C. Connell McCluskey

GLOBAL RESULTS FOR AN EPIDEMIC MODEL WITH VACCINATION THAT EXHIBITS BACKWARD BIFURCATION

Vaccination of both newborns and susceptibles is included in a transmission model for a disease that confers immunity. The interplay of the vaccination strategy together with the vaccine efficacy and waning is studied. In particular, it is shown that a backward bifurcation leading to bistability can occur. Under mild parameter constraints, compound matrices are used to show that each orbit limits to an equilibrium. In the case of bistability, this global result requires a novel approach since there is no compact absorbing set.

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.

GLOBAL STABILITY OF AN SIR 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.

TWO-GROUP INFECTION AGE MODEL INCLUDING AN APPLICATION TO NOSOCOMIAL INFECTION ∗

In this article we analyze the global asymptotic behavior of a two-group SI (susceptible--infected) epidemic model with age of infection. We prove that the model exhibits the traditional threshold behavior where the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than one, and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than one. We conclude the paper by presenting an application to nosocomial infections. Moreover some numerical simulations are presented for this application.

Global stability for an SEI model of infectious disease with immigration

We study an SEI model of disease transmission with immigration into all three classes. For incidence, we allow for a nonlinear response to the number of infectives, including mass action and saturating incidence as special cases. There is no disease-free equilibrium and therefore no basic reproduction number. For all parameter values, the only equilibrium is an endemic equilibrium. Using a Lyapunov function, we show that this equilibrium is globally asymptotically stable.

Delay versus age-of-infection – Global stability

Abstract We consider an SIR model of disease transmission which has been formulated with delay in order to give a fixed duration of infectiousness. A Lyapunov functional that has worked for similar models does not work here. Instead, the model is shown to be a consequence of an age-of-infection model for which the same class of Lyapunov functionals has worked, resolving the global stability.

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.

Spatial spread of an epidemic through public transportation systems with a hub.

Abstract This article investigates an epidemic spreading among several locations through a transportation system, with a hub connecting these locations. Public transportation is not only a bridge through which infections travel from one location to another but also a place where infections occur since individuals are typically in close proximity to each other due to the limited space in these systems. A mathematical model is constructed to study the spread of an infectious disease through such systems. A variant of the next generation method is proposed and used to provide upper and lower bounds of the basic reproduction number for the model. Our investigation indicates that increasing transportation efficiency, and improving sanitation and ventilation of the public transportation system decrease the chance of an outbreak occurring. Moreover, discouraging unnecessary travel during an epidemic also decreases the chance of an outbreak. However, reducing travel by infectives while allowing susceptibles to travel may not be enough to avoid an outbreak.

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 epidemic model with applications to feline infectious peritonitis and tuberculosis

Abstract 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.