This project explores the how we can model the outbreak of an epidemie Assume we have a population with a fixed size N. There is an infection spreading through the population and we're going to divide the population into three groups: Susceptible, Infected and Recovered. At a given moment in time t,
S(t) = Number who are have not had the infection but are susceptible
1(t) = Number infected
R(t) = Number who have had the infection but have recovered
We will assume that once you have had the infection you cannot get it again. At any given moment in time s(t) +1(c)+ R(t) = N. Notice this means that ds + 4 + 4 = 0.
We make the following assumptions: .
• Those who are susceptible can become infected. Those who are infect can recover. Hence the number of susceptibles decreases and the number of recovered increases.
• The rate of change of S (with respect time) is directly proportional to both I and S.
• The rate of change of R (with respect to time) is directly proportional to I
These three assumptions (along with +*+ * = 0) imply that this system is governed by the following differential equations: ds -BSI dt d1 BSI- dt UR vi where 8 >0 is called the transmission rate and > >O is called the recovery rate.
Unfortunately we will not be able to find explicit solutions to these equations in general. However, we will be able prove various important properties of solutions to such a system.
a) The monotone convergence theorem implies that if a function f is monotone (increasing or decreasing) and bounded on [0, 1) then limit as t goes to infinity f(t) exists. Use this to show that limit as t goes to infinity S(t), limit as t goes to infinity I(t) and limit as t goes to infinity R(t) all exist. Denote these limits by S1, I1 and R1. This means the situation stabilizes over time. Hint: Consider S and R first and remember the total population size is N.
b) It is a fact that if limt!1 f0 (t) > 0 then limt!1 f(t) = 1. Use this to prove that I1 = 0. This means the epidemic will always end. Nice! Hint: Consider f(t) = R(t)