Disease Models

Simon Frost

Overview
SIR: susceptible → infected → recovered
SIS: susceptible → infected → susceptible
SEIR: susceptible → exposed → infected → recovered
Comparing prevalence across models
Modifying disease parameters
- Varying beta
- Varying infectious duration
Disease-induced mortality
Multiple diseases simultaneously
Summary

Overview

Starsim.jl ships with several built-in disease models: SIR, SIS, and SEIR. This vignette compares their dynamics, shows how to modify disease parameters, and demonstrates running multiple diseases simultaneously.

SIR: susceptible → infected → recovered

The classic SIR model. Recovered individuals gain permanent immunity.

using Starsim
using Plots

n_contacts = 10
beta = 0.5 / n_contacts

sim_sir = Sim(
    n_agents = 1_000,
    networks = RandomNet(n_contacts=n_contacts),
    diseases = SIR(beta=beta, dur_inf=4.0, init_prev=0.01),
    dt = 1.0,
    stop = 40.0,
    rand_seed = 42,
    verbose = 0,
)
run!(sim_sir)

n_sus_sir = get_result(sim_sir, :sir, :n_susceptible)
n_inf_sir = get_result(sim_sir, :sir, :n_infected)
n_rec_sir = get_result(sim_sir, :sir, :n_recovered)
tvec_sir = 0:length(n_sus_sir)-1

plot(tvec_sir, n_sus_sir, label="Susceptible", lw=2, color=:blue)
plot!(tvec_sir, n_inf_sir, label="Infected", lw=2, color=:red)
plot!(tvec_sir, n_rec_sir, label="Recovered", lw=2, color=:green)
xlabel!("Day")
ylabel!("Number of agents")
title!("SIR Dynamics")

SIS: susceptible → infected → susceptible

In SIS, infected individuals return to the susceptible state — there is no lasting immunity. This produces endemic equilibria.

sim_sis = Sim(
    n_agents = 1_000,
    networks = RandomNet(n_contacts=n_contacts),
    diseases = SIS(beta=beta, dur_inf=4.0, init_prev=0.01),
    dt = 1.0,
    stop = 40.0,
    rand_seed = 42,
    verbose = 0,
)
run!(sim_sis)

n_sus_sis = get_result(sim_sis, :sis, :n_susceptible)
n_inf_sis = get_result(sim_sis, :sis, :n_infected)
tvec_sis = 0:length(n_sus_sis)-1

plot(tvec_sis, n_sus_sis, label="Susceptible", lw=2, color=:blue)
plot!(tvec_sis, n_inf_sis, label="Infected", lw=2, color=:red)
xlabel!("Day")
ylabel!("Number of agents")
title!("SIS Dynamics (Endemic Equilibrium)")

SEIR: susceptible → exposed → infected → recovered

SEIR adds a latent (exposed) period before individuals become infectious. This delays the epidemic peak.

sim_seir = Sim(
    n_agents = 1_000,
    networks = RandomNet(n_contacts=n_contacts),
    diseases = SEIR(beta=beta, dur_exp=2.0, dur_inf=4.0, init_prev=0.01),
    dt = 1.0,
    stop = 40.0,
    rand_seed = 42,
    verbose = 0,
)
run!(sim_seir)

n_sus_seir  = get_result(sim_seir, :seir, :n_susceptible)
n_exp_seir  = get_result(sim_seir, :seir, :n_exposed)
n_inf_seir  = get_result(sim_seir, :seir, :n_infected)
n_rec_seir  = get_result(sim_seir, :seir, :n_recovered)
tvec_seir = 0:length(n_sus_seir)-1

plot(tvec_seir, n_sus_seir, label="Susceptible", lw=2, color=:blue)
plot!(tvec_seir, n_exp_seir, label="Exposed", lw=2, color=:orange)
plot!(tvec_seir, n_inf_seir, label="Infected", lw=2, color=:red)
plot!(tvec_seir, n_rec_seir, label="Recovered", lw=2, color=:green)
xlabel!("Day")
ylabel!("Number of agents")
title!("SEIR Dynamics")

Comparing prevalence across models

prev_sir  = get_result(sim_sir,  :sir,  :prevalence)
prev_sis  = get_result(sim_sis,  :sis,  :prevalence)
prev_seir = get_result(sim_seir, :seir, :prevalence)

tvec_sir2  = 0:length(prev_sir)-1
tvec_sis2  = 0:length(prev_sis)-1
tvec_seir2 = 0:length(prev_seir)-1

plot(tvec_sir2,  prev_sir,  label="SIR",  lw=2, color=:blue)
plot!(tvec_sis2,  prev_sis,  label="SIS",  lw=2, color=:red)
plot!(tvec_seir2, prev_seir, label="SEIR", lw=2, color=:green)
xlabel!("Day")
ylabel!("Prevalence")
title!("Prevalence Comparison: SIR vs SIS vs SEIR")

Modifying disease parameters

The transmission rate beta and infectious duration dur_inf are key drivers of epidemic size. Let’s explore their effect.

Varying beta

betas = [0.01, 0.05, 0.10]
p = plot(xlabel="Day", ylabel="Prevalence", title="Effect of Beta on SIR")

for (i, b) in enumerate(betas)
    sim = Sim(
        n_agents=1_000, networks=RandomNet(n_contacts=n_contacts),
        diseases=SIR(beta=b, dur_inf=4.0, init_prev=0.01),
        dt=1.0, stop=40.0, rand_seed=42, verbose=0,
    )
    run!(sim)
    prev = get_result(sim, :sir, :prevalence)
    plot!(p, 0:length(prev)-1, prev, label="β=$b", lw=2)
end
p

Varying infectious duration

durations = [2.0, 4.0, 8.0]
p = plot(xlabel="Day", ylabel="Prevalence", title="Effect of Infectious Duration on SIR")

for d in durations
    sim = Sim(
        n_agents=1_000, networks=RandomNet(n_contacts=n_contacts),
        diseases=SIR(beta=beta, dur_inf=d, init_prev=0.01),
        dt=1.0, stop=40.0, rand_seed=42, verbose=0,
    )
    run!(sim)
    prev = get_result(sim, :sir, :prevalence)
    plot!(p, 0:length(prev)-1, prev, label="dur_inf=$d", lw=2)
end
p

Disease-induced mortality

The p_death parameter sets the probability of death upon infection.

sim_lethal = Sim(
    n_agents = 1_000,
    networks = RandomNet(n_contacts=n_contacts),
    diseases = SIR(beta=beta, dur_inf=4.0, init_prev=0.01, p_death=0.02),
    dt = 1.0,
    stop = 40.0,
    rand_seed = 42,
    verbose = 0,
)
run!(sim_lethal)

n_alive = get_result(sim_lethal, :n_alive)
tvec = 0:length(n_alive)-1
plot(tvec, n_alive, label="Population (p_death=0.02)", lw=2, color=:purple)
xlabel!("Day")
ylabel!("Population size")
title!("SIR with Disease-Induced Mortality")

Multiple diseases simultaneously

Starsim supports running multiple diseases in the same simulation. Each disease operates independently on the same population.

sim_multi = Sim(
    n_agents = 1_000,
    networks = RandomNet(n_contacts=n_contacts),
    diseases = [
        SIR(name=:flu, beta=0.08, dur_inf=3.0, init_prev=0.01),
        SIS(name=:cold, beta=0.05, dur_inf=3.0, init_prev=0.02),
    ],
    dt = 1.0,
    stop = 40.0,
    rand_seed = 42,
    verbose = 0,
)
run!(sim_multi)

prev_flu  = get_result(sim_multi, :flu, :prevalence)
prev_cold = get_result(sim_multi, :cold, :prevalence)

tvec_flu  = 0:length(prev_flu)-1
tvec_cold = 0:length(prev_cold)-1

plot(tvec_flu,  prev_flu,  label="Flu (SIR)",  lw=2, color=:red)
plot!(tvec_cold, prev_cold, label="Cold (SIS)", lw=2, color=:blue)
xlabel!("Day")
ylabel!("Prevalence")
title!("Two Diseases in One Simulation")

Summary

Model	Compartments	Key feature
SIR	S → I → R	Permanent immunity; single epidemic wave
SIS	S → I → S	No immunity; endemic equilibrium
SEIR	S → E → I → R	Latent period delays peak

Increase beta or dur_inf to produce larger epidemics
Use p_death for disease-induced mortality
Multiple diseases can share the same population and network