Epidemic modeling

SIR Model Explorer

Adjust SIR parameters to simulate how an infectious disease propagates through a population. This simplified model assumes homogeneous mixing, constant parameters, and no births, deaths, or reinfections. Use it to understand basic concepts like R₀, peak infections, and epidemic duration.

Total population

Initial infected

Initial recovered

Transmission rate β

Average contacts per day × probability of transmission per contact.

Recovery rate γ

Typically 1 ÷ infectious period (e.g., 1/10 days = 0.1).

Simulation days

Key metrics

R₀ (β ÷ γ): 3.00
Peak infected: 156,057
Peak day: 54

Population status (day 180)

Susceptible: 26,662
Infected: 5
Recovered: 473,333

Growing outbreak (R₀ > 1)

Each infected person infects >1 other person. Intensify interventions (masking, distancing, vaccination) to reduce transmission.

Modify β (contact rate) to model interventions like distancing or masking.

Simulation Table

Day	Susceptible	Infected	Recovered
0	499,950	50	0
1	499,935	60	5
2	499,917	72	11
3	499,895	86	18
4	499,869	104	27
5	499,838	124	37
6	499,801	149	50
7	499,756	179	65
8	499,703	215	82
9	499,638	258	104
10	499,561	309	130
11	499,468	371	161
12	499,357	445	198
13	499,224	534	242
14	499,064	641	296
15	498,872	768	360
16	498,642	922	437
17	498,366	1,105	529
18	498,036	1,325	639
19	497,640	1,589	772
20	497,165	1,904	931
21	496,597	2,282	1,121
22	495,918	2,733	1,349
23	495,104	3,273	1,623
24	494,132	3,918	1,950
25	492,970	4,688	2,342
26	491,584	5,606	2,810
27	489,930	6,699	3,371
28	487,961	7,998	4,041
29	485,619	9,540	4,841
30	482,840	11,366	5,795
31	479,547	13,522	6,931
32	475,657	16,060	8,283
33	471,073	19,037	9,889
34	465,692	22,514	11,793
35	459,401	26,554	14,045
36	452,082	31,218	16,700
37	443,614	36,564	19,822
38	433,882	42,640	23,478
39	422,782	49,476	27,742
40	410,231	57,079	32,690
41	396,182	65,421	38,398
42	380,631	74,430	44,940
43	363,633	83,985	52,383
44	345,309	93,910	60,781
45	325,852	103,976	70,172
46	305,524	113,907	80,570
47	284,643	123,397	91,960
48	263,569	132,131	104,300
49	242,673	139,814	117,513
50	222,316	146,190	131,495
51	202,816	151,071	146,114
52	184,432	154,347	161,221
53	167,352	155,993	176,655
54	151,689	156,057	192,255
55	137,485	154,654	207,860
56	124,728	151,947	223,326
57	113,357	148,123	238,520
58	103,282	143,385	253,333
59	94,397	137,932	267,671
60	86,584	131,951	281,464

Showing first 61 days. Adjust “Simulation days” to view shorter or longer periods.

How to Use This Calculator

Estimate β (transmission) and γ (recovery)

β reflects contact frequency and infectiousness; γ is typically 1 ÷ duration of infectiousness.

Set initial conditions

Define how many individuals are currently infected or immune. The rest are assumed susceptible.

Interpret results as conceptual guidance

The SIR model is simplified. Real-world outbreaks depend on demographics, behavior, vaccines, and variant characteristics.

Formula

SIR differential equations:

dS/dt = −βSI ÷ N
dI/dt = βSI ÷ N − γI
dR/dt = γI

R₀ = β ÷ γ. Epidemic grows when R₀ > 1. Solutions here use a simple daily-step Euler approximation.

Full Description

Mathematical models such as SIR help visualize how infectious diseases spread and respond to interventions. β can decrease with masking, distancing, ventilation, or vaccination, while γ increases when treatment shortens infectious periods. Realistic models incorporate additional compartments (SEIR), age structure, mobility, and stochastic effects, but the SIR baseline remains a valuable conceptual tool.

Use this calculator for educational exploration. For policy or clinical decisions, rely on expert epidemiologic analyses, updated surveillance data, and validated models tailored to the pathogen of interest.

Frequently Asked Questions

What does β represent?

β is the transmission rate: contacts per day × probability of transmission per contact. Lower β by reducing contacts or improving protection.

How is γ determined?

γ is the recovery/removal rate. If infectious period averages 7 days, γ ≈ 1/7 ≈ 0.14.

Can I model waning immunity?

Waning immunity requires an SIRS or SEIRS model. This calculator assumes recovered individuals stay immune.

Why do results differ from real outbreaks?

Real epidemics involve heterogeneity, behavior change, vaccinations, seasonality, and stochastic effects that the basic SIR model omits.