The effect of immigration of infectives on disease-free equilibria.

Many disease transmission models exhibit a threshold behaviour based on the basic reproduction number [Formula: see text], where the disease-free equilibrium is locally asymptotically stable if [Formula: see text] and unstable if [Formula: see text]. However, if a system includes immigration of infected individuals, then there is no disease-free equilibrium. We consider how the disease-free equilibrium moves as the level of immigration of infected individuals is increased from 0, finding, under mild assumptions, that the disease-free equilibrium becomes an endemic equilibrium if [Formula: see text] and leaves the biologically relevant space (by having at least one coordinate become negative) if [Formula: see text].

Global stability for an SEI model of infectious disease with age structure and immigration of infecteds.

We study a model of disease transmission with continuous age-structure for latently infected individuals and for infectious individuals and with immigration of new individuals into the susceptible, latent and infectious classes. The model is very appropriate for tuberculosis. A Lyapunov functional is used to show that the unique endemic equilibrium is globally stable for all parameter values.

Disease dynamics for the hometown of migrant workers.

A recent paper by L. Wang, X. Wang J. Theoret. Biol. 300:100--109 (2012) formulated and studied a delay differential equation model for disease dynamics in a region where a portion of the population leaves to work in a different region for an extended fixed period. Upon return, a fraction of the migrant workers have become infected with the disease. The global dynamics were not fully resolved in that paper, but are resolved here. We show that for all parameter values and all delays, the unique equilibrium is globally asymptotically stable, implying that the disease will eventually reach a constant positive level in the population.

Global stability for an epidemic model with applications to feline infectious peritonitis and tuberculosis.

A general compartmental model of disease transmission is studied. The generality comes from the fact that new infections may enter any of the infectious classes and that there is an ordering of the infectious classes so that individuals can be permitted (or not) to pass from one class to the next. The model includes staged progression, differential infectivity, and combinations of the two as special cases. The exact etiology of feline infectious peritonitis and its connection to coronavirus is unclear, with two competing theories - mutation process vs multiple virus strains. We apply the model to each of these theories, showing that in either case, one should expect traditional threshold dynamics. A further application to tuberculosis with multiple progression routes through latency is also presented.

Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes.

We study a model of disease transmission with continuous age-structure for latently infected individuals and for infectious individuals. The model is very appropriate for tuberculosis. Key theorems, including asymptotic smoothness and uniform persistence, are proven by reformulating the system as a system of Volterra integral equations. The basic reproduction number R0 is calculated. For R0 < 1, the disease-free equilibrium is globally asymptotically stable. For R0 > 1, a Lyapunov functional is used to show that the endemic equilibrium is globally stable amongst solutions for which the disease is present. Finally, some special cases are considered.

Attractivity of coherent manifolds in metapopulation models.

The likelihood that coupled dynamical systems will completely synchronize, or become "coherent", is often of great applied interest. Previous work has established conditions for local stability of coherent solutions and global attractivity of coherent manifolds in a variety of spatially explicit models. We consider models of communities coupled by dispersal and explore intermediate regimes in which it can be shown that states in phase space regions of positive measure are attracted to coherent solutions. Our methods yield rigorous and practically useful coherence criteria that facilitate useful analyses of ecological and epidemiological problems.

Global stability of an epidemic model with delay and general nonlinear incidence.

An SIR model with distributed delay and a general incidence function is studied. Conditions are given under which the system exhibits threshold behaviour: the disease-free equilibrium is globally asymptotically stable if R0 is less than 1 and globally attracting if R0=1; if R0 is larger than 1, then the unique endemic equilibrium is globally asymptotically stable. The global stability proofs use a Lyapunov functional and do not require uniform persistence to be shown a priori. It is shown that the given conditions are satisfied by several common forms of the incidence function.

Effect of a sharp change of the incidence function on the dynamics of a simple disease.

We investigate two cases of a sharp change of incidencec functions on the dynamics of a susceptible-infective-susceptible epidemic model. In the first case, low population levels have mass action incidence, while high population levels have proportional incidence, the switch occurring when the total population reaches a certain threshold. Using a modified Dulac theorem, we prove that this system has a single equilibrium which attracts all solutions for which the disease is present and the population remains bounded. In the second case, an increase of the number of infectives leads to a mass action term being added to a standard incidence term. We show that this allows a Hopf bifurcation to occur, with periodic orbits being generated when a locally asymptotically stable equilibrium loses stability.

Global stability for an SEIR epidemiological model with varying infectivity and infinite delay.

A recent paper (Math. Biosci. and Eng. (2008) 5:389-402) presented an SEIR model using an infinite delay to account for varying infectivity. The analysis in that paper did not resolve the global dynamics for R0 >1. Here, we show that the endemic equilibrium is globally stable for R0 >1. The proof uses a Lyapunov functional that includes an integral over all previous states.

An sveir model for assessing potential impact of an imperfect anti-sars vaccine.

The control of severe acute respiratory syndrome (SARS), a fatal contagious viral disease that spread to over 32 countries in 2003, was based on quarantine of latently infected individuals and isolation of individuals with clinical symptoms of SARS. Owing to the recent ongoing clinical trials of some candidate anti-SARS vaccines, this study aims to assess, via mathematical modelling, the potential impact of a SARS vaccine, assumed to be imperfect, in curtailing future outbreaks. A relatively simple deterministic model is designed for this purpose. It is shown, using Lyapunov function theory and the theory of compound matrices, that the dynamics of the model are determined by a certain threshold quantity known as the control reproduction number (R(v)). If R(v) =/< 1, the disease will be eliminated from the community; whereas an epidemic occurs if R(v) > 1. This study further shows that an imperfect SARS vaccine with infection-blocking efficacy is always beneficial in reducing disease spread within the community, although its overall impact increases with increasing efficacy and coverage. In particular, it is shown that the fraction of individuals vaccinated at steady-state and vaccine efficacy play equal roles in reducing disease burden, and the vaccine must have efficacy of at least 75% to lead to effective control of SARS (assuming R(0) = 4). Numerical simulations are used to explore the severity of outbreaks when R(v) > 1.

Lyapunov functions for tuberculosis models with fast and slow progression.

The spread of tuberculosis is studied through two models which include fast and slow progression to the infected class. For each model, Lyapunov functions are used to show that when the basic reproduction number is less than or equal to one, the disease-free equilibrium is globally asymptotically stable, and when it is greater than one there is an endemic equilibrium which is globally asymptotically stable.

Global analysis of competition for perfectly substitutable resources with linear response.

We study a model of the chemostat with two species competing for two perfectly substitutable resources in the case of linear functional response. Lyapunov methods are used to provide sufficient conditions for the global asymptotic stability of the coexistence equilibrium. Then, using compound matrix techniques, we provide a global analysis in a subset of parameter space. In particular, we show that each solution converges to an equilibrium, even in the case that the coexistence equilibrium is a saddle. Finally, we provide a bifurcation analysis based on the dilution rate. In this context, we are able to provide a geometric interpretation that gives insight into the role of the other parameters in the bifurcation sequence.

Implication of Ariaal sexual mixing on gonorrhea.

Recent research on sexual mixing in populations of sub-Saharan Africa raises the question as to whether STDs can persist in these populations without the presence of a core group. A mathematical model is constructed for the spread of gonorrhea among the Ariaal population of Northern Kenya. A formula for the basic reproduction number R(0) (the expected number of secondary infections caused by a single new infective introduced into a susceptible population) is determined for this population in the absence of a core group. Survey data taken in 2003 on sexual behavior from the Ariaal population are used in the model which is formulated for their age-set system including four subpopulations: single and married, female and male. Parameters derived from the data, and other information from sub-Saharan Africa are used to estimate R(0). Results indicate that, even with the elevating effect of the age-set system, the disease should die out since R(0) < 1. Thus, the persistence of gonorrhea in the population must be due to factors not included in the model, for example, a core group of commercial sex workers or concurrent partnerships.

A model of HIV/AIDS with staged progression and amelioration.

An epidemic model with multiple stages of infection is studied. The model allows for infected individuals to move from advanced stages of infection back to less advanced stages of infection. A threshold parameter which determines the local stability of the disease free equilibrium is found and interpreted. Under conditions on the parameters, global stability is demonstrated using techniques involving compound matrices.