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Spreading speed and traveling waves for a multi-type SIS epidemic model. (English) Zbl 1126.35080

The authors consider a multiple-type SIS epidemic model that is given by \[ \frac{\partial y_i(x,t)}{\partial t}=(1-y_i(x,t))\sum_{j=1}^n\sigma_j\lambda_{ij}\int_\mathbb R y_j(x-u,t)p_{ij}(u) \,du-\mu_iy_i(x,t) \] \(1\leq i\leq n\), where \(y_i(t)\) is the proportion of individuals for the \(i\)th population at point \(x\) who where infected by time \(t\), \(\mu_i\geq 0\) is the combined death/emigration/recovery rate for infectious individuals, \(\sigma_i\geq 0\) is the population size of the \(i\)th population, \(\lambda_{ij}\geq 0\) is the infection rate of a type \(i\) susceptible by a type \(j\) infectious individual, and \(p_{ij}(u)\) is the corresponding contact distribution. They use the theory of asymptotic speeds of spread and monotone traveling waves for monotone semiflows to obtain the minimal spreading speed \(c^\ast\) for the given model. Furthermore they establish the existence of monotone traveling waves connecting the disease-free and endemic equilibira for \(c\geq c^\ast\). The existence result is derived by some comparison arguments.

MSC:

35Q80 Applications of PDE in areas other than physics (MSC2000)
92D30 Epidemiology
35R10 Partial functional-differential equations
92D25 Population dynamics (general)
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