Dynamics of an Epidemic

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This Demonstration shows the time behavior of solutions of the SIS model of an epidemic by means of a phase portrait. The values of the coefficients describing the system can be modified using the sliders. The population is constant and scaled to 1. If denotes the density of susceptible persons and the density of infected ones, the brown and gray lines are the zero isoclines for and , respectively, and the pink points represent the stationary states of the SIS system. Information about an outbreak or disappearance of the epidemic based on the computed value of the basic reproduction number is displayed.

Contributed by: Marcin Choiński (March 2017)
(The University of Warsaw)
Open content licensed under CC BY-NC-SA



Consider a population of constant size divided into two group of individuals: susceptible and infected.

Let and be the proportions of susceptible and infected individuals in the population: , be the birth/death ratio, be the infection coefficient and be the recovery rate. Scaling the population magnitude to 1 gives the model:

, .

The coefficient is called the basic reproduction number. If , then as , implying that the epidemic will dissipate and the population will return to the normal state. If , then as , meaning that the epidemic will continue.


[1] U. Foryś, "Modelowanie szczepień" (Vaccination modeling), University of Warsaw.

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