SIR epidemic simulation · Amirassyl Sagyndyk

The model

The population is split into three groups: susceptible S, infected I, and removed R.

dS/dt = -b·S·I / N
dI/dt =  b·S·I / N - k·I
dR/dt =  k·I

b is the infection rate and k the removal rate. The equations have no closed-form solution, so the solver steps them forward one day at a time: next value = current value + step × derivative.

Parameters

Parameter	Value	Source
Population	500,000 susceptible, 3 infected	Approximate size of Karaganda
Infection rate `b`	0.17	Mean transmission rate for Karaganda in Koichubekov et al. (2023)
Removal rate `k`	0.04	Baseline choice
Step size	1 day

The reported standard deviation of b is 0.075, so I also ran b = 0.10 and b = 0.25 to cover roughly one standard deviation either side, and k = 0.02 and k = 0.06 to see the effect of slower and faster removal. That gives five scenarios in total.

What it shows

A higher infection rate gives an earlier and taller peak of infections. A higher removal rate lowers and flattens the peak, and leaves more of the population never infected.

The baseline run:

Line chart of the baseline run (b = 0.17, k = 0.04) over 250 days. The susceptible curve falls from 500,000 in an S shape, the infected curve rises to a single peak a little after day 100 and falls back, and the removed curve climbs to nearly the whole population.

The higher infection rate, b = 0.25:

Line chart of the higher infection rate run (b = 0.25, k = 0.04) over 200 days. The infected curve peaks before day 75, earlier and higher than in the baseline run, and the susceptible curve drops to almost zero.

Assumptions and limits

Everyone mixes with everyone equally. There are no households, ages, or districts.
b and k stay constant, so lockdowns, vaccination, and behaviour change are not modelled.
The population is fixed: no births, deaths from other causes, or travel.
Euler’s method with a one-day step is a first-order approximation. A smaller step or a higher-order method would track the true solution more closely.

Reference

Koichubekov, B., Takuadina, A., Korshukov, I., Turmukhambetova, A., & Sorokina, M. (2023). Is it possible to predict COVID-19? Stochastic system dynamic model of infection spread in Kazakhstan. Healthcare, 11(5), 752.