Robust uniform persistence and competitive exclusion in a nonautonomous multi-strain SIR epidemic model with disease-induced mortality

A nonautonomous version of the SIR epidemic model in Ackleh and Allen (2003) is considered, for competition of $n$ infection strains in a host population. The model assumes total cross immunity, mass action incidence, density-dependent host mortality and disease-induced mortality. Sufficient conditions for the robust uniform persistence of the total population, as well as of the susceptible and infected subpopulations, are given. The first two forms of persistence depend entirely on the rate at which the population grows from the extinction state, respectively the rate at which the disease is vertically transmitted to offspring. We also discuss the competitive exclusion among the $n$ infection strains, namely when a single infection strain survives and all the others go extinct. Numerical simulations are also presented, to account for the situations not covered by the analytical results. These simulations suggest that the nonautonomous nature of the model combined with the disease induced mortality allow for many strains to coexist. The theoretical approach developed here is general enough to apply to other nonautonomous epidemic models.     

Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality

Stochastic differential equations that model an SIS epidemic with multiple pathogen strains are derived from a system of ordinary differential equations. The stochastic model assumes there is demographic variability. The dynamics of the deterministic model are summarized. Then the dynamics of the stochastic model are compared to the deterministic model. In the deterministic model, there can be either disease extinction, competitive exclusion, where only one strain persists, or coexistence, where more than one strain persists. In the stochastic model, all strains are eventually eliminated because the disease-free state is an absorbing state. However, if the population size and the initial number of infected individuals are sufficiently large, it may take a long time until all strains are eliminated. Numerical simulations of the stochastic model show that coexistence cases predicted by the deterministic model are an unlikely occurrence in the stochastic model even for short time periods. In the stochastic model, either disease extinction or competitive exclusion occur. The initial number of infected individuals, the basic reproduction numbers, and other epidemiological parameters are important determinants of the dominant strain in the stochastic epidemic model.     

An epidemic model with infector and exposure dependent severity

A stochastic epidemic model allowing for both mildly and severely infectious individuals is defined, where an individual can become severely infectious directly upon infection or if additionally exposed to infection. It is shown that, assuming a large community, the initial phase of the epidemic may be approximated by a suitable branching process and that the main part of an epidemic that becomes established admits a law of large numbers and a central limit theorem, leading to a normal approximation for the final outcome of such an epidemic. Effects of vaccination prior to an outbreak are studied and the critical vaccination coverage, above which only small outbreaks can occur, is derived. The results are illustrated by simulations that demonstrate that the branching process and normal approximations work well for finite communities, and by numerical examples showing that the final outcome may be close to discontinuous in certain model parameters and that the fraction mildly infected may actually increase as an effect of vaccination.     

Optimal treatment of an SIR epidemic model with time delay

In this paper the optimal control strategies of an SIR (susceptible–infected–recovered) epidemic model with time delay are introduced. In order to do this, we consider an optimally controlled SIR epidemic model with time delay where a control means treatment for infectious hosts. We use optimal control approach to minimize the probability that the infected individuals spread and to maximize the total number of susceptible and recovered individuals. We first derive the basic reproduction number and investigate the dynamical behavior of the controlled SIR epidemic model. We also show the existence of an optimal control for the control system and present numerical simulations on real data regarding the course of Ebola virus in Congo. Our results indicate that a small contact rate(probability of infection) is suitable for eradication of the disease (Ebola virus) and this is one way of optimal treatment strategies for infectious hosts.     

The dynamics of two viral infections in a single host population with applications to hantavirus

An SI epidemic model for a host with two viral infections circulating within the population is developed, analyzed, and numerically simulated. The model is a system of four differential equations which includes a state for susceptible individuals, two states for individuals infected with a single virus, one which is vertically transmitted and the other which is horizontally transmitted, and a fourth state for individuals infected with both viruses. A general growth function with density-dependent mortality is assumed. A special case of this model, where there is no coinfection and total cross immunity, is thoroughly analyzed. Several threshold values are defined which determine establishment of the disease and persistence at equilibrium for one or both of the infections within the host population. The model has applications to a hantavirus and an arenavirus that infect cotton rats. The hantavirus is transmitted horizontally whereas the arenavirus is transmitted vertically. It is shown through analysis and numerical simulations that both diseases can be maintained within a single host population, where individuals can be either infected with both viruses or with a single virus.     

Reinfection induced disease in a spatial SIRI model

Spatial models are widely used in epidemiology to investigate persistence and extinction of disease as well as their spatial patterns. One of the most important issues in studying epidemic models is the role of infection on the persistence and extinction of the disease. In this paper, we investigate a spatial susceptible–infected–recovered–infected model using cellular automata. We show that, in the regime where disease disappears in the susceptible–infected–recovered–susceptible model, spiral and target waves will emerge in the two-dimensional space due to the reinfection. The obtained results may point out that reinfection has great influence on the epidemic spreading, which enriches the findings of spatiotemporal dynamics in epidemic models.     

Some threshold and stability results for epidemic models with a density-dependent death rate.

Most classical models for infectious diseases assume that the birth and death rates of individuals and the meeting rates between susceptible and infected individuals do not depend on the total number of individuals in the population. While these assumptions are valid in some situations they are less valid in others. For example, for diseases in animal an insects populations competition for scarce resources might well mean that the death rate depends on the number of individuals. The present paper examines two epidemic models where the death rate is density dependent. For each model the possible equilibrium levels of disease incidence are determined and the stability of these equilibrium levels to small perturbations is discussed. The biological interpretation of these results is presented together with the results of some numerical simulations.     

The differential infectivity and staged progression models for the transmission of HIV

Recent studies of HIV RNA in infected individuals show that viral levels vary widely between individuals and within the same individual over time. Individuals with higher viral loads during the chronic phase tend to develop AIDS more rapidly. If RNA levels are correlated with infectiousness, these variations explain puzzling results from HIV transmission studies and suggest that a small subset of infected people may be responsible for a disproportionate number of infections. We use two simple models to study the impact of variations in infectiousness. In the first model, we account for different levels of virus between individuals during the chronic phase of infection, and the increase in the average time from infection to AIDS that goes along with a decreased viral load. The second model follows the more standard hypothesis that infected individuals progress through a series of infection stages, with the infectiousness of a person depending upon his current disease stage. We derive and compare threshold conditions for the two models and find explicit formulas of their endemic equilibria. We show that formulas for both models can be put into a standard form, which allows for a clear interpretation. We define the relative impact of each group as the fraction of infections being caused by that group. We use these formulas and numerical simulations to examine the relative importance of different stages of infection and different chronic levels of virus to the spreading of the disease. The acute stage and the most infectious group both appear to have a disproportionate effect, especially on the early epidemic. Contact tracing to identify super-spreaders and alertness to the symptoms of acute HIV infection may both be needed to contain this epidemic.     

The AIDS epidemic: profiles of development

As all HIV-infected subjects become virus carriers, the epidemic will not attain a "steady state" until the number of deletions (from death and other factors) equals or outnumbers that of new cases, i.e. each HIV-infected subject transmits the infection to only one subject in the course of his lifespan. A full stop of all spreading of HIV will most likely require worldwide vaccination. By simple mathematical models it is shown that calculation of the number of HIV infected individuals based on the number of AIDS cases is very uncertain. The ratio of HIV infected subjects to AIDS cases is greatly influenced by the length of the incubation period and the case doubling time. Since the growth of the epidemic is exponential, all efforts to control the epidemic should be continuously intensified as single measures will only retard the rate of spread. The effect of saturation/deletion on the number of susceptible individuals is insignificant in this phase of the epidemic, except in small groups at special risk.     

Outbreak analysis of an SIS epidemic model with rewiring

This paper is devoted to the analysis of the early dynamics of an SIS epidemic model defined on networks. The model, introduced by Gross et al. (Phys Rev Lett 96:208701, 2006), is based on the pair-approximation formalism and assumes that, at a given rewiring rate, susceptible nodes replace an infected neighbour by a new susceptible neighbour randomly selected among the pool of susceptible nodes in the population. The analysis uses a triple closure that improves the widely assumed in epidemic models defined on regular and homogeneous networks, and applies it to better understand the early epidemic spread on Poisson, exponential, and scale-free networks. Two extinction scenarios, one dominated by transmission and the other one by rewiring, are characterized by considering the limit system of the model equations close to the beginning of the epidemic. Moreover, an analytical condition for the occurrence of a bistability region is obtained.     

Intervention-Based Stochastic Disease Eradication

Disease control is of paramount importance in public health, with infectious disease extinction as the ultimate goal. Although diseases may go extinct due to random loss of effective contacts where the infection is transmitted to new susceptible individuals, the time to extinction in the absence of control may be prohibitively long. Intervention controls are typically defined on a deterministic schedule. In reality, however, such policies are administered as a random process, while still possessing a mean period. Here, we consider the effect of randomly distributed intervention as disease control on large finite populations. We show explicitly how intervention control, based on mean period and treatment fraction, modulates the average extinction times as a function of population size and rate of infection spread. In particular, our results show an exponential improvement in extinction times even though the controls are implemented using a random Poisson distribution. Finally, we discover those parameter regimes where random treatment yields an exponential improvement in extinction times over the application of strictly periodic intervention. The implication of our results is discussed in light of the availability of limited resources for control.     

The effect of risk-taking behaviour in epidemic models

We study an epidemic model that incorporates risk-taking behaviour as a response to a perceived low prevalence of infection that follows from the administration of an effective treatment or vaccine. We assume that knowledge about the number of infected, recovered and vaccinated individuals has an effect in the contact rate between susceptible and infectious individuals. We show that, whenever optimism prevails in the risk behaviour response, the fate of an epidemic may change from disease clearance to disease persistence. Moreover, under certain conditions on the parameters, increasing the efficiency of vaccine and/or treatment has the unwanted effect of increasing the epidemic reproductive number, suggesting a wider range of diseases may become endemic due to risk-taking alone. These results indicate that the manner in which treatment/vaccine effectiveness is advertised can have an important influence on how the epidemic unfolds.     

Bursty Communication Patterns Facilitate Spreading in a Threshold-Based Epidemic Dynamics

Records of social interactions provide us with new sources of data for understanding how interaction patterns affect collective dynamics. Such human activity patterns are often bursty, i.e., they consist of short periods of intense activity followed by long periods of silence. This burstiness has been shown to affect spreading phenomena; it accelerates epidemic spreading in some cases and slows it down in other cases. We investigate a model of history-dependent contagion. In our model, repeated interactions between susceptible and infected individuals in a short period of time is needed for a susceptible individual to contract infection. We carry out numerical simulations on real temporal network data to find that bursty activity patterns facilitate epidemic spreading in our model.     

A stochastic SIS epidemic with demography: initial stages and time to extinction

We study an open population stochastic epidemic model from the time of introduction of the disease, through a possible outbreak and to extinction. The model describes an SIS (susceptible–infective–susceptible) epidemic where all individuals, including infectious ones, reproduce at a given rate. An approximate expression for the outbreak probability is derived using a coupling argument. Further, we analyse the behaviour of the model close to quasi-stationarity, and the time to disease extinction, with the aid of a diffusion approximation. In this situation the number of susceptibles and infectives behaves as an Ornstein–Uhlenbeck process, centred around the stationary point, for an exponentially distributed time before going extinct.     

Moment Equations and Dynamics of a Household SIS Epidemiological Model

An SIS epidemiological model of individuals partitioned into households is studied, where infections take place either within or between households, the latter generally happening much less frequently. The model is explored using stochastic spatial simulations, as well as mathematical models which consist of an infinite system of ordinary differential equations for the moments of the distribution describing the proportions of individuals who are infectious among households. Various moment-closure approximations are used to truncate the system of ODEs to finite systems of equations. These approximations can sometimes lead to a system of ill-behaved ODEs which predict moments which become negative or unbounded. A reparametrization of the ODEs is then developed, which forces all moments to satisfy necessary constraints.Changing the proportion of contacts within and between households does not change the endemic equilibrium, but does affect the amount of time it takes to approach the fixed point; increasing the proportion of contacts within households slows the spread of the infection toward endemic equilibrium. The system of moment equations does describe this phenomenon, although less accurately in the limit as the proportion of between-household contacts approaches zero. The results indicate that although controlling the movement of individuals does not affect the long-term frequency of an infection with SIS dynamics, it can have a large effect on the time-scale of the dynamics, which may provide an opportunity for other controls such as immunizations to be applied.     

Fatal disease and demographic Allee effect: population persistence and extinction

If a healthy stable host population at the disease-free equilibrium is subject to the Allee effect, can a small number of infected individuals with a fatal disease cause the host population to go extinct? That is, does the Allee effect matter at high densities? To answer this question, we use a susceptible–infected epidemic model to obtain model parameters that lead to host population persistence (with or without infected individuals) and to host extinction. We prove that the presence of an Allee effect in host demographics matters even at large population densities. We show that a small perturbation to the disease-free equilibrium can eventually lead to host population extinction. In addition, we prove that additional deaths due to a fatal infectious disease effectively increase the Allee threshold of the host population demographics.     

Epidemic progression on networks based on disease generation time

We investigate the time evolution of disease spread on a network and present an analytical framework using the concept of disease generation time. Assuming a susceptible–infected–recovered epidemic process, this network-based framework enables us to calculate in detail the number of links (edges) within the network that are capable of producing new infectious nodes (individuals), the number of links that are not transmitting the infection further (non-transmitting links), as well as the number of contacts that individuals have with their neighbours (also known as degree distribution) within each epidemiological class, for each generation period. Using several examples, we demonstrate very good agreement between our analytical calculations and the results of computer simulations.     

A model of the Ebola epidemics in West Africa incorporating age of infection

A model of an Ebola epidemic is developed with infected individuals structured according to disease age. The transmission of the infection is tracked by disease age through an initial incubation (exposed) phase, followed by an infectious phase with variable transmission infectiousness. The removal of infected individuals is dependent on disease age, with three types of removal rates: (1) removal due to hospitalization (isolation), (2) removal due to mortality separate from hospitalization, and (3) removal due to recovery separate from hospitalization. The model is applied to the Ebola epidemics in Sierra Leone and Guinea. Model simulations indicate that successive stages of increased and earlier hospitalization of cases have resulted in mitigation of the epidemics.     

Complex dynamics of a predator–prey–parasite system: An interplay among infection rate,predator's reproductive gain and preference

Parasites are considered as an important factor in regulating their host populations through trait-mediated effects. On the other hand, predation becomes particularly interesting in host–parasite systems because predation can significantly alter the abundance of parasites and their host population. The combined effects of parasites and predator on host population and community structure therefore may have larger effect. Different field experiments confirm that predators consume disproportionately large number of infected prey in comparison to their susceptible counterpart. There are also substantial evidences that predator has the ability to distinguish prey that have been infected by a parasite and avoid such prey to reduce fitness cost. In this paper we study the predator–prey dynamics, where the prey species is infected by some parasites and predators consume both the susceptible and infected prey with some preference. We demonstrate that complexity in such systems largely depends on the predator's selectivity, force of infection and predator's reproductive gain. If the force of infection and predator's reproductive gain are low, parasites and predators both go to extinction whatever be the predator's preference. The story may be totally different in the opposite case. Survival of species in stable, oscillatory or chaotic states, and their extinction largely depend on the predator's preference. The system may also show two coexistence equilibrium points for some parameter values. The equilibrium with lower susceptible prey density is always stable and the equilibrium with higher susceptible prey density is always unstable. These results suggest that understanding the consequences of predator's selectivity or preference may be crucial for community structure involving parasites.     

Epidemics on Networks with Large Initial Conditions or Changing Structure

In this paper we extend previous work deriving dynamic equations governing infectious disease spread on networks. The previous work has implicitly assumed that the disease is initialized by an infinitesimally small proportion of the population. Our modifications allow us to account for an arbitrarily large initial proportion infected. This helps resolve an apparent paradox in earlier work whereby the number of susceptible individuals could increase if too many individuals were initially infected. It also helps explain an apparent small deviation that has been observed between simulation and theory. An advantage of this modification is that it allows us to account for changes in the structure or behavior of the population during the epidemic.