A simple SIS epidemic model with a backward bifurcation

Abstract

It is shown that an SIS epidemic model with a non-constant contact rate may have multiple stable equilibria, a backward bifurcation and hysteresis. The consequences for disease control are discussed. The model is based on a Volterra integral equation and allows for a distributed infective period. The analysis includes both local and global stability of equilibria.

Dynamical behaviour of epidemiological models with sub-optimal immunity and nonlinear incidence

“…Moreover, even if theoretically possible, it is unlikely that backwards bifurcations will be noticeable in the SI R model. In the case of the SI S model, oscillations cannot appear and this agrees with the results previously obtained in [24]. Figure 4 shows the equilibrium surfaces for the (T I ) system.…”

“…Mathematical models with nonlinear force of infection have been studied by many authors ( [14,22,12,24,1]). In several of these papers (see for example [14,22]) simple models with a particular nonlinear force of infection are proposed and analysed.…”

“…The authors analyse the model and give some examples in which backwards bifurcations or oscillations appear. In [24] a SI S model with a general nonlinear incidence function is proposed and it is proved the existence of multiple stable equilibria, backwards bifurcations and hysteresis, but in this case oscillations do not exist. The model consists on a Volterra integral equation which includes ordinary and delay differential equations as special cases.…”

“…In diseases that do not elicit significant protective immunity (such as chronic infections or in the SI S model), nonlinear incidence functions tend to generate bistable behaviour as a positive feedback maintains high or low endemicities depending on the initial infection density ( [17,5,24]). The same mechanism is also capable of sustaining a periodic epidemic behaviour ( [1,22]) if there is significant time delay in the replenishment of the susceptibility pool (such as in the SI R model where susceptibles are supplied by births).…”

Parameter Estimation of Some Epidemic Models. The Case of Recurrent Epidemics Caused by Respiratory Syncytial Virus

“…In the context of incidence rate, Capasso and Serio (1978), Gomes et al (2005), Hethcote et al (1989), Hethcote and van den Driessche (1991), Kyrychko and Blyuss (2005), Li and Muldowney (1995), Ruan and Wang (2003), van den Driessche and Watmough (2000), Wang (2006) and Xiao and Ruan (2007) among others, have studied and biologically interpreted epidemic models with nonlinear incidence rates departing from standard incidence rate. Korobeinikov and Maini (2005) established stability theorems for generalized incidence rates under standard biological hypotheses.…”