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Nonlinear incidence and stability of infectious disease models. (English) Zbl 1076.92048
Summary: We consider the impact of the form of the nonlinearity of the infectious disease incidence rate on the dynamics of epidemiological models. We consider a very general form of the nonlinear incidence rate (in fact, we assumed that the incidence rate is given by an arbitrary function $$f (S, I, N)$$ constrained by a few biologically feasible conditions) and a variety of epidemiological models. We show that under the constant population size assumption, these models exhibit asymptotically stable steady states.
Precisely, we demonstrate that the concavity of the incidence rate with respect to the number of infective individuals is a sufficient condition for stability. If the incidence rate is concave in the number of the infectives, the models we consider have either a unique and stable endemic equilibrium state or no endemic equilibrium state at all; in the latter case the infection-free equilibrium state is stable. For the incidence rate of the form $$g(I)h(S)$$, we prove global stability, constructing a Lyapunov function and using the direct Lyapunov method. It is remarkable that the system dynamics is independent of how the incidence rate depends on the number of susceptible individuals. We demonstrate this result using a SIRS model and a SEIRS model as case studies. For other compartment epidemic models, the analysis is quite similar, and the same conclusion, namely stability of the equilibrium states, holds

##### MSC:
 92D30 Epidemiology 35Q80 Applications of PDE in areas other than physics (MSC2000) 34D23 Global stability of solutions to ordinary differential equations
##### Keywords:
direct Lyapunov method; endemic equilibrium state
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