Consider the epidemic model governed by the following equation
with initial condition x = n at t = 0 . Here )t(x is the number of susceptibles at time ,t β is the contact rate. The population is assumed to be closed and homogeneously mixing. Let the contact rate be 0.002 and the number of susceptibles be 5000 initially
i) Find the density of the population when the rate of appearance of new cases is maximum.
ii) Find the time (in weeks) at which the rate of appearance of new cases is maximum.
iii) Obtain the maximum rate of appearance of new cases.
The given epidemic model is: \[ \frac{dx}{dt} = -\beta x (n + 1 - x) \] with the initial condition \( x(0) = n \), where: - \( x(t) \) is the number of susceptibles at time \( t \), - \( ___ ______ ______ __________ _________ ______ __________ __________ ____ _________ _______.
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