Background: Suppose we want to model the spread of an infectious disease through a population. In the previous assignment, we did this with a discrete model. Here, we'll formulate things as a continuous model. We'll keep track of the numbers of individuals susceptible to the disease S ( t ) , infected by the disease I ( t ) , and recovered from the disease R ( t ) at time t (measured in days). Infected individuals have a chance to transmit the disease upon contact with a susceptible individual at a rate > 0 . Infected individuals also recover from the infection at a rate > 0 . We assume that all N individuals in the population produce offspring at a rate b > 0 and these newborns add to the pool of susceptible individuals. Finally, individuals in all disease states can die due to natural causes at a rate equal to the birth rate b . This description can be captured by the following system of equations: d t d S = b N S I b S d t d I = S I I b I d t d R = I b R Problems Problem 1 [3 points]: Given that S , I , and R are measured in number of people and t is measured in days, what must be the units of b , , and ? Justify your answer. Problem 2 [3 points]: Let N ( t ) = S ( t ) + I ( t ) + R ( t ) be the total population size at any point in time t . If the total population size remains constant for all time (i.e., N ( t ) = N for all t , where N is a constant) we say we have a "closed population". Show that the population in our model is closed. (Hint: Think about the rate of change d t d N of the population size and what must be true for the population to be constant.).

1. Background: Suppose we want to model the spread of an infectious disease through a population.
In the previous assignment, we did this with a discrete model. Here, we'll formulate things as a
continuous model. We'll keep track of the numbers of individuals susceptible to the disease S ( t
) , infected by the disease I ( t ) , and recovered from the disease R ( t ) at time t (measured in
days). Infected individuals have a chance to transmit the disease upon contact with a susceptible
individual at a rate > 0 . Infected individuals also recover from the infection at a rate > 0 . We
assume that all N individuals in the population produce offspring at a rate b > 0 and these
newborns add to the pool of susceptible individuals. Finally, individuals in all disease states can
die due to natural causes at a rate equal to the birth rate b . This description can be captured by
the following system of equations: d t d S = b N S I b S d t d I = S I I b I d t d R = I b R Problems
Problem 1 [3 points]: Given that S , I , and R are measured in number of people and t is
measured in days, what must be the units of b , , and ? Justify your answer. Problem 2 [3 points]:
Let N ( t ) = S ( t ) + I ( t ) + R ( t ) be the total population size at any point in time t . If the total
population size remains constant for all time (i.e., N ( t ) = N for all t , where N is a constant) we
say we have a "closed population". Show that the population in our model is closed. (Hint: Think
about the rate of change d t d N of the population size and what must be true for the population
to be constant.)