Periodic Incidence in a Discrete-Time SIS Epidemic Model - Mathematical Methods and Models in Biomedicine - page 398

mechanisms that have been used to generate oscillations in epidemiological models
include external forcing, seasonality, time-dependent coefficients, periodic inci-
dence, age-structure, time-delay, and migration [ 1 , 4 , 5 , 8 - 12 , 16 - 19 , 22 , 26 , 28 - 31 ].
Following Castillo-Chavez and Yakubu, we assume that a disease invades
and subdivides the target population into two compartments: susceptibles
(non-infectives) and infectives. Prior to the time of disease invasion, the host
population is assumed to be asymptotically constant via either Beverton-Holt
or constant recruitment functions or growing geometrically or oscillatory via
the Ricker recruitment function [ 6 - 11 , 13 - 15 , 17 , 18 , 20 , 21 , 23 - 25 , 27 , 29 , 30 ].
The transition from susceptible to infective is a function of the T-periodic contact
rate β t = β t + T and the proportion of infectives (prevalence) in the population.

To simplify our analysis, we assume that the disease is nonfatal and individuals
(infected and susceptible) have equal probability of surviving one generation. Mild
viral infections, such as most infections from rhinoviruses (causative agents of the
common cold) are examples of such nonfatal infections. Our primary focus is on
the impact of periodic contact rate, asymptotically constant, geometric growth and
periodic demographic dynamics on the persistence or control of infectious diseases.
For many epidemiological models, the threshold parameter is the basic repro-
duction number,

ℜ 0 and used it to predict disease
persistence or extinction when the host population dynamics are asymptotically
constant or when the host population is growing at a geometric rate. That is,
potentially, by developing strategies that reduce

ℜ 0 . In this chapter, we compute

ℜ 0 to values less than 1, we can
combat infectious diseases with periodic incidence, when the host population dy-
namics is either asymptotically constant or under geometric growth. Castillo-Chavez
and Yakubu, in [ 8 , 10 ], showed that the demographic equation drives the disease
dynamics in discrete-time SIS models with constant contact rate. We illustrate that
when the host population is asymptotically constant, it is possible for the infective
and susceptible populations to exhibit oscillatory dynamics. That is, under periodic
contact rate, the demographic dynamics do not drive the disease dynamics. When
the demographic dynamics are oscillatory, we explore the relationship between
the demographic equation and the epidemic process as the demographic model
undergoes period-doubling bifurcations.
The chapter is organized as follows. In Sect. 2 , we introduce the demographic
equation for the study. The equation, a deterministic discrete-time model, describes
the dynamics of the host population before disease invasion [ 2 , 3 ]. The main model,
a discrete-time SIS epidemic model with periodic contact rate, is constructed in
Sect. 3 of the chapter. In Sect. 4 , the basic reproductive number, ℜ 0 ,isintro-

duced and used to predict the (uniform) persistence or extinction of the infective
population, where the population dynamics are asymptotically constant or under
geometric demographic dynamics. In Sect. 5 , we illustrate that it is possible for the
demographic dynamics to drive both the S - dynamics and I - dynamics as it undergoes
period-doubling bifurcations. The implications of our results are discussed in
Sect. 6 .