Longtime Behavior of Solutions of a SIS Epidemiological Model.

This paper investigates the longtime behavior of solutions of a susceptible-infectious-susceptible (SIS) epidemiological model, proposed to explain an epidemiological phenomenon that pathogen spread does not necessarily keep pace with its host invasion. When the rate of transmission of disease is small, the infected population will vanish. When the rate is of moderate size, a local infection of disease will spread with a speed that is smaller than that of the host population. For large disease transmission rates, the infection can keep pace with the host invasion, provided that the host does not propagate too fast. This paper also modifies a method originated from [F. Hamel et al., Netw. Heterog. Media, 8 (2013), pp. 275-289] to prove the logarithmic lag of the front of the infected population from that of the minimum speed traveling wave of a reduced equation obtained by setting the host population to its capacity. Our analysis can be used to study monostable scalar equations in inhomogeneous media.