. To investigate the role Tregs play in HIV infection, a delay differatial equation model was constructed to examine (1) the possible existence of two distinct Treg populations, normal (nTregs) and adaptive (aTregs), and (2) their respective effects in limiting viral load. Sensitivity analysis was performed to test parameter regimes that show the proportionality of viral load with adaptive regulatory populations and also gave insight into the importance of downregulation of CD4(+) cells by normal Tregs on viral loads. Through the inclusion of Treg populations in the model, a diverse array of viral dynamics was found. Specifically, oscillatory and steady state behaviors were both witnessed and it was seen that the model provided a more accurate depiction of the effector cell population as compared with previous models. Through further studies of adaptive and normal Tregs, improved treatments for HIV can be constructed for patients and the viral mechanisms of infection can be further elucidated.     

Modeling the synergy between HSV-2 and HIV and potential impact of HSV-2 therapy

Mounting evidence indicates that genital HSV-2 infection may increase susceptibility to HIV infection and that co-infection may increase infectiousness. Accordingly, antiviral treatment of people with HSV-2 may mitigate the incidence of HIV in populations where both pathogens occur. To better understand the epidemiological synergy between HIV and HSV-2, we formulate a deterministic compartmental model that describes the transmission dynamics of these pathogens. Unlike earlier models, ours incorporates gender and heterogeneous mixing between activity groups. We derive explicit expressions for the reproduction numbers of HSV-2 and HIV, as well as the invasion reproduction numbers via next generation matrices. A qualitative analysis of the system includes the local and global behavior of the model. Simulations reinforce these analytical results and demonstrate epidemiological synergy between HSV-2 and HIV. In particular, numerical results show that HSV-2 favors the invasion of HIV, may dramatically increase the peak as well as reducing the time-to-peak of HIV prevalence, and almost certainly has exacerbated HIV epidemics. The potential population-level impact of HSV-2 on HIV is demonstrated by calculating the fraction of HIV infections attributable to HSV-2 and the difference between HIV prevalence in the presence and absence of HSV-2. The potential impact of treating people with HSV-2 on HIV control is demonstrated by comparing HIV prevalence with and without HSV-2 therapy. Most importantly, we illustrate that the aforementioned aspects of the population dynamics can be significantly influenced by the sexual structure of the population.     

Mathematical Insights in Evaluating State Dependent Effectiveness of HIV Prevention Interventions

Mathematical models have been used to simulate HIV transmission and to study the use of preexposure prophylaxis (PrEP) for HIV prevention. Often a single intervention outcome over 10 years has been used to evaluate the effectiveness of PrEP interventions. However, different metrics express a wide variation over time and often disagree in their forecast on the success of the intervention. We develop a deterministic mathematical model of HIV transmission and use it to evaluate the public-health impact of oral PrEP interventions. We study PrEP effectiveness with respect to different evaluation methods and analyze its dynamics over time. We compare four traditional indicators, based on cumulative number or fractions of infections prevented, on reduction in HIV prevalence or incidence and propose two additional methods, which estimate the burden of the epidemic to the public-health system. We investigate the short and long term behavior of these indicators and the effects of key parameters on the expected benefits from PrEP use. Our findings suggest that public-health officials considering adopting PrEP in HIV prevention programs can make better informed decisions by employing a set of complementing quantitative metrics.     

The Epidemiology of HIV and Prevention Needs of Men Who Have Sex with Men in Abidjan,Cote d’Ivoire

To determine HIV prevalence and associated risk factors among men who have sex with men (MSM) in Abidjan, Côte d’Ivoire. We conducted a cross-sectional RDS survey of MSM in Abidjan from October 2011 to February 2012. Eligibility criteria included age ≥ 18 years and having had oral or anal sex with another man in the last 12 months. Weighted data analysis was conducted with RDSAT and SAS. We enrolled 603 participants, of whom 601 (99.7%) completed the questionnaire and 581 (96.7%) consented to HIV testing. HIV population prevalence was estimated as 18.0% (95% CI: 13.0-23.1); 86.4% (95% CI: 75.1-94.9) of HIV-positive MSM were unaware of their serostatus. In multivariable analysis, adjusting for age, education, and income, HIV infection was associated with unprotected sex at last sex with a woman, more than two male anal sex partners in last 12 months, inconsistent condom use during anal sex with a man, self-perceived risk of HIV, history of forced sex, history of physical abuse due to MSM status, and not receiving last HIV test result prior to study. HIV prevalence among MSM in Abidjan is more than four times as high as that of general population men. MSM engage in high-risk sexual behavior and most HIV-positive MSM are unaware of their serostatus. Greater access to HIV prevention, care, and treatment services targeted to MSM is necessary.     

Learning that circumcision is protective against HIV: risk compensation among men and women in Cape Town, South Africa

Objectives

We examined whether knowledge of the HIV-protective benefits of male circumcision (MC) led to risk compensating behavior in a traditionally circumcising population in South Africa. We extend the current literature by examining risk compensation among women, which has hitherto been unexplored.

Methods

We used data on Xhosa men and women from the 2009 Cape Area Panel Study. Respondents were asked if they had heard that MC reduces a man’s risk of contracting HIV, about their perceived risk of contracting HIV, and condom use. For each gender group we assessed whether risk perception and condom use differed by knowledge of the protective benefits of MC using bivariate and then multivariate models controlling for demographic characteristics, HIV knowledge/beliefs, and previous sexual behaviors. In a further check for confounding, we used data from the 2005 wave to assess whether individuals who would eventually become informed about the protective benefits of circumcision were already different in terms of HIV risk perception and condom use.

Results

34% of men (n = 453) and 27% of women (n = 690) had heard that circumcision reduces a man’s risk of HIV infection. Informed men perceived slightly higher risk of contracting HIV and were more likely to use condoms at last sex (p<0.10). Informed women perceived lower HIV risk (p<0.05), were less likely to use condoms both at last sex (p<0.10) and more generally (p<0.01), and more likely to forego condoms with partners of positive or unknown serostatus (p<0.01). The results were robust to covariate adjustment, excluding people living with HIV, and accounting for risk perceptions and condom use in 2005.

Conclusions

We find evidence consistent with risk compensation among women but not men. Further attention should be paid to the role of new information regarding MC, and drivers of HIV risk more broadly, in modulating sexual behavior among women.     

Stochastic theory of early viral infection: continuous versus burst production of virions

Viral production from infected cells can occur continuously or in a burst that generally kills the cell. For HIV infection, both modes of production have been suggested. Standard viral dynamic models formulated as sets of ordinary differential equations can not distinguish between these two modes of viral production, as the predicted dynamics is identical as long as infected cells produce the same total number of virions over their lifespan. Here we show that in stochastic models of viral infection the two modes of viral production yield different early term dynamics. Further, we analytically determine the probability that infections initiated with any number of virions and infected cells reach extinction, the state when both the population of virions and infected cells vanish, and show this too has different solutions for continuous and burst production. We also compute the distributions of times to establish infection as well as the distribution of times to extinction starting from both a single virion as well as from a single infected cell for both modes of virion production.     

Avoiding the bullies: The resilience of cooperation among unequals

Can egalitarian norms or conventions survive the presence of dominant individuals who are ensured of victory in conflicts? We investigate the interaction of power asymmetry and partner choice in games of conflict over a contested resource. Previous models of cooperation do not include both power inequality and partner choice. Furthermore, models that do include power inequalities assume a static game where a bully’s advantage does not change. They have therefore not attempted to model complex and realistic properties of social interaction. Here, we introduce three models to study the emergence and resilience of cooperation among unequals when interaction is random, when individuals can choose their partners, and where power asymmetries dynamically depend on accumulated payoffs. We find that the ability to avoid bullies with higher competitive ability afforded by partner choice mostly restores cooperative conventions and that the competitive hierarchy never forms. Partner choice counteracts the hyper dominance of bullies who are isolated in the network and eliminates the need for others to coordinate in a coalition. When competitive ability dynamically depends on cumulative payoffs, complex cycles of coupled network-strategy-rank changes emerge. Effective collaborators gain popularity (and thus power), adopt aggressive behavior, get isolated, and ultimately lose power. Neither the network nor behavior converge to a stable equilibrium. Despite the instability of power dynamics, the cooperative convention in the population remains stable overall and long-term inequality is completely eliminated. The interaction between partner choice and dynamic power asymmetry is crucial for these results: without partner choice, bullies cannot be isolated, and without dynamic power asymmetry, bullies do not lose their power even when isolated. We analytically identify a single critical point that marks a phase transition in all three iterations of our models. This critical point is where the first individual breaks from the convention and cycles start to emerge.     

Phylodynamic Inference for Structured Epidemiological Models

Coalescent theory is routinely used to estimate past population dynamics and demographic parameters from genealogies. While early work in coalescent theory only considered simple demographic models, advances in theory have allowed for increasingly complex demographic scenarios to be considered. The success of this approach has lead to coalescent-based inference methods being applied to populations with rapidly changing population dynamics, including pathogens like RNA viruses. However, fitting epidemiological models to genealogies via coalescent models remains a challenging task, because pathogen populations often exhibit complex, nonlinear dynamics and are structured by multiple factors. Moreover, it often becomes necessary to consider stochastic variation in population dynamics when fitting such complex models to real data. Using recently developed structured coalescent models that accommodate complex population dynamics and population structure, we develop a statistical framework for fitting stochastic epidemiological models to genealogies. By combining particle filtering methods with Bayesian Markov chain Monte Carlo methods, we are able to fit a wide class of stochastic, nonlinear epidemiological models with different forms of population structure to genealogies. We demonstrate our framework using two structured epidemiological models: a model with disease progression between multiple stages of infection and a two-population model reflecting spatial structure. We apply the multi-stage model to HIV genealogies and show that the proposed method can be used to estimate the stage-specific transmission rates and prevalence of HIV. Finally, using the two-population model we explore how much information about population structure is contained in genealogies and what sample sizes are necessary to reliably infer parameters like migration rates.     

Sex-structured HIV/AIDS model to analyse the effects of condom use with application to Zimbabwe

We present a sex-structured model for heterosexual transmission of HIV/AIDS in a community. The model is formulated using integro-differential equations, which are shown to be equivalent to delay differential equations with a time delay due to incubation period. The sex-structured HIV/AIDS model divides the population into a two sex-structure consisting of females and males. The threshold and equilibria for the model are determined and stabilities are examined. We extend the model to focus on the effects of condom use as a single-strategy approach in HIV prevention in the absence of any treatment. Initially we model the use of male condoms and further extend the model to incorporate the use of both female and male condoms. The model includes two primary factors in condom use to control HIV that are condom efficacy and compliance. The exposure risk of infection after each intervention is obtained. Basic reproductive numbers for these models are computed and compared to assess the effectiveness of male and female condom use in a community. The models are numerically analysed to assess the effectiveness of condom use on the transmission dynamics of HIV/AIDS using demographic and epidemiological parameters for Zimbabwe. The study demonstrates the use of sex-structured HIV/AIDS models in assessing the effectiveness of female and male condom use as a preventative strategy in a heterosexually active population. Z. Mukandavire would like to acknowledge financial support given by the National University of Science and Technology through a Staff Development Scholarship. The authors are grateful to Eagle Insurance Company of Zimbabwe for financial support.     

Modelling viral and CD4 cellular population dynamics in HIV: approaches to evaluate intervention strategies.

Computational models, such as in epidemiology, provide a powerful tool that can be used to systematically examine an array of dynamic interactions among populations as well as to evaluate altemate disease intervention strategies. The specific objectives in this study were to: a/ examine the interaction of cellular (CD4) and HIV population dynamics and evaluate the impact of the use of combination chemotherapies on viral and CD4 populations (Experiment #1), b/ demonstrate how modelling can be used to evaluate the impact of an intervention (condom use) on reducing the rate of HIV/AIDS (Experiment #2). In this study, we used state transition models and conducted simulation experiments to evaluate various alternatives for the control and/or prevention of HIV/AIDS. The result indicated that combination therapy (double or triple drug therapies) was very effective. The HIV viral population decreased rapidly and remained suppressed for years. On the other hand, the CD4 cell population increased above 400 cells per ml and was maintained above that level for many years. Mono-therapy was not as effective; although the viral load decreased rapidly, it increased to its original levels within a few months. Since condom use is one of the key interventions of HIV/AIDS, we evaluated its use in 25%, 50% and 75% of an adult, sexually active population. Increasing condom use by 50% and 75% above an estimated baseline of 25% reduced the incidence of AIDS by 53% in Blacks, 49% in Hispanics and 43% in Whites. The study shows how a cellular/molecular level model can be incorporated within a macro-epidemiologic systems dynamics model to evaluate a variety of scientific questions such as to see if cellular/molecular level interventions reduce morbidity and mortality rates in HIV.     

Age and Sex Structured Model for Assessing the Demographic Impact of Mother-to-Child Transmission of HIV/AIDS

Age and sex structured HIV/AIDS model with explicit incubation period is proposed as a system of delay differential equations. The model consists of two age groups that are children (0–14 years) and adults (15–49 years). Thus, the model considers both mother-to-child transmission (MTCT) and heterosexual transmission of HIV in a community. MTCT can occur prenatally, at labour and delivery or postnatally through breastfeeding. In the model, we consider the children age group as a one-sex formulation and divide the adult age group into a two-sex structure consisting of females and males. The important mathematical features of the model are analysed. The disease-free and endemic equilibria are found and their stabilities investigated. We use the Lyapunov functional approach to show the local stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The basic reproductive number (ℛ₀) for the model shows that the adult population is responsible for the spread HIV/AIDS epidemic, thus up-to-date developed HIV/AIDS models to assess intervention strategies have focused much on heterosexual transmission by the adult population and the children population has received little attention. We numerically analyse the HIV/AIDS model to assess the community benefits of using antiretroviral drugs in reducing MTCT and the effects of breastfeeding in settings with high HIV/AIDS prevalence ratio using demographic and epidemiological parameters for Zimbabwe.