The SIR Model

In order to study infectious diseases and develop strategies to mitigate their harm, epidemiologists use mathematical models. The SIR model uses differential equations to mathematically express the movement of individuals from susceptible to infected and finally to recovered class. The question this paper seeks to answer is: how do the elements of the SIR model affect predictions made about disease dynamics, from both a mathematical and epidemiological perspective?

In order to establish this, this paper uses numerical experiments to analyse the effect of parametrisation, errors, and the efficiency of Runge-Kutta numerical integration methods (Euler, ode23, ode45, rk78f) and discusses this by means of both R and Excel.

The numerical experiments yielded the following findings: the transmission and recovery rate affected the disease dynamics in 2 ways. In the case of β<γ, it flattened the infectious curve and lead to a less rapid decrease in the susceptible group. In the case of β>γ, it heightened the spread of infection and lead to a rapid loss of susceptibles.

The local error estimates for the Euler and ode23 solution yielded O(h²) and O(h³) respectively, and precision stability regions for both solutions were discerned. Both methods yield similar precision stability regions.

For low error constraints (10^-3), the ode23 method proved most efficient. Higher error constraints (10^-6, 10^-10) are most efficiently dealt with the ode45 and rk78f method, respectively.