The SIR Model is epidemic model used to model diseases on a fixed group of indiv

Question

The SIR Model is epidemic model used to model diseases on a fixed group of individuals. Think Chickenpox. In the model, we have the following functions: S (t) represents those susceptible to the number of individuals not yet infected with the disease at time t. I (t) represents those infected by the disease and who are contagious at time t. R (t) represents those who have recovered or otherwise been removed from the population: they aren't infected/contagious or susceptible to the disease at time t. N = S (t) + I (t) + R (t) represent the total population, and is treated as a fixed number in this model. Infected patient who die are represented by R (t) and this model does not take births or other changes in population into consideration The SIR model is further defined by the following set of differential equations: ds/dt = - beta s (t) I (t)/N, dI/dt = beta (t) I (t)/N - y I (t), dR/dt = y I (t) In the above model beta represents the rate at which people interact, y represents the recover/death rate, and 1/y represents the infection period. a) Explain in words why we can assume lim_t rightarrow infinity I (t) = 0 and lim rightarrow infinity d5/dt = lim rightarrow infinity dI/dt = lim dR/dt = 0 in the simple SIR Model above provided the infection period is less than 20 weeks. b) A statistic that is used to examine epidemic is the ratio of S (t) to R (t). Let H (t) = s(t)/R(t) Explain in words what lim t rightarrow infinity H (t) = 0 would tell us about the disease Explain in word what lim t rightarrow infinity H (t) =2 would tell us about the disease. If lim t rightarrow infinity does not exist, what would we know about the disease. c) Assuming a scaled population of N = 1,000, Use the Chain Rule to find dH/dt when S = 700, R = 250, and given constant beta 13.7 and y = 15.9.

Explanation / Answer

a) As given I(t) gives the infected Individuals and as the time goes beyond 20weeks these individuals are either recovered or removed from the population which leads to the count 0

so Limt I(t) = 0 when t tends to Infinity as the functions dS/dt, dI/dt, dR/dt all contains the function I(t) and when t tends to infinity the function I(t) is '0' so the whole function becomes zero

b) given H(t) = S(t) /R(t)

where H(t) gives the ration of people susceptible to the recovered people .

now Lim H(t) = 0 when t is infinity..this means that when the time goes on the people susceptible to the disease are either cured or removed from the population and so when the time is beyond the infection period the susceptible individuals becomes '0'

now  Lim H(t) = 1 when t is infinity..this means that when the time goes on the people susceptible to the disease are either cured or removed from the population and so when susecptible individuals are treated and recovered then the ratio becomes '1' and as the time increases the recovered people stay recovered.

now Lim H(t) = inifinty when t is infinity..this means that when the time goes on the people susceptible to the disease are never recovered and so when the time is beyond the infection period the recovered individual are '0' and so the limit goes to infinity

c) given N = 1000, S = 700 , R = 250 and beta = 13.7 and gamma = 15.9

now I = N-S-R => I = 1000-700-250 = 50

now dS/dt = -betaS(t)I(t)/N = -13.7(700)(50)/1000 = -479.5

dR/dt = gamma.I(t) = 15.9*50 = 79.5

now we know H(t) = S(t)/R(t) => dH/dt = dS/dt.R - dR/dt .S / R^2 = (-479.5)(250) - (79.5)(700) / 2502 = -2.8084

=> dH/dt = -2.8084

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