Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model

In this paper we consider the phenomenon of backward bifurcation in epidemic modelling illustrated by an extended model for Bovine Respiratory Syncytial Virus (BRSV) amongst cattle. In its simplest form, backward bifurcation in epidemic models usually implies the existence of two subcritical endemic equilibria for R 0 < 1, where R 0 is the basic reproductive number, and a unique supercritical endemic equilibrium for R 0 > 1. In our three-stage extended model we find that more complex bifurcation diagrams are possible. The paper starts with a review of some of the previous work on backward bifurcation then describes our three-stage model. We give equilibrium and stability results, and also provide some biological motivation for the model being studied. It is shown that backward bifurcation can occur in the three-stage model for small b, where b is the common per capita birth and death rate. We are able to classify the possible bifurcation diagrams. Some realistic numerical examples are discussed at the end of the paper, both for b small and for larger values of b

Assessing the effects of multiple infections and long latency in the dynamics of tuberculosis

In order to achieve a better understanding of multiple infections and long latency in the dynamics of Mycobacterium tuberculosis infection, we analyze a simple model. Since backward bifurcation is well documented in the literature with respect to the model we are considering, our aim is to illustrate this behavior in terms of the range of variations of the model's parameters. We show that backward bifurcation disappears (and forward bifurcation occurs) if: (a) the latent period is shortened below a critical value; and (b) the rates of super-infection and re-infection are decreased. This result shows that among immunosuppressed individuals, super-infection and/or changes in the latent period could act to facilitate the onset of tuberculosis. When we decrease the incubation period below the critical value, we obtain the curve of the incidence of tuberculosis following forward bifurcation; however, this curve envelops that obtained from the backward bifurcation diagram

Dynamic phenomena arising from an extended Core Group model

In order to obtain a reasonably accurate model for the spread of a particular infectious disease through a population, it may be necessary for this model to possess some degree of structural complexity. Many such models have, in recent years, been found to exhibit a phenomenon known as backward bifurcation, which generally implies the existence of two subcritical endemic equilibria. It is often possible to refine these models yet further, and we investigate here the influence such a refinement may have on the dynamic behaviour of a system in the region of the parameter space near R0 = 1. We consider a natural extension to a so-called core group model for the spread of a sexually transmitted disease, arguing that this may in fact give rise to a more realistic model. From the deterministic viewpoint we study the possible shapes of the resulting bifurcation diagrams and the associated stability patterns. Stochastic versions of both the original and the extended models are also developed so that the probability of extinction and time to extinction may be examined, allowing us to gain further insights into the complex system dynamics near R0 = 1. A number of interesting phenomena are observed, for which heuristic explanations are provided