Localized outbreaks in an S-I-R model with diffusion

Abstract

We investigate an SIRS epidemic model with spatial diffusion and nonlinear incidence rates. We show that for small diffusion rate of the infected class D I , the infected population tends to be highly localized at certain points inside the domain, forming K spikes. We then study three distinct destabilization mechanisms, as well as a transition from localized spikes to plateau solutions. Two of the instabilities are due to coarsening (spike death) and self-replication (spike birth), and have well-known analogues in other reaction-diffusion systems such as the Schnakenberg model. The third transition is when a single spike becomes unstable and moves to the boundary. This happens when the diffusion of the recovered class, D R becomes sufficiently small. In all cases, the stability thresholds are computed asymptotically and are verified by numerical experiments. We also show that the spike solution can transit into an plateau-type solution when the diffusion rates of recovered and susceptible class are sufficiently small. Implications for disease spread and control through quarantine are discussed.

Effects of social distancing and isolation on epidemic spreading modeled via dynamical density functional theory

“…Radial spreading is even observed in modern traffic networks on global scales, provided these are described in terms of effective distance 89 (see the first part of the methods section). Moreover, spatial pattern formation is known in other contexts from reaction–diffusion SIR models 90 , including variants where cross-diffusion is used to describe susceptibles avoiding infected persons 34 . Empirically, hotspots (caused by other mechanisms) were observed, e.g., for the spread of HIV in South Africa 91 or Dengue in Thailand 92 .…”

“…In the latter case, the diffusion constant D I of infected persons is reduced. This is a typical way of incorporating quarantine in diffusion models for epidemic spreading 90 .…”

“…Hence, a reduced amount of motion of infected individuals as a consequence of self-isolation can be implemented by decreasing their temperature. Also reducing Γ I can be appropriate if the disease itself affects the persons’ ability to move 90 , e.g., by affecting their legs or causing particularly severe symptoms.…”

Modelagem dinâmica da COVID-19 com aplicação a algumas cidades brasileiras

“…Zhen et al (2018) propõem modificações em um modelo SIR para incorporar a difusão do Laplaciano e o atraso espaço-temporal para modelar a transmissão de doenças transmissíveis. Gai et al (2020) investigam o modelo epidêmico SIRS com difusão espacial e taxas de incidência não lineares. Chowell et al (2020) mostraram estudos usando o Modelo de Crescimento Generalizado (GMG) e o Modelo Logístico Generalizado (GLM) aplicado a uma epidemia de Ebola.…”

“…Finalmente, ressalta-se que foram encontrados na literatura vários trabalhos que utilizam também variações do modelo SIR, na modelagem da COVID-19: além de Giordano et al 2020e Gai et al (2020) já referenciados, há os trabalhos do grupo Gdispen (2020), Franco (2020), Reis et al 2020, Schulz et al 2020, Wu et al (2020) e Villaverde e Jones (2020).…”