SEIR Epidemic Spread Model Calculator

The SEIR Model SEIR is a compartmental model used in epidemiology to simulate how infectious diseases spread through a population. The population is divided into four groups. Susceptible (S) can catch the disease. Exposed (E) are infected but not yet infectious, which is the incubation period. Infectious (I) can spread it. Recovered (R) are immune.

The Equations dS/dt = -beta * S * I / N. dE/dt = beta * S * I / N - sigma * E. dI/dt = sigma * E - gamma * I. dR/dt = gamma * I. Where beta is the transmission rate, sigma is the rate of progression from exposed to infectious (1/incubation period), gamma is the recovery rate (1/infectious period), and N is the total population.

R0 (Basic Reproduction Number) R0 represents how many people one infectious person will infect in a fully susceptible population. R0 = beta / gamma. If R0 > 1, the disease will spread. If R0 < 1, it will die out. Historical R0 values: measles (12-18), chickenpox (10-12), COVID-19 original strain (~2.5), seasonal influenza (1.2-1.4).

Herd Immunity Threshold The proportion of the population that needs to be immune to stop spread: HIT = 1 - 1/R0. For measles (R0=15): 93% need immunity. For COVID-19 original (R0=2.5): 60%. For seasonal flu (R0=1.3): 23%.

Overshoot: why more people are infected than the threshold requires Here is the result that surprises people, and it falls straight out of the model. Herd immunity stops growth, it does not stop the epidemic. When the susceptible pool drops past the threshold, infections start declining, but there are still a great many infectious people at that moment and every one of them goes on to infect someone. So the epidemic sails past the threshold before it stops.

At R0 = 2.5 the threshold is 60%, and an unmitigated epidemic ends with about 89% infected. That 29-point gap is the overshoot, and it is the whole argument for flattening a curve rather than letting it run: the same disease with the same R0 infects far fewer people if you slow it down enough that the decline starts closer to the threshold.

The final size obeys its own equation, independent of the incubation and infectious periods: the fraction eventually infected, x, solves x = 1 − e^(−R0·x). Only R0 goes into it. Changing the incubation period moves the peak and the timing but not the total.

Model Limitations SEIR assumes homogeneous mixing (everyone contacts everyone equally), no births/deaths during the epidemic, permanent immunity after recovery, and constant parameters. Real epidemics are more complex, but SEIR provides valuable insight into epidemic dynamics and the effects of interventions.