Modelling epidemics with variable contact rates
We are concerned with modelling an infectious disease in a population where the infected people suffer a disease-related mortality. We present a simple mathematical model for the effect of an endemic disease on a population. We use a compartmental model which divides the population into classes such as susceptible, infected, and immune individuals and use differential equations to model the spread of the disease between these classes. This model would be suitable for a disease where a typical individual starts off susceptible, at some stage catches the disease, and after a short infectious period becomes permanently immune. Such diseases are typically viral diseases and examples include measles, hepatitis, influenza, mumps, and chicken-pox. The paper aims to examine epidemic models where both the contact rate and the death rate depend on population density. We attempt to answer questions such as what conditions are necessary for the disease to persist and what are the implications of density dependence in these parameters for the growth or decay of the population in the presence of disease. We ask whether disease can turn a growing population into a steady one or even reverse its growth. Vaccination is also an important practical consideration in the control of a disease, so we aim also to ask what proportion of a population needs to be vaccinated to eliminate a disease. We also aim to compare our results with the results of simpler models. We first outline the basic compartmental differential equation model which we shall use and then perform an equilibrium and stability analysis for the model where the contact rate is independent of the population density but the background death rate is an increasing function of population density. We then reformulate the model in terms of the proportions of individuals susceptible, infected and immune and perform an equilibrium and stability analysis on this model. With this proportional model we use the methods of \textit{S. Busenberg} and \textit{P. Van den Driessche} [J. Math. Biol. 28, No. 3, 257-270 (1990; Zbl 0725.92021)]\ to extend the local stability results to global results and hence prove global results for the original model. Then we discuss how the results can be modified to the situation where the contact rate \(\beta (N)\) is a continuous monotone increasing function of the population size \(N\). Next we present some numerical simulations to back up our analytic results and finally we briefly summarize and discuss some of the results presented in this paper.
scientific article; zbMATH DE number 756548