Educational model. Simplified outbreaks with textbook-range parameters, for learning how epidemics and interventions work. Not a forecast or public-health planning tool.
Day 0Reff –Infectious –06:00

Key facts to remember

  • R₀ = secondary cases from one case in a fully susceptible population; Reff = R₀ × fraction susceptible. Outbreak grows while Reff > 1.
  • Herd immunity threshold = 1 − 1/R₀. Coverage needed = threshold ÷ vaccine efficacy. Measles (R₀ 12–18) needs ~93–95% immune: hence two MR doses and a 95% coverage target.
  • KEPI: BCG + OPV0 at birth; OPV, Penta, PCV10 and Rota at 6, 10, 14 weeks; IPV at 14 weeks (and 9 months); MR at 9 and 18 months; malaria vaccine at 6, 7, 9, 24 months in selected counties; HPV single dose for girls at 10.
  • Attack rate = new cases ÷ population at risk (× 100). CFR = deaths ÷ cases (× 100). Risk ratio = attack rate in exposed ÷ attack rate in unexposed.
  • Epi curve: one sharp peak within one incubation period = point source; successive growing waves = propagated (person-to-person).
  • Cholera: suspected case in an outbreak area = anyone with (or dying from) acute watery diarrhoea; confirm by culture or PCR. WASH stops transmission; prompt ORS/Ringer's lactate keeps CFR below 1%. Follow contacts for 5 days.
  • Malaria: vector control (ITNs, IRS) works best by killing adult mosquitoes, because transmission needs a mosquito to survive the ~10-day parasite development. Test with mRDT, treat with AL.
  • Contact tracing: list contacts, follow daily for the maximum incubation period (measles 21 days), refer anyone who becomes ill, report through the CHA to IDSR.

Compartmental models

Compartmental models split a population of size $N$ into groups by infection status and describe the flows between them with differential equations. The classic SIR model (Kermack and McKendrick, 1927) has Susceptible, Infectious and Removed people. In words: each day new infections happen at a rate proportional to how often susceptible and infectious people meet ($S \times I/N$), and a fixed fraction of infectious people recover. $S$, $I$ and $R$ are the numbers of people in each group and $t$ is time in days:

$$\frac{dS}{dt}=-\beta\frac{SI}{N},\qquad \frac{dI}{dt}=\beta\frac{SI}{N}-\gamma I,\qquad \frac{dR}{dt}=\gamma I$$

$\beta$ is the transmission rate (effective contacts per infectious person per day) and $\gamma = 1/D$ is the recovery rate for an average infectious period $D$. The lab uses SEIR(D)+V: an Exposed (latent) stage of mean length $1/\sigma$, a split of removals into recovered and dead, and a vaccinated-and-protected group:

$$\begin{aligned}\frac{dS}{dt}&=-\beta_c\frac{SI}{N} & \frac{dE}{dt}&=\beta_c\frac{SI}{N}-\sigma E & \frac{dI}{dt}&=\sigma E-\gamma I\\ \frac{dR}{dt}&=(1-f)\,\gamma I & \frac{dD}{dt}&=f\,\gamma I & V&=c\,\varepsilon N\ (\text{constant})\end{aligned}$$

where $\beta_c$ is the transmission rate after control measures, $\sigma$ the rate of becoming infectious (1/latent period), $D$ in these equations counts deaths (not the infectious period), $f$ is the case fatality ratio, $c$ the vaccine coverage and $\varepsilon$ the efficacy (an "all-or-nothing" vaccine: a fraction $\varepsilon$ of vaccinees are fully protected, the rest are not).

The basic reproduction number

$$R_0=\frac{\beta}{\gamma}=\beta D$$

$R_0$ is the average number of secondary cases caused by one infectious person in a fully susceptible population. In SEIR without deaths during the latent period it is still $\beta/\gamma$: the latent period changes the speed of the epidemic, not how many people each case infects. As immunity builds up, the effective reproduction number falls:

$$R_\text{eff}(t)=R_c\,\frac{S(t)}{N},\qquad R_c=R_0\times(\text{effect of interventions})$$

Cases grow while $R_\text{eff}>1$ and shrink once $R_\text{eff}<1$. In SIR the number infectious peaks exactly when $S/N = 1/R_c$; in SEIR it is the total currently infected ($E+I$) that peaks there, with $I$ alone peaking slightly later.

Herd immunity threshold

$$\text{HIT}=1-\frac{1}{R_0},\qquad c_\text{crit}=\frac{1-1/R_0}{\varepsilon}$$

If more than the HIT fraction is immune, $R_\text{eff}<1$ from the start and an introduced case cannot start a large outbreak. Measles ($R_0 \approx 15$) needs about 93% immune, which is why two-dose coverage of 95% is the target. With an imperfect vaccine, the required coverage is the HIT divided by efficacy, and may exceed 100%: then vaccination alone cannot achieve herd immunity.

Final size relation

Without interventions changing over time, the fraction of the initially susceptible population $s_0$ that is still susceptible at the end, $s_\infty$, satisfies

$$\ln\frac{s_0}{s_\infty}=R_c\,(s_0-s_\infty)\quad\Rightarrow\quad z=1-e^{-R_0 z}\ \ \text{(fully susceptible, } z=\text{attack rate)}$$

For $R_0 = 2$ about 80% are eventually infected; for $R_0 = 1.5$, 58%. The epidemic "overshoots" the herd immunity threshold because many people are still infectious when $R_\text{eff}$ passes 1.

The epidemic curve and early growth

Early on, $S\approx N$ and cases grow exponentially at rate $r$. For SEIR, $r$ solves $(1+r/\sigma)(1+r/\gamma) = R_0$; for SIR, simply $r=\gamma(R_0-1)$. The doubling time is $\ln 2/r$. The curve rises, peaks when the susceptible fraction has fallen to $1/R_c$, and falls as fewer susceptibles remain.

Interventions in this model

  • Masks and distancing multiply transmission by $(1-m)$.
  • School closure removes school contacts, assumed here to be 20% of transmission ($\times 0.8$) in the equation model. In the town, children simply stay home.
  • Isolation of detected cases: a fraction $q$ of cases is isolated after a testing delay $d$, so they transmit for only $d/D$ of their infectious period: $\times\,[1-q(1-\min(1, d/D))]$. Fast testing matters as much as finding cases.
  • Contact tracing quarantines the household contacts of isolated cases, catching some infections before they spread. In the equations this is approximated as $\times(1-0.4q)$.
  • Vaccination removes protected people from $S$ before the outbreak.

Hospital load is approximated as a fixed share of people currently infectious. When demand exceeds beds, the case fatality ratio of the patients without a bed is doubled, a simple way to show why "flattening the curve" saves lives.

Agent-based versus compartmental models

The town view simulates every person: households, school, market, offices, and the bus stage where commuters meet. Infection passes only between people who are physically close, with a per-contact probability calibrated so that a typical case infects $R_0$ others. Agent-based models capture clustering (households saturate quickly), randomness (an outbreak can die out by chance) and targeted measures, at the cost of more assumptions. Compartmental models assume homogeneous mixing but give clean, analysable results. Real public-health modelling uses both.

CFR versus IFR. The case fatality ratio divides deaths by diagnosed cases; the infection fatality ratio divides by all infections, including mild ones never tested. The lab treats every infection as a case.

1. Measles vaccination target

  1. $R_0 = 15$ gives $\text{HIT} = 1 - 1/15 = 0.933$ (93.3%).
  2. With 97% efficacy (two doses), coverage needed $= 0.933/0.97 = 0.962$ (96.2%).
  3. With one dose at about 93% efficacy, $0.933/0.93 = 1.003$: more than 100%, so even universal one-dose coverage would not quite reach herd immunity.

Try it: Measles preset, efficacy 97%, then coverage 95% and 97%. Watch whether the outbreak takes off.

2. Final size for R₀ = 2.8

  1. Solve $z = 1 - e^{-2.8z}$ by iteration: start $z=0.9$: $1-e^{-2.52} = 0.920$; $1-e^{-2.576} = 0.924$; $1-e^{-2.587}=0.925$.
  2. So about 92.5% of a fully susceptible population would be infected without any measures. The SEIR model in the lab gives the same answer.
  3. The HIT is only $1-1/2.8 = 64\%$; the remaining 28 percentage points are the "overshoot".

3. Combining interventions

  1. COVID-19 preset: $R_0 = 2.8$, infectious period 7 days.
  2. Masks/distancing 20%: $\times 0.8$. Isolate 50% of cases after 2 days: $\times[1 - 0.5(1-2/7)] = \times 0.643$.
  3. $R_c = 2.8 \times 0.8 \times 0.643 = 1.44$. Still above 1, but the peak is much lower and later.
  4. Add contact tracing ($\times 0.8$): $R_c = 1.15$. Add school closure ($\times 0.8$): $R_c = 0.92 < 1$: the outbreak shrinks.

4. Ebola: why isolation works

  1. $R_0 = 1.8$, infectious period 10 days.
  2. Isolating 70% of cases after 2 days: $\times[1-0.7(1-0.2)] = \times 0.44$, so $R_c = 0.79$.
  3. With a 6-day testing delay: $\times[1 - 0.7(1-0.6)] = \times 0.72$, so $R_c = 1.30$ and the outbreak continues. Speed of detection is decisive.

5. When does the epidemic peak?

  1. Infections peak when $R_\text{eff} = 1$, i.e. $S/N = 1/R_c$.
  2. For influenza with $R_0 = 1.5$, the peak comes when 1/1.5 = 67% are still susceptible: 33% have already been infected or are in the latent stage.

Challenges are checked against the model currently loaded. Set up loads the starting conditions.

Guided experiments

  1. Chance extinction. In the town, set R₀ = 1.5 and press restart several times. Some outbreaks fizzle out after a few cases even though R₀ > 1. The equation model can never show this.
  2. The latent period sets the speed. In the SEIR view, change the latent period from 2 to 10 days. The peak moves later but the final size stays the same.
  3. Overshoot. With no measures, compare the total infected with the herd immunity threshold shown in the results box.
  4. Schools. In the town, close schools and watch the school empty; compare the peak with and without closure.
  5. Testing speed. With 60% isolation, slide the testing delay from 0 to 7 days and watch Rc rise.

Outbreak scenarios

Four teaching outbreaks. Load scenario sets up the lab (scroll up to watch it), then use the measures to meet the target. Each scenario checks itself as you move the controls. Numbers are textbook-range teaching values, not data for any real place.

KEPI routine immunisation schedule and herd immunity

AgeVaccinesProtects against
Birth (or first contact)BCG (0.05 mL intradermal, right upper arm); OPV 0Severe childhood TB (miliary, meningitis); polio
6 weeksOPV 1; Penta 1; PCV10 1; Rota 1Penta = diphtheria, tetanus, whole-cell pertussis, hepatitis B, Haemophilus influenzae type b. PCV10: pneumococcal pneumonia, meningitis, otitis media. Rota: rotavirus diarrhoea (Rotavac, 3 oral doses).
10 weeksOPV 2; Penta 2; PCV10 2; Rota 2
14 weeksOPV 3; Penta 3; PCV10 3; Rota 3; IPV 1
6 monthsVitamin A 100 000 IU; malaria vaccine 1*Malaria vaccine (RTS,S, now transitioning to R21): clinical and severe malaria in young children
7 monthsMalaria vaccine 2*
9 monthsMeasles–rubella (MR) 1; IPV 2; malaria vaccine 3*; yellow fever†Measles, rubella (and congenital rubella syndrome); polio; yellow fever
12 months onwardVitamin A 200 000 IU every 6 months to 59 monthsVitamin A deficiency; lowers measles deaths
18 monthsMR 2Catches the ~15% not protected by one dose at 9 months
24 monthsMalaria vaccine 4*Booster
10 years (girls)HPV, single dose (since late 2025)Cervical cancer

* Malaria vaccine only in selected malaria-endemic counties (first rolled out in the Lake region). † Yellow fever in designated high-risk counties. A hepatitis B birth dose is being piloted but is not yet routine nationally. Schedules change: always check the current KEPI chart and the child's Mother and Child Health Handbook. Sources checked: Kenya MoH/National Vaccines and Immunization Programme announcements (single-dose HPV, October 2025), WHO/UNICEF estimates of national immunization coverage (WUENIC, 2025 revision), WHO AFRO and Gavi reports on the malaria and rotavirus vaccines.

Is coverage high enough for herd immunity?

BCG has no herd-immunity threshold in this sense: it protects infants from severe TB but does little to stop adults transmitting. Hib and PCV give some herd protection by reducing carriage. IPV prevents paralysis but gives weak gut immunity, which is why oral polio vaccine is still used to stop transmission.

Contact tracing and isolation: the community health promoter's role

Community health promoters (CHPs) and their community health assistant (CHA) are the first link in Kenya's Integrated Disease Surveillance and Response (IDSR) system. In the lab, the Contact tracing toggle quarantines the households of isolated cases; in the community it looks like this:

  1. Recognise and report. A person meeting the community case definition (e.g. fever with rash for measles; acute watery diarrhoea for cholera) is referred to the link facility and reported at once to the CHA, who alerts the sub-county disease surveillance coordinator.
  2. Protect yourself and the household. Hand hygiene; chlorinated water and safe handling of vomit and stool for cholera; a mask and good ventilation for respiratory illness.
  3. Isolate the case as advised. Measles: keep away from others until 4 days after the rash appears. Cholera: treatment at an ORS point or cholera treatment centre, household disinfection. Respiratory outbreaks: separate room, mask, as per current guidance.
  4. List the contacts. Household members, carers, schoolmates, people who shared water, food, a minibus or a church service: name, age, phone, village, date of last contact and vaccination status.
  5. Follow up every contact daily for the maximum incubation period after the last exposure, asking about symptoms and recording visits (registers or eCHIS).
  6. Act on what you find. A contact who becomes ill is now a suspected case: refer, report and list their contacts. Refer unvaccinated measles contacts for MR vaccine (within 72 hours of exposure it can prevent disease).
  7. Close and report. When the follow-up period ends without symptoms, discharge the contact and submit the summary to the CHA.
DiseaseFollow contacts for
Cholera5 days
Measles21 days
Ebola, Marburg21 days
Mpox21 days
COVID-19 (as used in 2020–21)14 days

Follow-up calculator

OSCE station: outbreak investigation

10:007 marks · pass mark 70% (5 of 7)

Teaching case. In mid-March, after heavy rain, a health centre serving three villages (A: population 400; B: 250; C: 350) sees a rise in adults and children with profuse watery diarrhoea. Stool samples are sent to the county laboratory. A household survey finds that 300 of the 1000 residents had collected drinking water from water kiosk K in the week before the outbreak. You are the clinician on the rapid response team.

1. Which is the best working case definition for this investigation?

2. Apply your definition to the line list: tick the patients who are cases. The epidemic curve builds from your ticks. Swipe the table sideways to see symptoms, water source and outcome.

3. What does the shape of the epidemic curve suggest?

4. Attack rate in village A (%, 1 decimal place)

%

5. Risk ratio for drinking kiosk K water (attack rate in kiosk users ÷ attack rate in non-users, 1 decimal place)

×

6. Case fatality ratio among the cases (%, 1 decimal place)

%

7. What is the most urgent next public-health action?

Typical textbook values. Real values vary with setting, population and strain.

DiseaseR₀Latent / incubationInfectious periodHIT (1 − 1/R₀)Transmission
Measles12–18~10–12 d~8 d92–94%Airborne
COVID-19 (ancestral)2.5–3~5 d~7 d60–67%Respiratory droplets, aerosols
Seasonal influenza1.3–21–2 d~4–5 d23–50%Respiratory droplets
Cholera~1.1–2.7 (setting-dependent)2 h–5 d~5 d symptomatic shedding~50%Faecal–oral: water, food
Malaria (P. falciparum)<5 in highlands to >100 in holoendemic areas~10–14 d in humans + ~10 d in mosquitoweeks if untreatedn/a (partial immunity)Anopheles mosquito bite
Ebola virus disease1.5–22–21 d (mean ~9)~7–10 d33–50%Contact with body fluids

The lab's case fatality ratios (COVID-19 0.8%, measles 0.3%, influenza 0.1%, Ebola 50%, cholera 10% without rehydration falling to 0.8% with ORS points, malaria 0.3%) are illustrative; they depend strongly on age, health care and nutrition.