In a Model of Epidemics, Let $y ( t )$ , in Thousands, Be the Number of Infected Individuals in

question 93

Essay

In a model of epidemics, let $y ( t )$ , in thousands, be the number of infected individuals in the
population at time $t$ , in days. If we assume that the infection spreads to all those who are
susceptible, one possible solution for $y ( t )$ is given by $\frac { d y } { d t } = k ( p - y ) \cdot y$ where $k$ is a positive
constant which measures the rate of infection and P , in thousands, is the total population in
this situation.(a) Determine the solution of this differential equation if $y ( 0 ) = 1$ .(b) Discuss what $y ( 0 ) = 1$ means.(c) As the time increases without bound, what happens to $y$ ? (That is, what does $\lim _ { t \rightarrow \infty } y$ mean?)
(d) Sketch the solution of the differential equation in part (a).

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