Math 326 Feb 5 Model

Math 326, Feb 5 - Epidemic!

Build the model from 2.1, Simple Epidemics. We use a model called the Kermack-McKendrick, or SIR. The population is separated into three groups, Susceptibles (who haven't caught the disease yet), Infectives (who have the disease) and Removals (who have healed). Each time period, some of the Susceptibles becomes Infectives (they catch the disease), and some Infectives become Removals (they recover).

The rate at which Susceptibles become Infectives depends on the Contact Probability, which is the chance a particular Susceptible is exposed to a particular Infective. The number of new Infectives each period is given by:

Susceptibles becoming Infectives = Susceptibles * (1-(1-Contact Probability)^(Infectives))

The rate at which Infectives become Removals depend on the Healing rate. This is the fraction of Infectives which become healed each time period.

Infectives becoming Removals = Infectives * Healing Rate

Use for starting values Population = 1000, Contact Probability = 0.002, Healing Rate = 1 and Initial Infectives = 3 (so initial Susceptibles should be 997).

Construct a graph with Susceptibles, Infectives and Removals. Make sure the graph cover enough time for the epidemic to settle down.

Additional Questions

Introduce scroll bars for Contact Rate (0 to 0.5, steps of 0.001) and Healing Rate (0 to 1, steps of 0.01). See how changing these affects the graph.
Change the model so that the population increases over time. A number of new citizens, proportional to total population, is added to the Susceptibles each period. Start with a population growth rate of 0.008 per period (so 0.008*(Total Pop) are added to Susceptibles in the next step). This growing population can produce waves of epidemic relapses. You may have to greatly increase the time period covered in the graph to see the waves.
Along with the population growth, suppose that the disease is sometimes fatal. Now each period, some Infectives become Removals and some Infectives become Casualties (some may remain infected). What kind of fatalities rates will lead to an 80% drop in population before the population begins to recover? Make sure your (Healing Rate + Death Rate)<=1, otherwise we would have some people both recovering and dying.
Imagine there is a two island nations. At first, everyone lives on the first island, Paniku. However, instead of letting infected citizens stay on Paniku, they are forced to move to Sanitori. Infecteds still have that one, first period on Paniku to spread the epidemics, then they are forced to move, even if they have recovered by the next period. Assume the populations of both islands have population growth as before at a rate of 0.008. Vary Contact, Heal and Death rates to see the long term effects on the island populations.