Question

(10 pt) (i) Consider the spread of an infectious disease through a population. Let $S(t) > 0$, $I(t) \ge 0$ and $R(t) \ge 0$ denote the number of susceptible (not yet infected), infected, and recovered (and now immune) individuals respectively. Suppose that each infected case comes in contact with another individual an average of $\beta > 0$ times per day. Further, a fixed fraction $\gamma > 0$ of infected individuals recover per day from the disease. Assume that the number of individuals born per day, denoted $\mu$, is equal to the number of individual deaths per day, and that deaths are proportional to the number of people in each state. Figure 1 shows a block diagram of the system. $\beta$ $\gamma$ $S(t)$ $I(t)$ $R(t)$ Figure 1: Block diagram of simple epidemiological model. Write down a system of differential equations that models this process. Is this system linear or nonlinear?

          (10 pt) (i) Consider the spread of an infectious disease through a population. Let $S(t) > 0$, $I(t) \ge 0$ and $R(t) \ge 0$ denote the number of susceptible (not yet infected), infected, and recovered (and now immune) individuals respectively. Suppose that each infected case comes in contact with another individual an average of $\beta > 0$ times per day. Further, a fixed fraction $\gamma > 0$ of infected individuals recover per day from the disease. Assume that the number of individuals born per day, denoted $\mu$, is equal to the number of individual deaths per day, and that deaths are proportional to the number of people in each state. Figure 1 shows a block diagram of the system.
$\beta$
$\gamma$
$S(t)$
$I(t)$
$R(t)$
Figure 1: Block diagram of simple epidemiological model.
Write down a system of differential equations that models this process. Is this system linear or nonlinear?

$(10 pt) (i) Consider the spread of an infectious disease through a population. Let S(t) > 0, I(t) ≥ 0 and R(t) ≥ 0 denote the number of susceptible (not yet infected), infected, and recovered (and now immune) individuals respectively. Suppose that each infected case comes in contact with another individual an average of β > 0 times per day. Further, a fixed fraction γ > 0 of infected individuals recover per day from the disease. Assume that the number of individuals born per day, denoted μ, is equal to the number of individual deaths per day, and that deaths are proportional to the number of people in each state. Figure 1 shows a block diagram of the system. β γ S(t) I(t) R(t) Figure 1: Block diagram of simple epidemiological model. Write down a system of differential equations that models this process. Is this system linear or nonlinear?$

Added by Nicholas J.

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