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Spreading speed for an epidemic model with time delay in a patchy environment. (English) Zbl 1480.34102

The system of differential delay equations is considered \[ \begin{array}{rcl} \frac{S_j(t)}{dt} & = & d\,\mathcal{D}_j[S](t)+\overline{\mu}(1-S_j(t))-\beta\frac{S_j(t)I_j(t)}{S_j(t)+I_j(t)} \\ \frac{I_j(t)}{dt} & = & D\,\mathcal{D}_j[I](t)-(r+\overline{\mu})I_j(t)+\frac{\mu}{2\pi}\sum_{m=-\infty}^{\infty}\tilde{\beta}(j+m)\left[\frac{\beta S_mI_m}{S_m+I_m}\right](t-\tau) \end{array} \tag{1} \] where \(S_j (t)\) and \(I_j (t), j\in\mathbb Z,\) represent the densities of the susceptible and infective individuals at time \(t\) in the \(j\)-th patch, the positive constants \(d\) and \(D\) stand for their respective diffusion rates. Additionally, the coefficient \(\tilde{\beta}(m)\ge0\) satisfies \(\sum_{-\infty}^{\infty}\tilde{\beta}(m)=1\), the operator \(\mathcal{D}\) is defined as \({\mathcal D}_j [f](t) = f_{j+1}(t) + f_{j-1}(t) - 2f_{j} (t)\), and \(\tau, \beta, \overline{\mu}, r\) are positive constants.
An estimate for the spreading speed in system (1) is given. The analysis is done by using a simpler auxiliary system and estimating the lower and upper bounds of the \(S\)-component. It is also based on some previously known results.

MSC:

34K31 Lattice functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D30 Epidemiology
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