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Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models. (English) Zbl 1161.92046
Summary: The dynamics of simple discrete-time epidemic models without disease-induced mortality are typically characterized by global transcritical bifurcations. We prove that in corresponding models with disease-induced mortality a tiny number of infectious individuals can drive an otherwise persistent population to extinction. Our model with disease-induced mortality supports multiple attractors. In addition, we use a W. E. Ricker [Stock recruitment. J. Fish Res. Board Canada II 5, 559–623 (1954)] recruitment function in an SIS model and obtain a three component discrete Hopf (Neimark-Sacker) cycle attractor coexisting with a fixed point attractor. The basin boundaries of the coexisting attractors are fractal in nature, and the example exhibits sensitive dependence of the long-term disease dynamics on the initial conditions. Furthermore, we show that in contrast to corresponding models without disease-induced mortality, the disease-free state dynamics do not drive the disease dynamics.
MSC:
92D30Epidemiology
37G15Bifurcations of limit cycles and periodic orbits
37G35Attractors and their bifurcations
39A11Stability of difference equations (MSC2000)
37N25Dynamical systems in biology
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