SEIR model — compartments, rates and the R₀ threshold
The four compartments with the rate on every arrow: force of infection βI/N out of S, σ = 1/5.2 d⁻¹ through the latent period, γ = 1/7 d⁻¹ into recovery, and what follows from them — R₀ = β/γ, Rₜ = R₀S/N falling as susceptibles are used up, a peak at S/N = 1/R₀ and a herd-immunity threshold of 1 − 1/R₀.
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{
"nodes": [
{ "id": "susceptible", "type": "sphere", "label": "S — 999 susceptible, no immunity", "x": 40, "y": 40, "width": 150, "height": 104, "fill": "#dbeafe", "stroke": "#1d4ed8", "bold": true },
{ "id": "exposed", "type": "sphere", "label": "E — infected, not yet infectious", "x": 215, "y": 40, "width": 150, "height": 104, "fill": "#dbeafe", "stroke": "#1d4ed8", "bold": true },
{ "id": "infectious", "type": "sphere", "label": "I — 1 infectious and shedding", "x": 390, "y": 40, "width": 150, "height": 104, "fill": "#dbeafe", "stroke": "#1d4ed8", "bold": true },
{ "id": "recovered", "type": "sphere", "label": "R — recovered and assumed immune", "x": 565, "y": 40, "width": 150, "height": 104, "fill": "#dbeafe", "stroke": "#1d4ed8", "bold": true },
{ "id": "r0", "type": "gauge-chart", "label": "R₀ = 12.0 (β/γ) — measles-like", "x": 740, "y": 40, "width": 150, "height": 104, "fill": "#ccfbf1", "stroke": "#0f766e", "bold": true },
{ "id": "herd", "type": "shield", "label": "Herd immunity 1 − 1/R₀ = 92%: HIT = 91.7 %", "x": 915, "y": 40, "width": 150, "height": 104, "fill": "#fce7f3", "stroke": "#be185d", "bold": true },
{ "id": "contacts", "type": "network-globe", "label": "β = 1.7148 /d — contacts × transmission", "x": 40, "y": 200, "width": 150, "height": 104, "fill": "#e0f2fe", "stroke": "#0369a1", "bold": true },
{ "id": "sigma", "type": "clock", "label": "σ = 0.1923 /d, i.e. 1/5.2 d⁻¹ latent", "x": 215, "y": 200, "width": 150, "height": 104, "fill": "#fef3c7", "stroke": "#b45309", "bold": true },
{ "id": "gamma", "type": "clock", "label": "γ = 0.1429 /d, i.e. 1/7 d⁻¹ infectious", "x": 390, "y": 200, "width": 150, "height": 104, "fill": "#fef3c7", "stroke": "#b45309", "bold": true },
{ "id": "population", "type": "hexagon", "label": "N = 1000 — a closed population", "x": 565, "y": 200, "width": 150, "height": 104, "fill": "#e0f2fe", "stroke": "#0369a1" },
{ "id": "isolation", "type": "shield-check", "label": "Isolation shortens the infectious period", "x": 740, "y": 200, "width": 150, "height": 104, "fill": "#e0f2fe", "stroke": "#0369a1" },
{ "id": "vaccination", "type": "syringe", "label": "Vaccination moves S straight to R", "x": 915, "y": 200, "width": 150, "height": 104, "fill": "#e0f2fe", "stroke": "#0369a1", "bold": true },
{ "id": "odes", "type": "clipboard", "label": "dS/dt = −βSI/N; dE/dt = βSI/N − σE", "x": 40, "y": 360, "width": 150, "height": 104, "fill": "#f1f5f9", "stroke": "#475569", "bold": true },
{ "id": "lambda", "type": "arrow-right", "label": "Force of infection λ = βI/N", "x": 215, "y": 360, "width": 150, "height": 104, "fill": "#fef3c7", "stroke": "#b45309", "bold": true },
{ "id": "rt", "type": "line-chart", "label": "Rₜ = R₀ · S/N falls as S is depleted", "x": 390, "y": 360, "width": 150, "height": 104, "fill": "#ccfbf1", "stroke": "#0f766e", "bold": true },
{ "id": "peak", "type": "target", "label": "The epidemic peaks when S/N = 1/R₀", "x": 565, "y": 360, "width": 150, "height": 104, "fill": "#ccfbf1", "stroke": "#0f766e", "bold": true },
{ "id": "births", "type": "chevron", "label": "Births top up S at rate µN", "x": 740, "y": 360, "width": 150, "height": 104, "fill": "#ede9fe", "stroke": "#6d28d9" },
{ "id": "waning", "type": "arrow-left", "label": "Waning immunity returns R to S", "x": 915, "y": 360, "width": 150, "height": 104, "fill": "#fef3c7", "stroke": "#b45309" },
{ "id": "mixing", "type": "cross", "label": "Homogeneous mixing is the weak assumption", "x": 215, "y": 520, "width": 150, "height": 104, "fill": "#fee2e2", "stroke": "#b91c1c", "bold": true },
{ "id": "latency", "type": "clipboard", "label": "Incubation is clinical; latency is infectious", "x": 390, "y": 520, "width": 150, "height": 104, "fill": "#f1f5f9", "stroke": "#475569" },
{ "id": "deaths", "type": "bucket", "label": "A fatal disease needs a fifth compartment", "x": 565, "y": 520, "width": 150, "height": 104, "fill": "#f1f5f9", "stroke": "#475569" }
],
"edges": [
{ "id": "e1", "source": "susceptible", "target": "exposed", "label": "βSI/N — new infections", "thickness": 2.5 },
{ "id": "e2", "source": "exposed", "target": "infectious", "label": "σE — the latent period ends", "thickness": 2.5 },
{ "id": "e3", "source": "infectious", "target": "recovered", "label": "γI — recovery", "thickness": 2.5 },
{ "id": "e4", "source": "infectious", "target": "lambda", "label": "the I in βI/N — this is the loop", "style": "dashed" },
{ "id": "e5", "source": "lambda", "target": "exposed", "label": "λ·S new exposures per day", "style": "dashed" },
{ "id": "e6", "source": "contacts", "target": "lambda", "style": "dashed" },
{ "id": "e7", "source": "sigma", "target": "exposed", "label": "sets how long E lasts", "style": "dashed" },
{ "id": "e8", "source": "gamma", "target": "infectious", "label": "sets how long I lasts", "style": "dashed" },
{ "id": "e9", "source": "population", "target": "susceptible", "label": "N = S + E + I + R stays constant", "style": "dashed" },
{ "id": "e10", "source": "contacts", "target": "r0", "label": "R₀ = β/γ", "style": "dashed" },
{ "id": "e11", "source": "gamma", "target": "r0", "style": "dashed" },
{ "id": "e12", "source": "r0", "target": "herd", "label": "1 − 1/R₀", "thickness": 2.5 },
{ "id": "e13", "source": "r0", "target": "rt", "thickness": 2.5 },
{ "id": "e14", "source": "rt", "target": "peak", "label": "the peak is where Rₜ crosses 1", "thickness": 2.5 },
{ "id": "e15", "source": "herd", "target": "vaccination", "label": "sets the coverage target", "style": "dashed" },
{ "id": "e16", "source": "vaccination", "target": "recovered", "label": "immunity without infection", "thickness": 2.5 },
{ "id": "e17", "source": "vaccination", "target": "susceptible", "label": "removes them from S", "style": "dashed" },
{ "id": "e18", "source": "isolation", "target": "gamma", "label": "raises γ, so lowers R₀", "style": "dashed" },
{ "id": "e19", "source": "odes", "target": "susceptible", "label": "the equations the arrows encode", "style": "dashed" },
{ "id": "e20", "source": "latency", "target": "exposed", "style": "dashed" },
{ "id": "e21", "source": "mixing", "target": "contacts", "label": "real contact networks are clustered", "style": "dashed" },
{ "id": "e22", "source": "infectious", "target": "deaths", "style": "dashed" },
{ "id": "e23", "source": "births", "target": "susceptible", "label": "µN — switched off in this closed run" },
{ "id": "e24", "source": "recovered", "target": "waning" },
{ "id": "e25", "source": "waning", "target": "susceptible", "label": "why measles never quite goes away", "style": "dashed" }
]
}