Epidemiology: An SIRS Model with Births/Deaths
by Mike Martin

On this page, plots of three different populations, susceptible (S), infected (I), and recovered (R), are generated. We assume that an infected individual can recover and will become susceptible again after some time recovering. The recovery rate is parameterized by γ and recovered individuals become susceptible again at a rate of v   β is the average number of adequate contacts made by an infected individual per time.   The parameter b is the birth rate or death rate (which are assumed to be equal).   There is a constant total number, N, of susceptible, infected, and recovered individuals:

S (t) + I (t) + R (t) = N

The solutions are also projected into the SI-phase plane along with background stream lines. The equations are of the form:

S ' ( t ) = - (β / N) S (t) I (t) + b I (t) + b R (t) + v R (t)

I ' ( t ) = (β / N) S (t) I (t) - γ I (t) - b I (t)

R ' ( t ) = γ I (t) - b R (t) - v R (t)

S(0) = S0
I(0) = I0
R(0) = R0

The populations, S, I, and R, are usually in units relative to a large population, say in thousands or millions; alternatively, one could normalize the populations and let the two populations represent the percentage of the total population.   The user inputs positive values of the parameters β, γ, v, and b.    Also, grid bounds, time bounds, and the initial conditions must be specified.  Note that the initial conditions are added together to obtain the value of the total population.   Note also that the populations are only meaningful if they are non-negative, so this model is only valid for non-negative time values that produce non-negative population values.



Enter in the value of the parameters and select the plotting bounds, then press the "Plot/Evaluate" button.

β =  

γ =  

v =  

b =  


S (0) =  

I (0) =  

R (0) =  


0   ≤   t   ≤  


  ≤   S   ≤  

  ≤   I   ≤  


References:

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