Let's explore what happens when the contact rate is a periodic
function of time. For example, for a childhood disease we might
imagine that
is higher during the school year and lower during
the summer. Following the paper ``Asymptotic behavior in a deterministic
model'' by H.W. Hethcote, Bull. Math. Biol. 35:607-614, 1973,
let's take
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global N alpha alpha = 1; N = 1; options = odeset('MaxStep',0.01); [T,Y] = ode45('func_SIS',[0 10],[0.2 0],options); figure(1) plot(T,Y(:,1)); xlabel('t'); ylabel('I'); for i=1:1000 %also plot a scaled version of beta tt(i) = 0.01*i; bb(i) = beta(tt(i)); end hold on; plot(tt,0.1*bb,'r');
Next, func_SIS.m:
function dy = func_SIS(t,y) global N alpha I = y(1); tau = y(2); dI = (beta(tau)*N - alpha)*I - beta(tau)*I^2; dtau = 1; dy = [dI;dtau];
Finally, beta.m:
function r = beta(t) r = 2 - 1.8*cos(5*t);
Figure 1 shows the numerical solution for this system with
and
in blue, and
in red. It is seen that
decays to zero with a few wiggles in the transient.
Figure 2 shows the numerical solution with
and
,
again with
in red. Here
settles into a periodically
oscillating state with period equal to the period of
; note that
the maximum of
does not occur at the same time as the maximum of
.
To understand these results, Hethcote defines the average reproduction
number as
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