SIR Models and Epidemic Spread

Epidemiological Models: From SIR to COVID-19

Mathematical epidemiology allows us to predict the dynamics of infectious diseases, assess the effectiveness of interventions, and justify public measures. The COVID-19 pandemic vividly demonstrated both the power and limitations of these models.

Basic SIR Model (Kermack-McKendrick, 1927)

The population N is divided into three compartments:

• S (Susceptible): can become infected
• I (Infected): infectious
• R (Recovered): recovered or dead, immune

Equations:

dS/dt = −βSI/N

dI/dt = βSI/N − γI

dR/dt = γI

Parameter breakdown:

• β — contact rate × transmission probability. βI/N — “force of infection” for a single S-individual
• γ — rate of recovery: 1/γ = mean duration of infectious period
• S+I+R = N = const (neglecting demography)

Basic Reproduction Number R₀

Central characteristic of infection: R₀ = β/γ — the average number of secondary cases produced by one infected individual in a fully susceptible population.

Threshold condition: dI/dt = I(βS/N − γ) > 0 if and only if S > γN/β = N/R₀. An epidemic is possible as long as S > N/R₀.

Herd immunity threshold: to suppress the epidemic, S must be reduced below N/R₀, i.e., immunize a fraction p* = 1 − 1/R₀ of the population.

Examples of R₀: measles 12–18, diphtheria 6–7, COVID-19 Wuhan strain ≈ 2.4, omicron ≈ 8–15, influenza ≈ 1.5.

Final Size Hat-Trick

How many people will ultimately be infected? The epidemic’s final size R∞ = N − S∞ is determined from the transcendental equation:

S∞ = S₀ · exp(−R₀(1 − S∞/N))

Approximation for small I₀: R∞/N ≈ 1 − exp(−R₀·R∞/N) → R∞ is solved numerically. For R₀=2.5: R∞/N ≈ 0.89 (89% will be infected without interventions).

SIR Extensions

SEIR: add E (Exposed — incubation period). dE/dt = βSI/N − σE (σ — rate of completion of incubation). More realistic for many diseases.

SIR with demography: births μN and deaths μS, μI, μR → endemic equilibrium. The disease does not disappear, but becomes endemic.

Age structure: different β and γ for different age groups. POLYMOD contact matrix: ageij × β = frequency of contact of age group i with j. Key for COVID-19: fatality is sharply higher in the elderly.

Network SIR models: instead of “complete mixing” — network structure (scale-free, small world). R₀_network = ⟨k²⟩/(⟨k⟩·τ), where τ — recovery time. In scale-free networks, there is no threshold!

COVID-19: Ferguson et al. Model

The group of Neil Ferguson (Imperial College, 2020) created an agent-based model of COVID-19 with 66 million agents, representing the population of the United Kingdom.

Main parameters: R₀ ≈ 2.4 (initial Wuhan strain in the UK), incubation period 5.1 days, proportion of asymptomatic ≈ 50%, infectious period 7 days.

Scenarios and forecasts: without intervention → 510,000 deaths in the UK over 2 years. Full suppression (lockdown, testing, quarantine) → 20,000 deaths. Mitigation (self-isolation of the sick) → 260,000 deaths.

These calculations directly influenced the UK government’s decision to introduce lockdown on March 23, 2020.

Infodemic: Viral Spread of Disinformation

Disinformation spreads faster than truth (Vosoughi et al., Science 2018): false news on Twitter spreads 6 times faster than truthful news, reaching more than 10× more users. Reason: emotional charge (shock, anger) → higher probability of repost. SIR analogs for information on social networks — “infodemic models”.

Assignment: Implement an SEIR model with COVID-19 omicron parameters: R₀=10, σ=1/3 (incubation 3 days), γ=1/5 (infectious period 5 days), N=10 million, I₀=10. (1) Plot the “epidemic wave” curve (I(t)). (2) Calculate peak ICU load (10% severe cases out of I). (3) Introduce vaccination: 40% of the population immunized before the start (S₀ = 0.6N). How does the peak change? (4) Introduce “non-pharmaceutical interventions”: from week 4 β decreases by 30%. Effect?