Dynamics of Infectious Diseases

Using Lotka-Volterra equations? PredatorPrey VS

Full model Susceptible Infectious Removed ab c dNdN ddd where a is the infection rate b is the removal rate of infectives c is the rate of individuals losing immunity d is the mortality rate

Reduced model (Classic Kermack- McKendrick Model) Susceptible Infectious Removed ab where a is the infection rate b is the removal rate of infectives

} > 0 if S 0 > < 0 if S 0 < “THRESHOLD EFFECT” S(t) +I (t) + R(t) = N We can set the initial conditions as S(0)=S 0 > 0, I(0) =I 0 > 0, R(0) =0

Integrating the equation, = constant = I 0 + S 0 – ρ ln S 0

b is the removal rate from the infective class and is measured in unit (1/time) Thus, the reciprocal (1/b) is the average period of infectivity. is the fraction of population that comes into contact with an infective individual during the period of infectiveness The fraction is also known as infection’s contact rate, or intrinsic reproductive rate of disease.

R 0 is the basic reproduction rate of the infection, that is the number of infections produced by one primary infection in a whole susceptible population.

Modelling venereal disease Susceptible, S Infectious, I a b where a,a* is the infection rate b,b* is the removal rate of infectives Susceptible, S* Infectious, I* b* a* Female Male

Since we have the condition S(t)+I(t)=N and S*(t)+I*(t)=N*, we can simplify the equations to Equating both equations to zero, we can obtain the steady states

AIDS (Autoimmune Deficiency Syndrome) Susceptible X Infectious Y Natural Death AIDS A Seropositive Z (non-infectious) Disease induced Death Natural Death Natural Death B