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Some results for an SEIR epidemic model with density dependence in the death rate. (English) Zbl 0805.92025
The model investigated in this paper combines the features of models studied earlier by the author [IMA J. Math. Appl. Med. Biol. 7, No. 1, 1- 26 (1990; Zbl 0751.92014)] and {\it F. Rinaldi} [ibid., No. 2, 69-75 (1990; Zbl 0728.92020)], including both a class of latent infected individuals and a density-dependent death rate. The spread of the infection is described by a system of ordinary differential equations. The model assumes that the number of contacts of a single individual per unit time is proportional to the number of individuals in the population (not constant) and that an additional death rate is suffered by infective individuals. It takes into account also vaccination of susceptible individuals. First the results are discussed for a constant death rate and second for a per capita death rate which depends on the number of individuals in the population. It is found that although the equilibrium results have the same qualitative form the local stability results are different from the results obtained earlier for the more specific models. There are three possible steady states: one where the population is extinct, one where the population maintains itself at a constant level and the disease is extinct, and one with disease present. It is possible for this third equilibrium to exist and be unstable, and the present model also allows for three equilibria to exist and each being unstable. Numerical work and simulation show that we can have cycles of infection prevalence with increasing amplitude, a constant amplitude, and a decreasing amplitude, depending on the parameter values of the model. In this paper the term `incubation period’ should read `latent period’, `incubating individuals’ should read `latent infected individuals’, and `infected individuals’ should read `infective individuals’.