Instructions: Please read the following background information and answer all ques- tions.  You may handwrite or type (e.g., Microsoft Word, LaTeX) your solutions, but if they are handwritten please make sure they are legible.  You are welcome to discuss this assignment with others in the class, but you must write up your own solutions; if your work too closely matches the work of another student, this will be treated as an act of academic dishonesty and appropriate measures will be taken. All work for this assignment is due by 11:59pm on Monday, March 6, 2023 through Crowdmark.  Late work is subject to the late work policy described in the official course outline.

Final Submission: Your final submission will be done through Crowdmark. You will need to upload your solutions for each question individually, either by scanning or taking a picture of your work. You will also need to upload the plot you make for Problem 9 and a screenshot of the R code used to generate that plot.

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 β  > 0.  Infected individuals also recover from the infection at a rate γ > 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:

dS

= bN − βSI − bS                                                    (1a)

dI

= βSI − γI − bI                                                    (1b)

dR

= γI − bR.                                                         (1c)

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, β, and γ? 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 of the population size and what must be true for the population to be constant.)

Problem 3 [4 points]: We could start applying our usual techniques to analyze this model, but since it’s a three-dimensional model that could get messy. Instead, let’s start by seeing if we can reduce the complexity of our model in any way to aid in our analysis. Since we have a closed population, we don’t actually need to track the number of recovered individuals R(t).  At any point in time, we know R(t) = N − S(t) − I(t) so as long as we track S(t) and I(t) we can always compute R(t) if desired. This allows us to drop one of the equations from our model and reduce things to the following two-dimensional model:

dS

= bN − βSI − bS (2a)

dt

dI

= βSI − γI − bI . (2b)

dt

When we have a closed population, we can also “nondimensionalize” our model by replacing our state variables S and I with the unitless state variables u = and v = . We can then derive differential equations for these new state variables by using the differential equations for our old state variables: = and = . Find the nondimensionalized version of our original system by finding the differential equations and . This new system should be written entirely in terms of the new state variables u and v; the old state variables S and I should not appear anywhere in the equations. When you are finished, your system should look as follows (but you’ll need to fill in the steps to arrive at these equations):

= b(1 − u) − βNuv (3a)

dt

= βNuv − (γ + b)v . (3b)

dt

Problem 4 [4 points]: In the last problem, we nondimensionalized our state variables. We can also nondimensionalize our time variable by changing to a new unitless time variable τ = (γ + b)t.  We can then use the chain rule to find = and = .  Rewrite our model by computing and . When you are done, your system should look as follows (but you’ll need to fill in the steps to arrive at these equations):

= B(1 − u) − R0uv (4a)

dτ

dv

where B = and R0  = . (4b)