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Model the spread of a hypothetical virus in a closed population of 10,000 indivi...

Mar 10, 2024

Model the spread of a hypothetical virus in a closed population of 10,000 individuals using a setof differential equations and calculate the time to reach the epidemic peak and the maximumnumber of infected individuals at this peak. The unique characteristic of this virus is thatindividuals who recover from it gain no lasting immunity and can be immediately reinfected. Thisscenario modifies the traditional SlR (Susceptible, Infected, Recovered) model to reflect theimmediate loss of immunity.

Solution by Steps

step 1

To model the spread of the virus, we use a modified SIR model that accounts for the lack of immunity after recovery, allowing individuals to move from the recovered class back to the susceptible class

step 2

The differential equations provided by the asksia-ll calculator are:

\frac{ds}{dt} = -\beta s i

\frac{di}{dt} = \beta s i - \gamma i

\frac{dr}{dt} = \gamma i

. Since recovered individuals can be immediately reinfected, we need to add a term that moves individuals from

r

back to

s

step 3

The modified equations will be:

\frac{ds}{dt} = -\beta s i + \sigma r

\frac{di}{dt} = \beta s i - \gamma i

\frac{dr}{dt} = \gamma i - \sigma r

, where

\sigma

is the rate at which recovered individuals lose immunity and become susceptible again

step 4

To find the maximum number of infected individuals, we look for the critical points where

\frac{di}{dt} = 0

. This occurs when

i(t) = 0

or when

s(t) = \frac{\gamma}{\beta}

step 5

Since

i(t) = 0

corresponds to no infected individuals, we focus on the condition

s(t) = \frac{\gamma}{\beta}

to find the peak of the epidemic

step 6

To find the time to reach the peak, we would need to solve the system of differential equations with initial conditions and find the time

t

when

s(t) = \frac{\gamma}{\beta}

. This typically requires numerical methods as the system does not have a closed-form solution

Answer

The time to reach the epidemic peak and the maximum number of infected individuals cannot be determined analytically and would require numerical solutions of the modified SIR model equations with appropriate initial conditions.

Key Concept

Modified SIR Model with No Lasting Immunity

Explanation

The traditional SIR model is modified to account for the immediate loss of immunity by adding a term that moves recovered individuals back to the susceptible class. The peak of the epidemic is found when the number of susceptible individuals equals the ratio of the recovery rate to the infection rate,

\frac{\gamma}{\beta}

. Numerical methods are needed to solve the system and find the time to reach the peak and the maximum number of infected individuals.