4. Recall the SIR infectious disease model d.S dt -aSI, dI dt dR B1 dt satisfied...

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4. Recall the SIR infectious disease model d.S dt -aSI, dI dt dR B1 dt satisfied by a community with a fixed total population S + I + R consisting of sus- ceptibles (S), infectives (I), and removals/individuals recovered with full immunity (R). The parameters a and 3 are assumed to be positive constants which depend on the nature of the disease. = aSI - BI, (2) (a) Define the reproductive ratio r = (a/3)S and the scaled time variable 7 = Bt. Rewrite the system (2) as an equivalent system of differential equations ex- pressing dS/dr, dI/dr, and dR/dr in terms of r and I. (b) Similar to the reproductive ratio r = (a/3)S, define scaled versions of I and R as h= (a/B)I and q = (a/3)R, respectively. Rewrite the system of differ- ential equations that you obtained in part (a) as an equivalent system that expresses dr/dr, dh/dr, and dq/dr in terms of r and h. (c) Let g=r-1- lnr. Show that g + h is constant. (Hint: Differentiate with respect to 7 and use the equations from your system in part (b) to simplify the result.) Conclude that for all 720, g(T) +h(T) = g(0) + h(0). (3) (d) Prove that for any positive real r, r-1-lnr ≥ 0, with equality if and only if r = 1. (Hint: Use calculus to show that the left hand side of the inequality has a unique global minimum at r= 1.) Conclude from (3) that the maximum possible value of h occurs when the reproductive ratio r = 1, and is equal to To 1-In(ro) +h(0), where ro = r(0) = (a/3)S(0) is the initial reproductive ratio, commonly referred to as the basic reproductive number. 4. Recall the SIR infectious disease model d.S dt -aSI, dI dt dR B1 dt satisfied by a community with a fixed total population S + I + R consisting of sus- ceptibles (S), infectives (I), and removals/individuals recovered with full immunity (R). The parameters a and 3 are assumed to be positive constants which depend on the nature of the disease. = aSI - BI, (2) (a) Define the reproductive ratio r = (a/3)S and the scaled time variable 7 = Bt. Rewrite the system (2) as an equivalent system of differential equations ex- pressing dS/dr, dI/dr, and dR/dr in terms of r and I. (b) Similar to the reproductive ratio r = (a/3)S, define scaled versions of I and R as h= (a/B)I and q = (a/3)R, respectively. Rewrite the system of differ- ential equations that you obtained in part (a) as an equivalent system that expresses dr/dr, dh/dr, and dq/dr in terms of r and h. (c) Let g=r-1- lnr. Show that g + h is constant. (Hint: Differentiate with respect to 7 and use the equations from your system in part (b) to simplify the result.) Conclude that for all 720, g(T) +h(T) = g(0) + h(0). (3) (d) Prove that for any positive real r, r-1-lnr ≥ 0, with equality if and only if r = 1. (Hint: Use calculus to show that the left hand side of the inequality has a unique global minimum at r= 1.) Conclude from (3) that the maximum possible value of h occurs when the reproductive ratio r = 1, and is equal to To 1-In(ro) +h(0), where ro = r(0) = (a/3)S(0) is the initial reproductive ratio, commonly referred to as the basic reproductive number.