Mathematical Models of Epidemics

Reference: Mathematical Models of Epidemics} by Jan Medlock, University of Washington.

Susceptible-Infective (SI) model

Functions: susceptible (healthy) individuals S(t) and infective (sick) individuals I(t).

  

Assumptions:

1. No birth, death, immigration, emmigration, latency, recovery

2. Large constant population

3. Homogeneous mixing

4. Infection rate proportional to infectives, probability r

  

Model:

$$\frac{d S}{d t} = - r I(t) S(t)$$ (1)

$$\frac{d I}{d t} = + r I(t) S(t)$$ (2)

Note that N = S(t) + I(t) is a constant. This proves the equivalence to our previous simple model, the logistic growth model,

$$\frac{d I}{d t} = + r I(t)( N - I(t) )$$ (3)

Susceptible-Infective-Recovered (SIR) model

Functions: susceptible (healthy) individuals S(t), infective (sick) individuals I(t) and the recovered individuals R(t).

  

Assumptions: same as in the SI model except that the infective individuals recover from sickness with probability g.

  

Model:

$$\frac{d S}{d t} = - r I(t) S(t)$$ (4)

$$\frac{d I}{d t} = + r I(t) S(t) - g I(t)$$ (5)

$$\frac{d R}{d t} = + g I(t)$$ (6)

SIR model with death

Assumptions: same as in the SIR model except that all individuals may die of natural causes, probability d1, or due to the sickness, probability d2.

  

Model:

$$\frac{d S}{d t} = - r I(t) S(t) - d_1 S(t)$$ (7)

$$\frac{d I}{d t} = + r I(t) S(t) - g I(t) - d_2 I(t)$$ (8)

$$\frac{d R}{d t} = + g I(t) - d_1 R(t)$$ (9)