Periodic Incidence in a Discrete-Time SIS Epidemic Model - Mathematical Methods and Models in Biomedicine - page 411

1.2

[6.0, 1.2]

I-Dynamics

k

6

[1.0, 0.0]

Fig. 4 In Example 5.1 , the infected population dynamics follows the total population dynamics
and undergoes period-doubling bifurcation route to chaos as k is varied between 1 and 6 . On the

horizontal axis, 1 ≤ k ≤ 6 and on the vertical axis, 0 ≤ I ≤ 1 . 2

In Example 5.1 , the recruitment function is the Ricker model and, as in Example 4.1 ,
infection is modeled as a Poisson process [ 8 - 11 ]. When 0

2
1 − γ =

20
9 ,

< k < then

N ∞ = 1. As guaranteed by Theorem 4.1 , the disease persists

(see Fig. 3 ). As k is varied between 2 and 6, both the S -dynamics and I -dynamics
follow the N -dynamics as it undergoes period-doubling bifurcations route to chaos.
In Figs. 2 - 4 ,when5

k and ℜ 0 = 7 . 416 >

by zooming on the graphs, we obtain that the
S -dynamics and I -dynamics follow the N -dynamics on a period-4 cycle attractor.
However, N

. 8 ≤ k ≤ 6 ,

I implies that the amplitudes of oscillations in the S -dynamics
and I -dynamics are different from that of the N -dynamics whenever the disease is
endemic.

= S +

6 Conclusion and Discussion

In this chapter, we used an extension of the discrete-time SIS epidemic model
framework of Castillo-Chavez and Yakubu to study the relationship between
periodic infectious disease incidence and host population dynamics. When the host
population is asymptotically constant or growing at a geometric rate we computed
ℜ 0 and used it to predict disease persistence or extinction. In this case, we obtained
that the transmission rates as well as survival and recovery rates are critical model
parameters for the persistence or control of the disease.