Signature
lambda = beta * I / N; F_SI = lambda * S
| Inputs | Definition | Unit |
|---|---|---|
beta | Rate of contacts sufficient for transmission per person, the WHO guide's p times c | per day |
I | Number of infectious people at the time, zero or above | people |
N | Total population, S plus I plus R in a closed SIR model, above zero | people |
S | Number of susceptible people at the time, zero or above | people |
lambda | Rate at which each susceptible person is infected | per day |
|---|---|---|
F_SI | New infections per unit time, the flow out of S and into I | people per day |
Function
Compartment flow balance function of a compartmental model
Maps the numbers of people in the compartments of a population model, the flows between compartments per unit time and any inflow from or outflow to outside the model to the rate of change of each compartment, or, after a time step of length dt, to the next set of compartment counts. The flows can be fixed rates times compartment sizes, as in a static model, or can depend on the number infectious through the force of infection, which makes the model dynamic and nonlinear. A Markov cohort update (HE-FN-MM-001, HE-FM-MM-001) applies one fixed transition matrix to a closed cohort, the agent-based update HE-FN-ABM-001 with the per-contact infection probability HE-FM-ABM-001 follows individuals instead of pools, and a per-cycle probability from a constant rate follows HE-FM-TP-001. Reproduction numbers of the SIR model and of stratified models are in HE-FN-BRN-001 (HE-FM-BRN-001 to HE-FM-BRN-005 and HE-CF-BRN-001), and the final size of an epidemic and R_0 estimated from it are in HE-FN-ATR-001, HE-CF-ATR-001 and HE-FM-ATR-005, so none of them is restated here. Notation follows the Compartmental Model article.
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Implementations
Excel
Force of infection and new infection flow from named cells
With the transmission rate in TransRate, the numbers infectious and susceptible in InfectiousNow and SusceptibleNow and the population in PopTotal, the first cell returns the force of infection, held in ForceInf, and the second the flow of new infections, held in NewInfRate.
=TransRate*InfectiousNow/PopTotal; =ForceInf*SusceptibleNow
Assumptions
Homogeneous mixing and contacts that do not grow with population size
Every susceptible person is equally likely to meet every infectious person, and the number of contacts per person does not depend on population size, the standard incidence that Hethcote judges more realistic for human diseases than simple mass action. Mixing that differs between groups is handled by HE-FM-CMPM-003. lambda and beta share one time unit.
Force of infection read as a rate and converted to a probability only through the hazard
lambda is a rate, so lambda times dt is close to the probability of infection in a step only when it is small. Under a constant hazard the probability is 1 minus exp(minus lambda dt) by HE-FM-TP-001, which the article applies to its day 1, with lambda of 0.003 per day giving 0.0029955 and 29.66 infections among 9,900 susceptibles rather than 29.7.
Worked examples
Article's day-1 force of infection and new infections without vaccination
With beta of 0.3 per day, 100 infectious people in an illustrative population of 10,000 and 9,900 susceptibles, the force of infection is 0.003 per day and new infections are 29.7 a day, the article's day-1 figures.
beta = 0.3; I = 100; N = 10000; S = 9900; lambda = 0.003; F_SI = 29.7
Article's day-2 force of infection without a programme
After day 1 the article has 119.7 infectious and 9,870.3 susceptible people, so the force of infection rises to 0.003591 per day and new infections to about 35.44 a day.
beta = 0.3; I = 119.7; N = 10000; S = 9870.3; lambda = 0.003591; F_SI = 35.4442
No infectious people gives no force of infection
With nobody infectious the force of infection and the flow of new infections are zero whatever the transmission rate, a limiting case for checking an implementation.
beta = 0.3; I = 0; N = 10000; S = 9900; lambda = 0; F_SI = 0
Common errors
Frequency-dependent transmission rate entered into the mass action form
Entering the article's beta of 0.3 per day into beta S I without dividing by N gives 297,000 new infections a day in a population of 10,000 instead of 29.7. Hethcote shows that the mass action coefficient eta equals beta/N, here 0.00003 per person per day, so a coefficient taken from one form is rescaled before use in the other.
Susceptible count used in place of the whole population in the force of infection
Dividing by S instead of N gives 0.3 times 100/9,900, about 0.00303, and 30.0 infections on the article's day 1, 1% too many, and the error grows as susceptibles are used up. With 5,000 susceptibles left and 100 infectious the wrong force is 0.006, twice the correct 0.003 (computed here for illustration).
Sources
WHO guide on the force of infection and the incidence of new cases
World Health Organization. WHO guide for standardization of economic evaluations of immunization programmes, 2nd edition (WHO/IVB/19.10). Geneva, WHO, 2019. Section 6.1.2, Incidence and the force of infection, which defines the force of infection lambda as the probability per unit of time that a susceptible person becomes infected, the per-susceptible rate of infection, and gives the incidence of new cases as lambda X = p c (Y/N) X, with X the number of susceptible people, Y the number of infectious people, N the total population, p the probability of transmission given contact and c the number of contacts.
Hethcote on standard incidence against simple mass action
Hethcote HW. The mathematics of infectious diseases. SIAM Review. 2000;42(4):599-653. Section 2.1, Formulating Epidemiology Models (pages 602 and 603), which defines the standard incidence (beta I/N) S as the number of new cases per unit time, with beta the average number of adequate contacts of a person per unit time, compares it with the simple mass action law eta I S, for which beta = eta N, and concludes that the standard incidence is more realistic for human diseases.
van den Driessche on mass action transmission and the force of infection beta I
van den Driessche P. Reproduction numbers of infectious disease models. Infectious Disease Modelling. 2017;2(3):288-303. Section 2.1, SIR epidemic model, which models transmission by mass action with the term beta S I, beta being the transmission rate constant and beta I the force of infection, in the equations dS/dt = minus beta S I, dI/dt = beta S I minus gamma I and dR/dt = f gamma I.
Canonical Identity
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