Asymptotic Profiles of the Steady States for an SIS Epidemic Patch Model

Abstract

Spatial heterogeneity, habitat connectivity, and rates of movement can have large impacts on the persistence and extinction of infectious diseases. These factors are shown to determine the asymptotic profile of the steady states in a frequency-dependent SIS (susceptible-infectedsusceptible) epidemic model with n patches in which susceptible and infected individuals can both move between patches. Patch differences in local disease transmission and recovery rates characterize whether patches are low-risk or high-risk, and these differences collectively determine whether the spatial domain, or habitat, is low-risk or high-risk. The basic reproduction number R 0 for the model is determined. It is then shown that when the disease-free equilibrium is stable (R 0 < 1) it is globally asymptotically stable, and that when the disease-free equilibrium is unstable (R 0 > 1) there exists a unique endemic equilibrium. Two main theorems link spatial heterogeneity, habitat connectivity, and rates of movement to disease persistence and extinction. The first theorem relates the basic reproduction number to the heterogeneity of the spatial domain. For low-risk domains, the disease-free equilibrium is stable (R 0 < 1) if and only if the mobility of infected individuals lies above a threshold value, but for high-risk domains, the disease-free equilibrium is always unstable (R 0 > 1). The second theorem states that when the endemic equilibrium exists, it tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. This limiting disease-free equilibrium has a positive number of susceptible individuals on all low-risk patches and can also have a positive number of susceptible individuals on some, but not all, high-risk patches. Sufficient conditions for whether high-risk patches in the limiting disease-free equilibrium have susceptible individuals or not are given in terms of habitat connectivity, and these conditions are illustrated using numerical examples. These results have important implications for disease control.

Limiting the spread of disease through altered migration patterns

“…Allen et al [2] studied a frequency-dependent SIS metapopulation model, which in our notation is given by…”

A model for the spread of an SIS epidemic in a human population

“…They showed that, while the population is at an endemic level, increased travel rates of infectives can cause disease extinction in all patches. In a similar setting, Allen et al [3] showed for a frequency-dependent SIS deterministic model that, while the population is at an endemic level and if infectious individuals travel between the patches but the rate of travel for susceptible individuals approaches 0, then, contrary to what is expected, the endemic equilibrium approaches a disease-free equilibrium. McCormack and Allen [141] studied an SIR and an SIS model in both deterministic and stochastic settings, assuming disease independent travel rates.…”

“…The major drawback of these models is the absence of mobility, in both deterministic and stochastic settings. This is overcome for deterministic models in the studies made by Bailey [15,Section 7.33], Hethcote [98], Sattenspeil and Dietz [187], Allen et al [3], McCormack and Allen [141], van den Driessche [202], Salmani and van den Driessche [183], Arino and van den Driessche [9,10,11], Arino et al [7,8], Wang and Zhao [210], Wang and Mulone [209], Jin and Wang [107], Kuniya and Muroya [124], and Muroya et al [145]. The authors who have contributed on the stochastic side include Ball [22], Ball and Clancy [24,25], Clancy [50,52], Keeling et al [112,114], Sani et al [185], Lahodny Jr and Allen [131] and Neal [152].…”

“…Consequently, the infection rate of a susceptible individual at patch j and time t is β j m j /N, where β j = κ j p > 0 is the disease transmission rate. In contrast to the above assumption on the contact rate, the SIS model studied in [131,3] has a constant contact rate, which leads their model to have frequency-dependent transmission rates.…”

“…The model formulated in this study is closely related to the frequency-dependent SIS models studied in [131,3]; I will say more about the connection as I build the model.…”