23. Daniel Bemouli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox. which at that time was a major threat to public health....
23. Daniel Bemouli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox. which at that time was a major threat to public health. His model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity a. Find all of the critical p there are no critical points i and two critical points if a O b. Draw the phase line each critical point is asy Consider the cohort of individuals born in a given year (t 0), and let n(t) be the number of these individuals surviving r years later Let x(t) be the number of members of this cobort who have not had O c. In each case sketch s the ry-plane. smallpox by year and who are therefore still susceptible. Let 3 be the rate at which susceptibles contract smallpox, and let v be the rate Note: If we plot the location of t at which people who contract smallpox die from the disease. Finally, the ay-plane, we obtain Figure2 let a(t) be the death rate from all causes other than smallpox. Then diagram for equation (29). Th dx/dr, the rate at which the number of susceptibles declines, is given saddle - node bifurcation. This by of second-order systems dx dr (25) The first term on the righi-hand side of equation (25) is the rate at which susceptibles contract smallpox, and the second term is the rate at which they die from all other causes. Also dn dr (26) -2 ls where dn/dt is the death rate of the entire cobort, and the two terms on the right-hand side are the death rates due to smallpox and to all other causes, respectively aLetzx/n, and show that z satisfies the initial value FIGURE 2.5.10 -32(1 -z), z(0)-1. (27) dr Observe that the initial value problem (27) does not depend on 25. Consider the equation b. Find z(t) by solving equation (27). c. Bernoulli estimated that v -β = 1/8. Using these dr O a. Again consider the values, determine the proportion of 20-year-olds who have not had smallpox. case find the critical pois whether each critical poi or unstable. Note: On the basis of the model just described and the best mortality data available at the time, Bemoulli calculated that if deaths due to smallpox could be eliminated (v0), then approximately 3 years b. In each case sketd could be added to the average life expectancy (in 1760) of 26 years, 7 months. He therefore supported the inoculation program. Bifurcation Points. For an equation of the form the ry-plane. c. Draw the bifurcat plot the location of the c Note: For equation (30) the pitchfork bifurcation. Your (28) where a is a real parameter, the critical points (equilibrium solutions) usually depend on the value 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 solutions may be either 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 undergoing an abrupt change. For example, in fluid mechanics a smooth (laminar) flow may break up and become turbulent. Or an axially loaded column may suddenly buckle and exhibit a large 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. Problems 24 through 26 describe three types of bifurcations that can occur in simple equations of the form (28) 26. Consider the equation dr a. Again consider the case find the critical po whether each critical p or unstable. b. In each case sketch c. Draw the bifurcatio Note: Observe that for equ critical points for a < 0 and For a < 0 the equilibrium
23. Daniel Bemouli's work in 1760 had the goal of appraising the effectiveness of a controversial inoculation program against smallpox. which at that time was a major threat to public health. His model applies equally well to any other disease that, once contracted and survived, confers a lifetime immunity a. Find all of the critical p there are no critical points i and two critical points if a O b. Draw the phase line each critical point is asy Consider the cohort of individuals born in a given year (t 0), and let n(t) be the number of these individuals surviving r years later Let x(t) be the number of members of this cobort who have not had O c. In each case sketch s the ry-plane. smallpox by year and who are therefore still susceptible. Let 3 be the rate at which susceptibles contract smallpox, and let v be the rate Note: If we plot the location of t at which people who contract smallpox die from the disease. Finally, the ay-plane, we obtain Figure2 let a(t) be the death rate from all causes other than smallpox. Then diagram for equation (29). Th dx/dr, the rate at which the number of susceptibles declines, is given saddle - node bifurcation. This by of second-order systems dx dr (25) The first term on the righi-hand side of equation (25) is the rate at which susceptibles contract smallpox, and the second term is the rate at which they die from all other causes. Also dn dr (26) -2 ls where dn/dt is the death rate of the entire cobort, and the two terms on the right-hand side are the death rates due to smallpox and to all other causes, respectively aLetzx/n, and show that z satisfies the initial value FIGURE 2.5.10 -32(1 -z), z(0)-1. (27) dr Observe that the initial value problem (27) does not depend on 25. Consider the equation b. Find z(t) by solving equation (27). c. Bernoulli estimated that v -β = 1/8. Using these dr O a. Again consider the values, determine the proportion of 20-year-olds who have not had smallpox. case find the critical pois whether each critical poi or unstable. Note: On the basis of the model just described and the best mortality data available at the time, Bemoulli calculated that if deaths due to smallpox could be eliminated (v0), then approximately 3 years b. In each case sketd could be added to the average life expectancy (in 1760) of 26 years, 7 months. He therefore supported the inoculation program. Bifurcation Points. For an equation of the form the ry-plane. c. Draw the bifurcat plot the location of the c Note: For equation (30) the pitchfork bifurcation. Your (28) where a is a real parameter, the critical points (equilibrium solutions) usually depend on the value 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 solutions may be either 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 undergoing an abrupt change. For example, in fluid mechanics a smooth (laminar) flow may break up and become turbulent. Or an axially loaded column may suddenly buckle and exhibit a large 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. Problems 24 through 26 describe three types of bifurcations that can occur in simple equations of the form (28) 26. Consider the equation dr a. Again consider the case find the critical po whether each critical p or unstable. b. In each case sketch c. Draw the bifurcatio Note: Observe that for equ critical points for a