Question

Consider a population of size N. In the SIR model of epidemics the number of susceptible individuals, S(t), and infected individuals, I(t), at timet (measured in days) are governed by the equations: dt While S(t) is close to N and I(t) is close to zero the equations are approximated by where I(0) = 1o and S(0) = N – Io. A) Give the solution to the approximate model equations above (Egns.(3)-(4), along with initial conditions) for S(t) and I(t). Hint: find I(t) first then use it to find S(t). B) Let N = 107 and Io = 10. After 10 days it is measured that I(10) = 40 and S(10) = N - 100. Find a-b given this information. (Hint: you do not need to find both a and b separately.) C) Using the above, what is the doubling time of the number of infected? D) How long until there are 1280 infected? 10,240 infected?

Answer 1

Solution la da je I 1(0)-I. dt S(O) = N-I. c then de 9:57 - dle ab dt I On integrating: too ln I (t) = (a-but tence - Ilt)= a elabot Using I 10J = Io => G = Io - Il+) = Io element Now, de a - (1o e ) 30 ds = - ale element dit 1 on integrating : STHz -azo el + C2 = S() = -ado el munt the Using Slo) = N-I. - N-Io = - a to tz >> G = N-Zo +9 Zo M a-b a-b a-b Therefore, Sit a lo + a 1. b-a + N- 9-5

N=107 Then - 1. = 10 I 110) = 40 Ilt)= Io e 194 I (10) = 10 e Solo » enu= (a-b) 106 =) la-b = 1oo lnul 10 I (t) = 80 {": 71103=40 When infected get doubled then then Ilt) = Io e lloeren) + Ilt) = 1o ellnul.to I (H) = 4 I. ethio Using an opee I (+) = 80, 1. = 10 - 80 = 4x10 ethio as et/10 = 2 as the = lna t = 10. ln 2] - D When I (t) = 1280 , then I(t)= uxio e 10 - etho = 1280 { using Io=10 . 40 = = = = ln 32 + = 10 ln as t = 50 luz]

When then tio Iltl= 10240 I (t) = 40.00 = + = en to you = t = 10lw 256 =) t = 70 ln al