The effect of incidence functions on the dynamics of a quarantine/isolation model with time delay

Abstract

The problem of the asymptotic dynamics of a quarantine/isolation model with time delay is considered, subject to two incidence functions, namely standard incidence and the Holling type II (saturated) incidence function. Rigorous qualitative analysis of the model shows that it exhibits essentially the same (equilibrium) dynamics regardless of which of the two incidence functions is used. In particular, for each of the two incidence functions, the model has a globally asymptotically stable disease-free equilibrium whenever the associated reproduction threshold quantity is less than unity. Further, it has a unique endemic equilibrium when the threshold quantity exceeds unity. For the case with the Holling type II incidence function, it is shown that the unique endemic equilibrium of the model is globally asymptotically stable for a special case. The permanence of the disease is also established for the model with the Holling type II incidence function. Furthermore, it is shown that adding time delay to and/or replacing the standard incidence function with the Holling type II incidence function in the corresponding autonomous quarantine/isolation model with standard incidence (considered in Safi and Gumel (2010) [10]) does not alter the qualitative dynamics of the autonomous system (with respect to the elimination or persistence of the disease). Finally, numerical simulations of the model with standard incidence show that the disease burden decreases with increasing time delay (incubation period). Furthermore, models with time delay seem to be more suitable for modeling the 2003 SARS outbreaks than those without time delay.

Keywords: Equilibria; Isolation; Non-linear incidence; Quarantine; Stability; Time delay.