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:

I_{t+1} = I_{t} - γ I_{t} - b I_{t} + (β / N) I_{t} S_{t}

R_{t+1} = R_{t} - b R_{t} + γ I_{t} + p S_{t}

The model helps to describe the dynamics between the population levels of three different populations -- Susceptibles ( S_{t} ), Infected ( I_{t} ), and Recovered ( R_{t} ). 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)]. S_{1}, I_{1}, and R_{1} 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 S_{t} is displayed in the graph in BLUE, the sequence I_{t} is given in BROWN, and the sequence R_{t} is given in PURPLE.

Enter in the values of the non-negative parameters (described above), then press the "Evaluate" button.

p =

β =

b =

γ =

T =

S_{1} =

I_{1} =

R_{1} =

References:

To be added.