Dynamics of a Susceptible-Exposed-Infected-Recovered (SEIR) Epidemic Model with Time Delay

This Demonstration illustrates the effect of time delay on the dynamics of an SEIR epidemic model.
In this model [1] the host population is partitioned into four classes: the susceptible, exposed, infectious, and recovered, with , , , and denoting the fraction of each class; the disease spreads through direct contact, and a host stays in a latent period after contact with an infected host before becoming infective; an infectious host may die from the disease or recover with acquired immunity.
The system is governed by the following equations:
,
,
,
where is time, is the time delay that a susceptible host must be in contact with an infectious host to be considered exposed, represents both the birth and death rate, is the rate at which the exposed become infective, is the rate at which the infective recover, and is the contact rate at which susceptibles come into contact with the infective. The system is solved with , , and , , , . If we take the total population , the value of can be obtained from the other three values. The system has a stable equilibrium when at . When is increased, this equilibrium is approached asymptotically; further increases of cause the trajectories to reach first periodic and then chaotic oscillations.

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Reference
[1] C. Xu and M. Liao, "Stability and Bifurcation Analysis in a SEIR Epidemic Model with Nonlinear Incidence Rates," International Journal of Applied Mathematics, 41(3), 2011.
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