A Discrete Epidemic Model (SIR) II (with vaccinations)
by Mike Martin
This page generates plots and data for a discrete SIR Model with a constant total population, births and deaths proportional to population sizes, and with vaccinations. The model equations are:
It+1 = It - γ It - b It + (β / N) It St
Rt+1 = Rt - b Rt + γ It + p St
The model helps to describe the dynamics between the population levels of three different populations -- Susceptibles ( St ), Infected ( It ), and Recovered ( Rt ). N is the total population -- assumed to be a constant. p is the proportion vaccinated. β is the average number of successful contacts (resulting in infection) made by individuals during the time t and t + 1. β S / N is the proportion of contacts by one infected individual that result in an infection of a susceptible individual. β S I / N is the total number of contacts by the infected class that result in infection. The probability of a birth is equal to the probability of a death and is parameterized by b. γ is the probability of recovery. T is the last generation considered with the simulation. The basic reproduction number is the number of secondary infections caused by one infectious individual during the individual's infectious period; for this model the basic reproduction number is β b / [(γ + b)(p + b)]. S1, I1, and R1 are the initial conditions. In order for the populations to be non-negative (and meaningful) we impose the conditions that
0 < p + β < 1
The sequence St is displayed in the graph in BLUE, the sequence It is given in BROWN, and the sequence Rt is given in PURPLE.
Enter in the values of the non-negative parameters (described above), then press the "Evaluate" button.
To be added.