Question

Let P(t) represent the number of people who, at time t, are infected with a certain disease. Let N denote the total number of people in the population. Assume that the spread of the disease can be modeled by the initial value problem:

dP/dt = k(N − P)P, P(0) = P0.

At time t = 0, when 100,000 member of a population of 500,000 are known to be infected, medical authorities intervene with medical treatment. As a consequence of this intervention, the rate factor k is no longer constant but varies with time as k(t) = 2e −t − 1, where time is measured in months and k(t) represents the rate of infection per month per 100,000 people. Initially, as the effects of medical intervention begin to take hold, k(t) remains positive and the disease continues to spread. Eventually, however, the effects of medical treatment cause k(t) to become negative and the number of infected individuals then decreases

Answer 1

Solutiom* Given Data Conside 1 a initial value PYoblem, d t Let s solve the above initial vaue Problem Since , this is a SePavate differentia eayua tiom So, lets wite in the fom (N-P)P dt Now, integ ating ơn both sides. (NI-P)P dt (N-P) F

IN-P)P To complete the inte we need to separa te the functi ơn, Sowe want to find constants A and B So that So, weneed the terms of Po lt nom)- al toma七ch; a) hich İmplies that AB and AN plies that A (N-P)P NP N (N-P) Going back to the SolUHơn

N-P)P N)で Wc cam also Simpityh Al-P Thus, the Solution is, Now, we Can Plu the initai valu e to find the Val

첸( ) so, it we Plug in, we jct tn Now we. soive to P tot am e N-P PNKt P N-P N-B NKt N-P N-Po Now, we mtiy by N-P N EL Poe(N-P) N- PD ekt

we can eft hand side NKt P Ne P/N NKト There fove, the explicit eq/uation jk