Answer to Question #104556 in Differential Equations for mm

Question

The rate of increase of susceptible AIDS victims is proportional to the number of susceptible
persons and number of infected persons. If there are 0
S susceptible persons and 1 infected
person at a time to then i) set up the equation for the spread of the disease ii) solve the resulting
equation iii) give a physical interpretation to the same by plotting the epidemic curve iv) write the
limitations of the model.

Answer 1

i)

i - number of infected persons at time t

s - number of susceptible persons at time t

k - konstant

t - time

s₀ - initial number of susceptible persons

i₀ - initial number of infected persons = 1

i+s=s₀+1

di/dt = k*i*s

ii)

di/dt = k*i*(s₀+1-i)

di/(k*i*s₀+k*i-k*i²)=dt

(ln(i)-ln(1-i+s₀))/(k+ks₀)+c=t (c - constant)

at t=0 i=1

c=ln(s₀)/(k+k*s₀)

(ln(i)-ln(1-i+s₀)+ln(s₀))/(k+k*s₀)=t

i = exp[k t + k s₀ t](s₀+1)/(exp[k t + k s₀ t]+s0)

iii)

Let s₀=1000, k=1 then i(t) would be

iv)

Limitations:

does not take into account mortality, fertility, virus mutations, etc