Get answer: Epidemics. The use of mathematical methods to study the spread of contagious

Chapter 2, Problem 24

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Epidemics. The use of mathematical methods to study the spread of contagious diseases goes back at least to some work by Daniel Bernoulli in 1760 on smallpox. In more recent years many mathematical models have been proposed and studied for many different diseases.16 22 through 24 deal with a few of the simpler models and the conclusions that can be drawn from them. Similar models have also been used to describe the spread of rumors and of consumer products.Daniel Bernoullis work in 1760 had the goal of appraising the effectiveness of a controversialinoculation program against smallpox, which at that time was a major threat topublic health. His model applies equally well to any other disease that, once contractedand survived, confers a lifetime immunity.Consider the cohort of individuals born in a given year (t = 0), and let n(t) be thenumber of these individuals surviving t years later. Let x(t) be the number of members ofthis cohort who have not had smallpox by year t and who are therefore still susceptible.Let be the rate at which susceptibles contract smallpox, and let be the rate at whichpeople who contract smallpox die from the disease. Finally, let (t) be the death rate fromall causes other than smallpox. Then dx/dt, the rate at which the number of susceptiblesdeclines, is given bydx/dt = [ + (t)]x. (i)The first term on the right side of Eq. (i) is the rate at which susceptibles contract smallpox,and the second term is the rate at which they die from all other causes. Alsodn/dt = x (t)n, (ii)where dn/dt is the death rate of the entire cohort, and the two terms on the right side arethe death rates due to smallpox and to all other causes, respectively.(a) Let z = x/n and show that z satisfies the initial value problemdz/dt = z(1 z), z(0) = 1. (iii)Observe that the initial value problem (iii) does not depend on (t).(b) Find z(t) by solving Eq. (iii).(c) Bernoulli estimated that = = 18 . Using these values, determine the proportion of20-year-olds who have not had smallpox.Note: On the basis of the model just described and the best mortality data available atthe time, Bernoulli calculated that if deaths due to smallpox could be eliminated ( = 0),then approximately 3 years could be added to the average life expectancy (in 1760) of 26years, 7 months. He therefore supported the inoculation program.Bifurcation Points. For an equation of the formdy/dt = f(a, y), (i)where a is a real parameter, the critical points (equilibrium solutions) usually depend on thevalue of a. As a steadily increases or decreases, it often happens that at a certain value of a,called a bifurcation point, critical points come together, or separate, and equilibrium solutionsmay either be lost or gained. Bifurcation points are of great interest in many applications,because near them the nature of the solution of the underlying differential equation is undergoingan abrupt change. For example, in fluid mechanics a smooth (laminar) flow may breakup and become turbulent. Or an axially loaded column may suddenly buckle and exhibit alarge lateral displacement. Or, as the amount of one of the chemicals in a certain mixture is increased,spiral wave patterns of varying color may suddenly emerge in an originally quiescent fluid. 25 through 27 describe three types of bifurcations that can occur in simpleequations of the form (i).