Rational behavioral response and the transmission of STDs

The susceptible-infected (SI) model is extended by allowing for individual optimal choices of self-protective actions against infection, where agents differ with respect to preferences and costs of self-protection. It is shown that a unique endemic equilibrium prevalence exists when the basic reproductive number of a STD is strictly greater than unity, and that the disease-free equilibrium is the unique steady state equilibrium when the basic reproductive number is less than or equal to one. Unlike in models that take individual behavior as given and fixed, the endemic equilibrium prevalence need not vary monotonically with respect to the basic reproductive number. Specifically, with endogenously determined self-protective behavior, a reduction in the basic reproductive number may in fact increase the endemic equilibrium prevalence. The global stability of the endemic steady state is established for the case of a homogeneous population by showing that, for any non-zero initial disease prevalence, there exists an equilibrium path which converges to the endemic steady state.     

Public avoidance and epidemics: insights from an economic model

In this paper, we present a mathematical model of infectious disease transmission in which people can engage in public avoidance behavior to minimize the likelihood of acquiring an infection. The framework employs the economist's theory of utility maximization to model people's decision regarding their level of public avoidance. We derive the reproductive number of a disease which determines whether an endemic equilibrium exists or not. We show that when the contact function exhibits saturation, an endemic equilibrium must be unique. Otherwise, multiple endemic equilibria that differ in disease prevalence can coexist, and which one the population gets to depends on initial conditions. Even when a unique endemic equilibrium exists, people's preferences and the initial conditions may determine whether the disease will eventually die out or become endemic. Public health policies that increase the recovery rate or encourage self-quarantine by infected people can be beneficial to the community by lowering disease prevalence. However, it is also possible for these policies to worsen the situation and cause prevalence to rise since these measures give people less incentive to engage in public avoidance behavior. We also show that implementing policies that result in a higher level of public avoidance behavior in equilibrium does not necessarily lower prevalence and can result in more infections.     

Global dynamics of an SEIR epidemic model with saturating contact rate

Heesterbeek and Metz [J. Math. Biol. 31 (1993) 529] derived an expression for the saturating contact rate of individual contacts in an epidemiological model. In this paper, the SEIR model with this saturating contact rate is studied. The basic reproduction number R0 is proved to be a sharp threshold which completely determines the global dynamics and the outcome of the disease. If R0 < or =1, the disease-free equilibrium is globally stable and the disease always dies out. If R0 > 1, there exists a unique endemic equilibrium which is globally stable and the disease persists at an endemic equilibrium state if it initially exists. The contribution of the saturating contact rate to the basic reproduction number and the level of the endemic equilibrium is also analyzed.     

Modelling the effect of treatment and behavioral change in HIV transmission dynamics

In this paper we analyze a model for the HIV-infection transmission in a male homosexual population. In the model we consider two types of infected individuals. Those that are infected but do not know their serological status and/or are not under any sort of clinical /therapeutical treatment, and those who are. The two groups of infectives differ in their incubation time, contact rate with susceptible individuals, and probability of disease transmission. The aim of this article is to study the roles played by detection and changes in sexual behavior in the incidence and prevalence of HIV. The analytical results show that there exists a unique endemic equilibrium which is globally asymptotically stable under a range of parameter values whenever a detection /treatment rate and an indirect measure of the level of infection risk are sufficiently large. However, any level of detection/ treatment rate coupled with a decrease of the transmission probability lowers the incidence rate and prevalence level in the population. In general, only significant reductions in the transmission probability (achieved through, for example, the adoption of safe sexual practices) can contain effectively the spread of the disease.     

A Susceptible-infected Epidemic Model with Voluntary Vaccinations

An susceptible-infected epidemic model with endogenous behavioral changes is presented to analyze the impact of a prophylactic vaccine on disease prevalence. It is shown that, with voluntary vaccination, whether an endemic equilibrium exists or not does not depend on vaccine efficacy or the distribution of agent-types. Although an endemic equilibrium is unique in the absence of a vaccine, the availability of a vaccine can lead to multiple endemic equilibria that differ in disease prevalence and vaccine coverage. Depending on the distribution of agent-types, the introduction of a vaccine or, if one is available, a subsidy for vaccination can increase disease prevalence by inducing more risky behavior.I would like to thank one of the editors of the journal, Alan Hastings, for his comments and suggestions.     

Spatial patterns in a discrete-time SIS patch model

How do spatial heterogeneity, habitat connectivity, and different movement rates among subpopulations combine to influence the observed spatial patterns of an infectious disease? To find out, we formulated and analyzed a discrete-time SIS patch model. Patch differences in local disease transmission and recovery rates characterize whether patches are low-risk or high-risk, and these differences collectively determine whether the spatial domain, or habitat, is low-risk or high-risk. In low-risk habitats, the disease persists only when the mobility of infected individuals lies below some threshold value, but for high-risk habitats, the disease always persists. When the disease does persist, then there exists an endemic equilibrium (EE) which is unique and positive everywhere. This EE tends to a spatially inhomogeneous disease-free equilibrium (DFE) as the mobility of susceptible individuals tends to zero. The limiting DFE is nonempty on all low-risk patches and it is empty on at least one high-risk patch. Sufficient conditions for the limiting DFE to be empty on other high-risk patches are given in terms of disease transmission and recovery rates, habitat connectivity, and the infected movement rate. These conditions are also illustrated using numerical examples.     

Qualitative analysis of an HIV transmission model.

In 1988, a multiple-group model for HIV transmission with preferred mixing was proposed by Jacquez and coworkers. In the present paper, the work done by Jacquez et al. is extended. It is shown that the stability modulus of the Jacobian matrix at the no-disease equilibrium is a threshold for this model. Furthermore, if the no-disease equilibrium is unstable, the number of infected individuals will remain above a certain positive level regardless of initial levels; that is, the disease will persist uniformly. The stability of the endemic equilibrium in the case of restricted mixing is also studied. A series of sufficient conditions for local and global asymptotic stability of the endemic equilibrium are stated.     

The spread and persistence of infectious diseases in structured populations

A basic assumption of many epidemic models is that populations are composed of a homogeneous group of randomly mixing individuals. This is not a realistic assumption. Most actual populations are divided into a number of subpopulations, within which there may be relatively random mixing, but among which there is nonrandom mixing. As a consequence of the structuring of the population, there are several sources of heterogeneity within populations that can affect the course of an infection through the population. Two of these sources of heterogeneity are differences in contact number between subpopulations, and differences in the patterns of contact among subpopulations. A model for the spread of a disease in such a population is described. The model considers two levels of interaction: interactions between individuals within a subpopulation because of geographic proximity, and interactions between individuals of the same or different subpopulations because of attendance at common social functions. Because of this structure, it is possible to analyze with the model both heterogeneity in contact number and variation in the patterns of contact. A stability analysis of the model is presented which shows that there is a unique threshold for disease maintenance. Below the threshold the disease goes extinct, and the equilibrium is globally asymptotically stable. Above the threshold, the extinction equilibrium is unstable, and there is a unique endemic equilibrium. The analysis presents a sufficient condition for disease maintenance, which determines critical subpopulation sizes above which the disease cannot go extinct. The condition is a simple inequality relating the removal rate of infectives to the infection rate of susceptibles. In addition, bounds on the actual threshold and the effect of symmetry in the interaction matrix on the threshold are presented.     

Stability and Bifurcations in an Epidemic Model with Varying Immunity Period

An epidemic model with distributed time delay is derived to describe the dynamics of infectious diseases with varying immunity. It is shown that solutions are always positive, and the model has at most two steady states: disease-free and endemic. It is proved that the disease-free equilibrium is locally and globally asymptotically stable. When an endemic equilibrium exists, it is possible to analytically prove its local and global stability using Lyapunov functionals. Bifurcation analysis is performed using DDE-BIFTOOL and traceDDE to investigate different dynamical regimes in the model using numerical continuation for different values of system parameters and different integral kernels.     

Global stability of an SIR epidemic model with time delays

An SIR disease transmission model is formulated under the assumption that the force of infection at the present time depends on the number of infectives at the past. It is shown that a disease free equilibrium point is globally stable if no endemic equilibrium point exists. Further the endemic point (if it exists) is globally stable with respect to the whole state space except the neighborhood of the disease free state.Research partly supported by the Ministry of Education, Science and Culture, Japan, Grant 05640256     

Stochastic state transitions give rise to phenotypic equilibrium in populations of cancer cells

Cancer cells within individual tumors often exist in distinct phenotypic states that differ in functional attributes. While cancer cell populations typically display distinctive equilibria in the proportion of cells in various states, the mechanisms by which this occurs are poorly understood. Here, we study the dynamics of phenotypic proportions in human breast cancer cell lines. We show that subpopulations of cells purified for a given phenotypic state return towards equilibrium proportions over time. These observations can be explained by a Markov model in which cells transition stochastically between states. A prediction of this model is that, given certain conditions, any subpopulation of cells will return to equilibrium phenotypic proportions over time. A second prediction is that breast cancer stem-like cells arise de novo from non-stem-like cells. These findings contribute to our understanding of cancer heterogeneity and reveal how stochasticity in single-cell behaviors promotes phenotypic equilibrium in populations of cancer cells.     

A mathematical model for Chagas disease with infection-age-dependent infectivity

In this paper we develop a mathematical model for Chagas disease with infection-age-dependent infectivity. The effects of vector and blood transfusion transmission are considered, and the infected population is structured by the infection age (the time elapsed from infection). The authors identify the basic reproduction ratio R0 and show that the disease can invade into the susceptible population and unique endemic steady state exists if R0 > 1, whereas the disease dies out if R0 is small enough. We show that depending on parameters, backward bifurcation of endemic steady state can occur, so even if R0 < 1, there could exist endemic steady states. We also discuss local and global stability of steady states.     

Diseases with chronic stage in a population with varying size

An epidemiological model of hepatitis C with a chronic infectious stage and variable population size is introduced. A non-structured baseline ODE model which supports exponential solutions is discussed. The normalized version where the unknown functions are the proportions of the susceptible, infected, and chronic individuals in the total population is analyzed. It is shown that sustained oscillations are not possible and the endemic proportions either approach the disease-free or an endemic equilibrium. The expanded model incorporates the chronic age of the individuals. Partial analysis of this age-structured model is carried out. The global asymptotic stability of the infection-free state is established as well as local asymptotic stability of the endemic non-uniform steady state distribution under some additional conditions. A numerical method for the chronic-age-structured model is introduced. It is shown that this numerical scheme is consistent and convergent of first order. Simulations based on the numerical method suggest that in the structured case the endemic equilibrium may be unstable and sustained oscillations are possible. Closer look at the reproduction number reveals that treatment strategies directed towards speeding up the transition from acute to chronic stage in effect contribute to the eradication of the disease.     

Epidemiological models with prevalence dependent endogenous self-protection measure

A simple mathematical model for human disease epidemics that takes the human learning behaviour and self-protective measures into account is proposed and investigated. We have analysed the effect of endogenous self-protective measures with respect to the prevalence level of the disease and conversely. In the model it is assumed that people start reacting against contracting a disease with self-protective measures whenever they are informed about the disease and when the burden of the disease is in a recognizable stage. It is shown that increasing the average effectiveness of self-protective measures is more important in decreasing the prevalence of a disease than increasing the proportion of individuals in a population into which awareness is created.     

Differential susceptibility epidemic models

We formulate compartmental differential susceptibility (DS) susceptible-infective-removed (SIR) models by dividing the susceptible population into multiple subgroups according to the susceptibility of individuals in each group. We analyze the impact of disease-induced mortality in the situations where the number of contacts per individual is either constant or proportional to the total population. We derive an explicit formula for the reproductive number of infection for each model by investigating the local stability of the infection-free equilibrium. We further prove that the infection-free equilibrium of each model is globally asymptotically stable by qualitative analysis of the dynamics of the model system and by utilizing an appropriately chosen Liapunov function. We show that if the reproductive number is greater than one, then there exists a unique endemic equilibrium for all of the DS models studied in this paper. We prove that the endemic equilibrium is locally asymptotically stable for the models with no disease-induced mortality and the models with contact numbers proportional to the total population. We also provide sufficient conditions for the stability of the endemic equilibrium for other situations. We briefly discuss applications of the DS models to optimal vaccine strategies and the connections between the DS models and predator-prey models with multiple prey populations or host-parasitic interaction models with multiple hosts are also given.This research was partially supported by the Department of Energy under contracts W-7405-ENG-36 and the Applied Mathematical Sciences Program KC-07-01-01.     

Spread of disease with transport-related infection and entry screening

An SIQS model is proposed to study the effect of transport-related infection and entry screening. If the basic reproduction number is below unity, the disease free equilibrium is locally asymptotically stable. There exists an endemic equilibrium which is locally asymptotically stable if the reproduction number is larger than unity. It is shown that the disease is endemic in the sense of permanence if and only if the endemic equilibrium exists. Entry screening is shown to be helpful for disease eradication since it can always have the possibility to eradicate the disease led by transport-related infection and furthermore have the possibility to eradicate disease even when the disease is endemic in both isolated cities.     

Static behavioral effects on gonorrhea transmission dynamics in a MSM population

An SIS/SAS model of gonorrhea transmission in a population of highly active men-having-sex-with-men (MSM) is presented in this paper to study the impact of safe behavior on the dynamics of gonorrhea prevalence. Safe behaviors may fall into two categories—prevention and self-awareness. Prevention will be modeled via consistent condom use and self-awareness via STD testing frequency. Stability conditions for the disease free equilibrium and endemic equilibrium are determined along with a complete analysis of global dynamics. The control reproductive number is used as a means for measuring the effect of changes to model parameters on the prevalence of the disease. We also find that appropriate intervention would be in the form of a multifaceted approach at overall risk reduction rather than tackling one specific control individually.     

Threshold dynamics in an SEIRS model with latency and temporary immunity

A disease transmission model of SEIRS type with distributed delays in latent and temporary immune periods is discussed. With general/particular probability distributions in both of these periods, we address the threshold property of the basic reproduction number \(R_0\) and the dynamical properties of the disease-free/endemic equilibrium points present in the model. More specifically, we 1. show the dependence of \(R_0\) on the probability distribution in the latent period and the independence of \(R_0\) from the distribution of the temporary immunity, 2. prove that the disease free equilibrium is always globally asymptotically stable when \(R_0<1\) , and 3. according to the choice of probability functions in the latent and temporary immune periods, establish that the disease always persists when \(R_0>1\) and an endemic equilibrium exists with different stability properties. In particular, the endemic steady state is at least locally asymptotically stable if the probability distribution in the temporary immunity is a decreasing exponential function when the duration of the latency stage is fixed or exponentially decreasing. It may become oscillatory under certain conditions when there exists a constant delay in the temporary immunity period. Numerical simulations are given to verify the theoretical predictions.     

若干具有非线性传染力的传染病模型的稳定性分析

讨论了具有常数迁入和非线性传染力的三类传染病模型,即SIRI模型,SIRI框架下的DS模型及SIR框架下的DI模型。给出了它们基本再生数R0的表达式,证明了R0≤1时无病平衡点是全局稳定的,同时证明了如果地方病平衡点存在,则必是全局稳定的结果（从而必唯一）对第一和第三个模型还给出了R0＞1时地方病平衡点的存在唯一性。