Introduction

Catch the flu, fight it off, catch it again next season. Catch a cold, recover, catch another cold two months later. For many of the most common infectious diseases, recovery grants no lasting immunity — the immune system clears the infection, but leaves the host just as vulnerable as before.

Mathematical epidemiology captures this pattern with the SIS model (Susceptible → Infected → Susceptible). The population is divided into two compartments: those who can be infected (S, susceptible) and those who currently are infected (I, infectious). When someone recovers they drop straight back into S, not into a protected class.

The result is a feedback loop that never ends. As long as the disease spreads fast enough relative to how quickly people recover, it does not die out — it settles. Prevalence stops falling and stops rising, locking in at a fixed fraction of the population that remains infected forever. This is the endemic equilibrium, and it is the model's central prediction.

The SIS model is the simplest tool that captures endemic persistence, and it draws a clean line between two futures: extinction (the disease fades to zero) or endemicity (it stays forever). Which future unfolds depends on a single number.

Try It

Adjust the sliders to set the transmission rate β\beta (how fast the disease spreads) and the recovery rate γ\gamma (how fast infected people clear the infection). The simulation runs the SIS differential equations in discrete time steps and plots the fraction of the population that is currently infected.

<!-- Controls for beta (transmission rate) and gamma (recovery rate) -->
<div class="controls">
  <div class="slider-row">
    <label for="beta">Transmission rate <span class="param-name">β</span></label>
    <input type="range" id="beta" min="0.1" max="1.0" step="0.01" value="0.4">
    <span id="beta-val" class="val-badge">0.40</span>
  </div>
  <div class="slider-row">
    <label for="gamma">Recovery rate <span class="param-name">γ</span></label>
    <input type="range" id="gamma" min="0.05" max="0.8" step="0.01" value="0.2">
    <span id="gamma-val" class="val-badge">0.20</span>
  </div>
  <div class="r0-badge" id="r0-badge">R₀ = 2.00</div>
</div>
<canvas id="chart" width="480" height="220"></canvas>
<div class="status" id="status"></div>
<div class="btns">
  <button id="run" type="button">Run simulation</button>
  <button id="reset" type="button" class="ghost">Reset</button>
</div>

/* Styles for the SIS endemic model simulator */
* { box-sizing: border-box; }
body { font-family: system-ui, sans-serif; color: #222; margin: 0; }
.controls { display: flex; flex-direction: column; gap: .5rem; margin-bottom: .7rem; }
.slider-row { display: flex; align-items: center; gap: .5rem; flex-wrap: wrap; }
label { font-size: .85rem; width: 8rem; flex-shrink: 0; }
.param-name { font-weight: 700; font-style: italic; }
input[type=range] { flex: 1; min-width: 120px; accent-color: #1d3557; }
.val-badge { font-size: .85rem; font-weight: 600; width: 3rem; text-align: right; color: #1d3557; }
.r0-badge { font-size: 1rem; font-weight: 700; color: #fff; background: #1d3557;
            display: inline-block; padding: .25rem .7rem; border-radius: 8px; margin-top: .2rem; align-self: flex-start; }
.r0-badge.endemic { background: #e63946; }
canvas { display: block; border: 1px solid #cdd9e3; border-radius: 8px; width: 100%; max-width: 480px; }
.status { font-size: 1rem; font-weight: 600; margin: .5rem 0; min-height: 1.4em; }
.status.extinct { color: #0a7d33; }
.status.endemic { color: #e63946; }
.btns { display: flex; gap: .5rem; flex-wrap: wrap; }
button { font: 600 14px system-ui, sans-serif; padding: .45rem .9rem; border: 1px solid #1d3557;
         background: #1d3557; color: #fff; border-radius: 8px; cursor: pointer; }
button.ghost { background: #fff; color: #1d3557; }

const i18nFmt = (s, v) => s.replace(/%(\w+)%/g, (_, k) => String(v[k] ?? ''));

// SIS model: di/dt = beta*i*(1-i) - gamma*i
// Endemic equilibrium i* = 1 - gamma/beta = 1 - 1/R0  (exists only when R0 > 1)
// Disease-free equilibrium i* = 0 is stable iff R0 <= 1

const STEPS = 200;   // number of discrete time steps to simulate
const DT = 0.5;      // step size in time units
const I0 = 0.01;     // initial fraction infected (1%)

// DOM element references
const betaSlider  = document.getElementById('beta');
const gammaSlider = document.getElementById('gamma');
const betaVal     = document.getElementById('beta-val');
const gammaVal    = document.getElementById('gamma-val');
const r0Badge     = document.getElementById('r0-badge');
const canvas      = document.getElementById('chart');
const statusEl    = document.getElementById('status');
const ctx         = canvas.getContext('2d');

function getParams() {
  return { beta: parseFloat(betaSlider.value), gamma: parseFloat(gammaSlider.value) };
}

// Euler integration of the SIS ODE over STEPS steps
function simulate(beta, gamma) {
  const series = [I0];
  let i = I0;
  for (let t = 1; t <= STEPS; t++) {
    const di = beta * i * (1 - i) - gamma * i;
    i = Math.max(0, Math.min(1, i + DT * di));
    series.push(i);
  }
  return series;
}

// Render the prevalence curve onto the canvas
function draw(series, beta, gamma) {
  const W = canvas.width, H = canvas.height;
  const pad = { top: 20, right: 20, bottom: 30, left: 44 };
  const w = W - pad.left - pad.right;
  const h = H - pad.top - pad.bottom;

  ctx.clearRect(0, 0, W, H);

  // horizontal grid lines and y-axis labels
  ctx.strokeStyle = '#e0e7ee';
  ctx.lineWidth = 1;
  for (let k = 0; k <= 4; k++) {
    const y = pad.top + (h / 4) * k;
    ctx.beginPath(); ctx.moveTo(pad.left, y); ctx.lineTo(pad.left + w, y); ctx.stroke();
    ctx.fillStyle = '#888'; ctx.font = '10px system-ui'; ctx.textAlign = 'right';
    ctx.fillText((1 - k / 4).toFixed(2), pad.left - 4, y + 4);
  }

  // dashed red line marking the endemic equilibrium when R0 > 1
  const R0 = beta / gamma;
  if (R0 > 1) {
    const endemic = 1 - 1 / R0;
    const ey = pad.top + h * (1 - endemic);
    ctx.strokeStyle = '#e63946'; ctx.lineWidth = 1.5; ctx.setLineDash([4, 3]);
    ctx.beginPath(); ctx.moveTo(pad.left, ey); ctx.lineTo(pad.left + w, ey); ctx.stroke();
    ctx.setLineDash([]);
    ctx.fillStyle = '#e63946'; ctx.font = '10px system-ui'; ctx.textAlign = 'left';
    ctx.fillText('I* = ' + endemic.toFixed(2), pad.left + 2, ey - 3);
  }

  // prevalence curve in dark blue
  ctx.strokeStyle = '#1d3557'; ctx.lineWidth = 2.5; ctx.setLineDash([]);
  ctx.beginPath();
  series.forEach((v, t) => {
    const x = pad.left + (t / STEPS) * w;
    const y = pad.top + h * (1 - v);
    t === 0 ? ctx.moveTo(x, y) : ctx.lineTo(x, y);
  });
  ctx.stroke();

  // x-axis label
  ctx.fillStyle = '#888'; ctx.font = '10px system-ui'; ctx.textAlign = 'center';
  ctx.fillText('0', pad.left, pad.top + h + 16);
  ctx.fillText(String(STEPS), pad.left + w, pad.top + h + 16);
  ctx.fillText('time steps', pad.left + w / 2, pad.top + h + 16);
}

function updateBadge(R0) {
  r0Badge.textContent = 'R₀ = ' + R0.toFixed(2);
  r0Badge.className = 'r0-badge' + (R0 > 1 ? ' endemic' : '');
}

function setStatus(text, cls) {
  statusEl.textContent = text;
  statusEl.className = 'status ' + cls;
}

// Run a simulation with current slider values and update the display
function run() {
  const { beta, gamma } = getParams();
  const R0 = beta / gamma;
  const series = simulate(beta, gamma);
  draw(series, beta, gamma);
  updateBadge(R0);
  const finalI = series[series.length - 1];
  if (R0 <= 1 || finalI < 1e-4) {
    setStatus(i18nFmt('R₀ = %r0% ≤ 1 — the disease dies out.', { r0: R0.toFixed(2) }), 'extinct');
  } else {
    const endemic = (1 - 1 / R0);
    setStatus(i18nFmt('R₀ = %r0% > 1 — endemic at I* ≈ %star% of the population.', { r0: R0.toFixed(2), star: endemic.toFixed(2) }), 'endemic');
  }
}

// Live-update numeric labels as sliders move
betaSlider.addEventListener('input', () => {
  betaVal.textContent = parseFloat(betaSlider.value).toFixed(2);
});
gammaSlider.addEventListener('input', () => {
  gammaVal.textContent = parseFloat(gammaSlider.value).toFixed(2);
});

document.getElementById('run').addEventListener('click', run);
document.getElementById('reset').addEventListener('click', () => {
  betaSlider.value = '0.4'; gammaSlider.value = '0.2';
  betaVal.textContent = '0.40'; gammaVal.textContent = '0.20';
  setStatus('', '');
  const { beta, gamma } = getParams();
  updateBadge(beta / gamma);
  draw(simulate(beta, gamma), beta, gamma);
});

// Draw the initial simulation on page load
run();
Result
Console

Watch what happens as you cross the threshold R0=β/γ=1R_0 = \beta / \gamma = 1. Below it the curve falls to zero — the disease dies out. Above it the curve rises and levels off at the endemic equilibrium I∗=1−γ/βI^* = 1 - \gamma/\beta, no matter where it started. That plateau is the model's signature: prevalence stabilizes, but never reaches zero.

The Real Complexity

The SIS model's dynamics are governed by a pair of differential equations. Because S+I=NS + I = N is constant, everything reduces to one equation for the infected fraction i=I/Ni = I/N:

didt=β i (1−i)−γ i=i(β(1−i)−γ)\frac{di}{dt} = \beta\, i\,(1 - i) - \gamma\, i = i\bigl(\beta(1-i) - \gamma\bigr)

Setting di/dt=0di/dt = 0 gives two equilibria:

  • Disease-free equilibrium: i∗=0i^* = 0 — the infection has vanished.
  • Endemic equilibrium: i∗=1−γ/β=1−1/R0i^* = 1 - \gamma/\beta = 1 - 1/R_0, valid only when R0>1R_0 > 1.

The basic reproduction number R0=β/γR_0 = \beta / \gamma is the average number of new infections one case generates in a fully susceptible population. It is the single number that determines the model's fate:

  • If R0≤1R_0 \leq 1: the disease-free equilibrium is stable — any outbreak eventually dies out.
  • If R0>1R_0 > 1: the endemic equilibrium is stable — the disease persists at fraction i∗=1−1/R0i^* = 1 - 1/R_0 of the population.

Stability follows from linearizing around each equilibrium: the eigenvalue at i∗=0i^* = 0 is β−γ\beta - \gamma, which is negative exactly when R0<1R_0 < 1. The endemic equilibrium, when it exists, has a negative eigenvalue and is globally attracting on (0,1)(0,1).

This is a solved model: the analysis is exact and closed-form, and the threshold theorem has been rigorously proved. Unlike models on heterogeneous networks or with spatial structure, the homogeneous SIS model leaves nothing uncertain — its long-run behavior is entirely determined by R0R_0.

Where It Matters

Any spreading process where "recovery" returns agents to full vulnerability fits the SIS framework:

  • Bacterial STIs: gonorrhea and chlamydia produce no lasting immunity; the SIS model was the basis of Anderson & May's classic 1991 analysis of their endemic persistence.
  • Seasonal respiratory infections: the common cold cycles through populations year after year because coronaviruses and rhinoviruses generate short-lived immunity.
  • Computer network worms: a machine patched against one variant may be reinfected by a mutated strain; SIS dynamics on network graphs model botnet persistence and inform patch-deployment strategies.
  • Rumor and information spread: a person who forgets a rumor becomes susceptible to hearing it again — SIS-like dynamics appear in social-network contagion models.
  • Antimicrobial resistance: a hospital pathogen can recolonize patients who have cleared a previous infection, creating endemic reservoirs that SIS models help size and target.

The model's clean threshold — one number decides extinction or endemicity — makes R0R_0 the central target for public-health intervention. Reducing β\beta (through barrier methods, hygiene, or network quarantine) or increasing γ\gamma (through treatment) below the critical ratio R0=1R_0 = 1 is the universal goal, whether the "infection" is a bacterium, a virus, or a piece of malware. Compare this with the SIR model where immunity eventually burns out an epidemic even above the threshold.

Conclusion

The SIS model distills a profound truth into the simplest possible mathematics: without immunity, a disease that spreads faster than people recover will never leave. It will not explode without bound either — it will settle, precisely and predictably, at the fraction 1−1/R01 - 1/R_0 of the population.

That fraction is the endemic equilibrium, and reaching it does not require any clever computation — it is simply where the inflow of new infections exactly balances the outflow of recoveries. The model is analytically solved: the threshold R0=1R_0 = 1 is sharp, the equilibria are explicit, and their stability is global.

What makes the SIS model valuable is not sophistication but clarity. It tells us exactly which lever to pull — lower R0R_0 below one — and exactly how much the disease will shrink for each unit of reduction above that threshold. In a world where many diseases genuinely grant no immunity, that clarity is indispensable.

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