Force of Infection

Force of Infection in Dynamic Transmission Models

In an infectious disease model the force of infection, usually written $\lambda$, is the hazard of catching the infection faced by each person who is still susceptible, so multiplying it by the number of susceptible people gives the rate of new infections. Because that hazard moves with the number of people currently infectious, it is the term that lets a dynamic transmission model count the protection a vaccine or infection-control measure gives to people who never receive it. Health economists meet it as an input estimated from serology and as the route by which herd effects reach incremental costs and QALYs. A model that holds it fixed is static, whatever its structure.

A per-susceptible hazard tied to prevalence

The WHO guide on economic evaluation of immunisation programmes (2019) defines the force of infection as the probability per unit of time that a susceptible person becomes infected, in other words the per-susceptible rate of infection. It is a rate, measured per day or per year, so it behaves like any other hazard rate. The WHO guide writes new cases in a homogeneously mixing population as the force of infection acting on susceptible people.

$$\lambda(t) = p\,c\,\frac{Y(t)}{N(t)}, \qquad \text{new infections per unit time} = \lambda(t)\,X(t)$$

where $\lambda(t)$ is the force of infection at time $t$, $p$ is the probability of transmission per contact, $c$ is the number of contacts a person makes per unit time, $Y(t)$ is the number infectious, $N(t)$ is the total population and $X(t)$ is the number susceptible. The product $pc$ is the transmission rate $\beta$ of the SIR model, $Y/N$ is the prevalence of infectiousness and $\lambda X$ is incidence in people per unit time.

The ISPOR-SMDM task force on dynamic transmission modelling (Pitman and colleagues, 2012) notes that the force of infection changes over time and feeds back into its own future value, which makes transmission dynamics nonlinear. The form above is frequency dependent. Under density-dependent transmission the hazard is $\beta' Y$, where $\beta'$ is a per-pair transmission rate, so doubling the population at the same prevalence doubles it.

Constant or responsive, the static and dynamic choice

Edmunds, Medley and Nokes divide models of vaccination benefit into those that assume a constant force of infection, so immunisation leaves the per-susceptible rate unchanged, and those in which it depends on the number infectious at each moment, so immunisation lowers the hazard on people not immunised. With cases of infection as the outcome, they show that the constant model always underestimates the cost-effectiveness of an immunisation programme, except at the extremes where nobody or everybody is immunised. Jit and Brisson note that a single birth cohort model usually has a fixed force of infection and is therefore static, as is any Markov cohort model applying a fixed infection risk each cycle.

Pitman and colleagues accept a static model when the targeted population or the intervention effect is too small to change the force of infection. Their mechanisms map onto the equation. Mass vaccination lowers the proportion susceptible, school closure lowers contact rates, antivirals shorten the duration of infectiousness and antiretrovirals lower the probability of transmission per act. Jit and Brisson add that dynamic models are unnecessary for interventions in people against rabies and West Nile virus, whose environmental reservoirs are largely unaffected by the proportion of people infected.

Age groups, contact matrices and targeting

Real populations do not mix at random. Models therefore split the force of infection by age or risk group and sum the contributions from each group.

$$\lambda_i(t) = q \sum_{j} c_{ij}\,\frac{I_j(t)}{N_j}$$

where $\lambda_i(t)$ is the force of infection on susceptible people in group $i$, $q$ is the probability of transmission per contact, $c_{ij}$ is the mean daily number of contacts a person in group $i$ has with group $j$, $I_j(t)$ is the number infectious in group $j$ and $N_j$ is its size. Contacts must balance, so $N_i c_{ij} = N_j c_{ji}$.

Hens and colleagues describe how transmission rates between age groups populate the WAIFW ("who acquires infection from whom") matrix, traditionally given an imposed structure inspired by social and schooling systems. The POLYMOD survey by Mossong and colleagues measured contacts directly, with 7,290 participants in eight European countries, who recorded 97,904 contacts during one day, and found contact patterns highly assortative with age. Pitman and colleagues note that the group optimally targeted may be the one contributing most to the force of infection rather than the one with the greatest disease burden, as when immunising younger people prevents influenza deaths in older people.

Estimating the force of infection from serology and notifications

Hens and colleagues trace estimation to Muench, whose 1934 catalytic model used current status data and who named the quantity the force of infection in 1959. With no part of the population free from exposure, a perfect test, transmission unchanged over time, negligible disease mortality and lifelong immunity, a constant force of infection gives the share seropositive by age as follows.

$$\pi(a) = 1 - e^{-\lambda a}$$

where $\pi(a)$ is the proportion seropositive at age $a$ in years and $\lambda$ is the constant annual force of infection. Fitting this curve by binomial maximum likelihood, Hens and colleagues estimated 0.104 per year for rubella from a UK survey of males in 1986 and 1987 and 0.053 per year for parvovirus B19 in Belgium. The constant form suited rubella but not parvovirus B19, so later methods let the hazard vary with age, often in piecewise constant bands.

The WHO guide regards seroprevalence data as generally the most accurate way of determining age-specific susceptibility and lists two assumptions behind cross-sectional estimates, that transmission has not changed over time and that disease-specific and background mortality are not significant. Hepatitis A breaks the first, since better hygiene lowered incidence and moved it to older ages. Case notifications are typically underreported, and the guide notes that the degree is often impossible to estimate.

Worked example with two age groups and a school programme

All numbers in this example are illustrative. A town has 20,000 children and 80,000 adults, with 200 infectious people in each group. Children have 10 daily contacts with children and 4 with adults, and adults have 1 with children and 8 with adults, which balances because 20,000 children times 4 equals 80,000 adults times 1. The probability of transmission per contact is 0.03.

1. Daily hazards before the programme. Each group's force of infection sums over the groups it contacts.

$$\lambda_c = 0.03 \times \left(10 \times \frac{200}{20000} + 4 \times \frac{200}{80000}\right) = 0.0033, \qquad \lambda_a = 0.03 \times \left(1 \times \frac{200}{20000} + 8 \times \frac{200}{80000}\right) = 0.0009$$

where $\lambda_c$ and $\lambda_a$ are the daily forces of infection on susceptible children and adults. Children face almost four times the adult hazard, because more of their contacts are with children, among whom prevalence is four times higher.

2. Weekly cycle probabilities. Under a constant hazard the 7-day probability is as follows.

$$p = 1 - e^{-7\lambda}$$

where $p$ is the weekly probability. For children $p = 1 - e^{-0.0231} = 0.02284$, and for adults $p = 1 - e^{-0.0063} = 0.00628$. Multiplying the rate by 7 would overstate both (see rate-to-probability conversion).

3. A programme halves the number of infectious children. Suppose a school vaccination or exclusion policy brings infectious children down to 100 while adults stay at 200.

$$\lambda_c = 0.03 \times \left(10 \times \frac{100}{20000} + 4 \times \frac{200}{80000}\right) = 0.0018, \qquad \lambda_a = 0.03 \times \left(1 \times \frac{100}{20000} + 8 \times \frac{200}{80000}\right) = 0.00075$$

where the symbols are as in step 1. The adult hazard falls by a sixth although no adult was reached.

4. Adult infections in the next week. There are 60,000 susceptible adults. A static model keeps the adult hazard at 0.0009, giving 60,000 times 0.006280, or 376.8 infections. A dynamic model uses 0.00075, so $p = 1 - e^{-0.00525} = 0.005236$ and the count is 60,000 times 0.005236, or 314.2. The difference is 62.6 adult infections averted in one week, all indirect. At an illustrative GBP 150 per adult case, the static count misses about GBP 9,396 of averted cost.

5. Interpretation. The static model credits the programme only with child infections. A full model recalculates both hazards every step, so fewer adult infections this week mean fewer infectious adults next week and the gap compounds.

What a lower force of infection does to costs and outcomes

The indirect protection in the example is one of the vaccination externalities, and Brisson and Edmunds found that the bulk of economic evaluations of vaccination used models unable to capture it. Jit and Brisson explain that vaccination raises the average age at infection and lengthens the period between epidemic cycles. Under a constant hazard and ignoring mortality, the mean age at infection $A$ is about $\lambda^{-1}$, so the rubella estimate of 0.104 per year implies about 9.6 years, and halving the hazard would double it. Low rubella coverage can thus push infection into child-bearing ages and raise the risk of congenital rubella syndrome, and Edmunds and colleagues note that such a shift can change cost-effectiveness when severity depends on age.

Jit and Brisson also describe a honeymoon period in which incidence falls very low and then rises sharply as susceptible people accumulate, citing a rubella outbreak in Greece in 1993 after partial vaccination. Pitman and colleagues add that more widely spaced epidemics may have economic value under discounting. Edmunds and colleagues advise that under most circumstances decision makers should not assume elimination will allow immunisation to stop.

Infection control in hospitals

The same quantity drives models of healthcare-associated infection, where patients acquire infection from other patients directly or through staff and the environment. In a systematic review of 96 such models, van Kleef and colleagues found that 64% evaluated infection-control effectiveness and that MRSA was the most commonly modelled infection, at 34%. Hand hygiene, the most studied measure, lowers transmission per contact and isolation lowers effective contacts, both through the force of infection. Only 35% of the models were calibrated and only 5% compared predictions with external data.

Uncertainty, calibration and trial evidence

Hens and colleagues judge that in vaccine effectiveness and cost-effectiveness studies the force of infection will rank high, if not first, among the most influential parameters. Transmission and mixing parameters are often calibrated to observed incidence or serology, and Pitman and colleagues warn that probabilistic sensitivity analysis can then be challenging, because many of them are correlated and the correlations need to be preserved. Jit and Brisson show the structural uncertainty that remains, since SIS, SIR and SIRS structures can all fit the same outcome data by adjusting the force of infection, yet a long-lasting vaccine has the greatest impact under SIS and the least under SIR.

Pitman and colleagues explain that individually randomised trials underestimate effectiveness, because the trial is a tiny fraction of the population and neither arm experiences a reduced force of infection. Vaccine efficacy from such a trial captures only the direct effect, leaving the indirect effect to the model.

Force of infection and neighbouring quantities

Nearby quantities measure something other than a hazard on each susceptible person. The basic reproduction number counts secondary cases per case and has no time unit, and the effective reproduction number is its value once immunity has built up. The contact rate is one factor in the hazard, and the compartmental model page sets out the wider structure.

Misreadings that distort evaluations

Most errors misstate how the hazard behaves. Three recur in vaccine and infection-control models.

1. Using the rate as a cycle probability. Cycle-based models need the exponential conversion, especially for long cycles or high hazards.
2. Carrying a pre-vaccination estimate forward. A serology-based estimate assumes transmission unchanged over time, which is exactly what a programme breaks.
3. Treating a lower force of infection as pure gain. A later age at infection and post-honeymoon outbreaks can add costs, as rubella shows.

Sources

• Brisson M, Edmunds WJ. Economic evaluation of vaccination programs: the impact of herd-immunity. Medical Decision Making. 2003;23(1):76-82. https://doi.org/10.1177/0272989X02239651
• Edmunds WJ, Medley GF, Nokes DJ. Evaluating the cost-effectiveness of vaccination programmes: a dynamic perspective. Statistics in Medicine. 1999;18(23):3263-3282. https://doi.org/10.1002/(SICI)1097-0258(19991215)18:23%3C3263::AID-SIM315%3E3.0.CO;2-3
• Hens N, Aerts M, Faes C, et al. Seventy-five years of estimating the force of infection from current status data. Epidemiology and Infection. 2010;138(6):802-812. https://doi.org/10.1017/S0950268809990781
• Jit M, Brisson M. Modelling the epidemiology of infectious diseases for decision analysis: a primer. PharmacoEconomics. 2011;29(5):371-386. https://doi.org/10.2165/11539960-000000000-00000
• Mossong J, Hens N, Jit M, et al. Social contacts and mixing patterns relevant to the spread of infectious diseases. PLoS Medicine. 2008;5(3):e74. https://doi.org/10.1371/journal.pmed.0050074
• Pitman R, Fisman D, Zaric GS, et al. Dynamic transmission modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-5. Value in Health. 2012;15(6):828-834. https://doi.org/10.1016/j.jval.2012.06.011
• van Kleef E, Robotham JV, Jit M, et al. Modelling the transmission of healthcare associated infections: a systematic review. BMC Infectious Diseases. 2013;13:294. https://doi.org/10.1186/1471-2334-13-294
• World Health Organization. WHO guide for standardization of economic evaluations of immunization programmes. 2nd edition (WHO/IVB/19.10). Geneva: WHO; October 2019. Section 6.1.2. https://iris.who.int/handle/10665/329389