21 in the pond. Is theif the pond is not tot the population y ofibut) in a given areaod, it is intuitivelypopulation may beven to extinction.ions involved in16ery."ume that the ratey: the more fishe that the rate atositive constant,le to harvest thestic equation is(20)the biologistbrium points,ymptoticallyhich fish cant E and thection of theield-efforty find thenstant rateisfies(21)e whento thesame value as Y in Problem 19d.be overexploited if y is reduced to a level below K/2.Epidemics. The use of mathematical methods to study the spread ofcontagious diseases goes back at least to some work by DanielBernoulli in 1760 on smallpox. In more recent years manymathematical models have been proposed and studied for manydifferent diseases. 17 Problems 21 through 23 deal with a few of thesimpler models and the conclusions that can be drawn from them.Similar models have also been used to describe the spread of rumorsand of consumer products.then x + y21. Suppose that a given population can be divided into two parts:those who have a given disease and can infect others, and thosewho do not have it but are susceptible. Let x be the proportion ofsusceptible individuals and y the proportion of infectious individuals;1. Assume that the disease spreads by contact betweensick and well members of the population and that the rate of spreaddy/dt is proportional to the number of such contacts. Further, assumethat members of both groups move about freely among each other, sothe number of contacts is proportional to the product of x and y. Sincex = 1 - y, we obtain the initial value problemTen=ay(1-y), y(0) = yo,(22)where a is a positive proportionality factor, and yo is the initialproportion of infectious individuals.a. Find the equilibrium points for the differential equation (22)and determine whether each is asymptotically stable, semistable,or unstable.ble.yuilibriumdydtb. Solve the initial value problem 22 and verify that theconclusions you reached in part a are correct. Show that y(1)1 as t → ∞, which means that ultimately the disease spreadsthrough the entire population.22. Some diseases (such as typhoid fever) are spread largely bycarriers, individuals who can transmit the disease but who exhibit noovert symptoms. Let x and y denote the proportions of susceptiblesand carriers, respectively, in the population. Suppose that carriers areidentified and removed from the population at a rate 3, sodydtto=-By.(23)product of x and y; thusSuppose also that the disease spreads at a rate proportional to thedxdt= -axy.(24)a. Determine y at any time t by solving equation (23) subject tothe initial condition y(0)smallthe raat whlet pdx/cbyThewhicat wwheonothNedasrссmBWir1

Question

21 in the pond. Is theif the pond is not tot the population y ofibut) in a given areaod, it is intuitivelypopulation may beven to extinction.ions involved in16ery.&#34;ume that the ratey: the more fishe that the rate atositive constant,le to harvest thestic equation is(20)the biologistbrium points,ymptoticallyhich fish cant E and thection of theield-efforty find thenstant rateisfies(21)e whento thesame value as Y in Problem 19d.be overexploited if y is reduced to a level below K/2.Epidemics. The use of mathematical methods to study the spread ofcontagious diseases goes back at least to some work by DanielBernoulli in 1760 on smallpox. In more recent years manymathematical models have been proposed and studied for manydifferent diseases. 17 Problems 21 through 23 deal with a few of thesimpler models and the conclusions that can be drawn from them.Similar models have also been used to describe the spread of rumorsand of consumer products.then x + y21. Suppose that a given population can be divided into two parts:those who have a given disease and can infect others, and thosewho do not have it but are susceptible. Let x be the proportion ofsusceptible individuals and y the proportion of infectious individuals;1. Assume that the disease spreads by contact betweensick and well members of the population and that the rate of spreaddy/dt is proportional to the number of such contacts. Further, assumethat members of both groups move about freely among each other, sothe number of contacts is proportional to the product of x and y. Sincex = 1 - y, we obtain the initial value problemTen=ay(1-y), y(0) = yo,(22)where a is a positive proportionality factor, and yo is the initialproportion of infectious individuals.a. Find the equilibrium points for the differential equation (22)and determine whether each is asymptotically stable, semistable,or unstable.ble.yuilibriumdydtb. Solve the initial value problem 22 and verify that theconclusions you reached in part a are correct. Show that y(1)1 as t → ∞, which means that ultimately the disease spreadsthrough the entire population.22. Some diseases (such as typhoid fever) are spread largely bycarriers, individuals who can transmit the disease but who exhibit noovert symptoms. Let x and y denote the proportions of susceptiblesand carriers, respectively, in the population. Suppose that carriers areidentified and removed from the population at a rate 3, sodydtto=-By.(23)product of x and y; thusSuppose also that the disease spreads at a rate proportional to thedxdt= -axy.(24)a. Determine y at any time t by solving equation (23) subject tothe initial condition y(0)smallthe raat whlet pdx/cbyThewhicat wwheonothNedasrссmBWir1

Accepted Answer

Solution:- we have to solve given differential equation