Long-time dynamics of an epidemic model with nonlocal diffusion and free boundaries

Abstract

<abstract><p>In this paper, we consider a reaction-diffusion epidemic model with nonlocal diffusion and free boundaries, which generalises the free-boundary epidemic model by Zhao et al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup> by including spatial mobility of the infective host population. We obtain a rather complete description of the long-time dynamics of the model. For the reproduction number $ R_0 $ arising from the corresponding ODE model, we establish its relationship to the spreading-vanishing dichotomy via an associated eigenvalue problem. If $ R_0 \le 1 $, we prove that the epidemic vanishes eventually. On the other hand, if $ R_0 > 1 $, we show that either spreading or vanishing may occur depending on its initial size. In the case of spreading, we make use of recent general results by Du and Ni <sup>[<xref ref-type="bibr" rid="b2">2</xref>]</sup> to show that finite speed or accelerated spreading occurs depending on whether a threshold condition is satisfied by the kernel functions in the nonlocal diffusion operators. In particular, the rate of accelerated spreading is determined for a general class of kernel functions. Our results indicate that, with all other factors fixed, the chance of successful spreading of the disease is increased when the mobility of the infective host is decreased, reaching a maximum when such mobility is 0 (which is the situation considered by Zhao et al. <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>).</p></abstract>

Long-time dynamics of a nonlocal epidemic model with free boundaries: spreading-vanishing dichotomy

“…These results were extended by Zhao et al [33] and Du et al [17] to the case of (1.6) with d 2 = ρ = 0, and by Chang and Du [14] for (1.6) with d 2 > 0, ρ ≥ 0 and the term a 12 h(t) g(t)…”

“…K(x − y)v(y, t)dy replaced by a 12 v(x, t). Moreover, the rate of accelerated spreading was given in [14] when…”

Threshold dynamics of a nonlocal dispersal SIS epidemic model with free boundaries

“…In addition, scholars have extensively investigated infectious disease models with nonlocal diffusion, such as the West Nile virus model [15], SIS epidemic model [17,44], and SIR reaction-diffusion model [18,45]. For other epidemic models with nonlocal diffusion, see references [9,39,41] and references therein.…”

Rate of accelerated expansion of the epidemic region in a nonlocal epidemic model with free boundaries

“…The basic model (1.4) has been refined and further developed along a number of different lines by many authors (e.g. [1,6,8,12,[14][15][16][19][20][21][22][23][24]), and we refer to the introduction section of [17] for a detailed account. We note in particular that (1.1) is capable of describing the evolution of the epidemic region [g(t), h(t)] as time changes.…”