Question

3. (13 points) In the beginning of an epidemic, the rate at which new infections occur is proportional to the product of the number of people infected at time t and the difference between the total population size and the number of people infected at time t. Let I be the total size of a population experiencing a new epidemic and let I(t) be the number of people infected at time t. Consider a population size of 1000 people and let r= .1 be a proportionality constant. (a) Develop a differential equation model that can be used to find the number of people infected during the beginning of an epidemic. (b) Find the equilibria of the model. (c) Classify the stability of the equilibria.

Answer 1

Page 1 An Saucos As Per given details a) Note that the difference between the Population and the number of infected People is N-I[t)= todd - IH) Hence, from the formulation given, we get that the differential equation would be dI & I(t) (6000-76L)) = I'(X) = 8I14) (1000 -1(t)) = 0.15I(l)(1000-14) 6) Note that at the equillibrium, we have I'(t)=0 > D.(. 14) (lopp -T4])= 0 => Ilt): 0,1000 C) Note that the graph of the image is given belowe from this, we can see that both of the equilibrium Points are stable equilibrio, as the Slopes tend to fall towards these points PLEASE HIT LIKE YOTE

0:51 | | | | | 0.5 | | | | 1 -0.5