Concept Architecture
Concept
Theoretically, Force of Infection is the instantaneous rate at which susceptible individuals acquire an infectious disease. It represents the hazard of infection experienced by susceptible members of a population and links the epidemiological state of the population to the probability of new infections. In health economics, the force of infection is a fundamental component of dynamic transmission models used to evaluate vaccination programmes, screening strategies and other interventions that alter disease transmission.
Mathematically, Force of Infection is represented as a hazard function determined by the transmission probability, contact rate and prevalence of infectious individuals within the population. It defines the rate at which susceptible individuals become infected and serves as the transition rate from the susceptible compartment to the infected compartment in compartmental transmission models. The force of infection may vary over time according to changing population immunity, intervention coverage or disease prevalence.
In practice, Force of Infection is estimated using epidemiological surveillance data, serological studies, contact surveys or model calibration. Dynamic transmission models calculate the force of infection at each time step to simulate disease spread under alternative intervention scenarios. Age-specific contact matrices and heterogeneous mixing patterns are frequently incorporated to improve the accuracy of transmission estimates and health economic evaluations.
Purpose
Used to quantify the instantaneous hazard of infection among susceptible individuals, enabling dynamic transmission models to estimate disease spread and evaluate the population-level impact of infectious disease interventions.
Mathematical Formulae
Primary Formula
Force of infection:
? = ?c(I/N)
where:
- ? is the force of infection
- ? is the probability of transmission per effective contact
- c is the contact rate
- I is the number of infectious individuals
- N is the total population.
Supporting Formulae
Probability of infection during time interval t:
p = 1 ? e^(??t)
Basic reproduction number:
R? = ?c/?
where:
- ? is the recovery rate.
Related Mathematical Methods
- Dynamic transmission modelling
- SIR models
- SEIR models
- Compartmental modelling
- Contact matrix analysis
- Ordinary differential equation modelling
- Model calibration
Example
A vaccination model evaluates seasonal influenza transmission.
The model assumes:
- Transmission probability per contact = 0.05
- Contact rate = 10 effective contacts per day
- Infectious individuals = 500
- Population size = 100,000
The force of infection is:
? = 0.05 ? 10 ? (500/100000) = 0.0025
The daily probability that a susceptible individual becomes infected is:
p = 1 ? e^(?0.0025) = 0.00250
or approximately 0.25%.
This value is updated throughout the simulation as the number of infectious individuals changes following vaccination.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| PRODUCT | =B2*C2*(D2/E2) | Calculate the force of infection from transmission parameters. |
| EXP | =1-EXP(-A2*B2) | Convert the force of infection into a cycle-specific infection probability. |
| SUMPRODUCT | =SUMPRODUCT(ContactMatrix,InfectiousVector) | Calculate weighted contact rates using age-specific mixing patterns. |
| INDEX | =INDEX(ContactMatrix,AgeGroup1,AgeGroup2) | Retrieve age-specific contact rates. |
| MMULT | =MMULT(ContactMatrix,InfectiousVector) | Apply contact matrices in age-structured transmission models. |
VBA (Optional)
Automate recalculation of the force of infection during each simulation cycle as disease prevalence, immunity and intervention coverage change over time.
Sources
- Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press; 1991.
- Diekmann O, Heesterbeek JAP, Roberts MG. The construction of next-generation matrices for compartmental epidemic models. Journal of the Royal Society Interface. 2010;7(47):873?885.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006.
- Drummond MF, Sculpher MJ, Claxton K, Stoddart GL, Torrance GW. Methods for the Economic Evaluation of Health Care Programmes. 4th ed. Oxford University Press; 2015.
- National Institute for Health and Care Excellence (NICE). Health Technology Evaluation Manual. Latest edition.
Related Concepts (2)
Library
Publications
4
Dynamic Transmission Modeling: A Report of the ISPOR-SMDM Modeling Good Research Practices Task Force-5 — Pitman, Fisman, Zaric, Postma, Kretzschmar, Edmunds & Brisson, Task Force Report 5 ed., 2012 (Value in Health / Medical Decision Making)
Best-practice guidance on dynamic transmission models for infectious disease, capturing indirect (herd) effects that static models cannot, and their use in cost-effectiveness analysis of vaccination and control programmes.
Journal ArticleView source →Applying Dynamic Simulation Modeling Methods in Health Care Delivery Research — The SIMULATE Checklist: Report of the ISPOR Simulation Modeling Emerging Good Practices Task Force — Marshall, Burgos-Liz, IJzerman, Osgood, Padula, Higashi, Wong, Pasupathy & Crown, Vol. 18, No. 1 ed., 2015 (Value in Health)
The first ISPOR dynamic-simulation good-practice report, introducing system dynamics, discrete event simulation and agent-based modelling for health care delivery problems and providing the SIMULATE checklist for their application.
Journal ArticleView source →Selecting a Dynamic Simulation Modeling Method for Health Care Delivery Research — Part 2: Report of the ISPOR Dynamic Simulation Modeling Emerging Good Practices Task Force — Marshall, Burgos-Liz, IJzerman, Crown, Padula, Wong, Pasupathy, Higashi & Osgood, Vol. 18, No. 2 ed., 2015 (Value in Health)
The second ISPOR dynamic-simulation report, giving decision guidance on choosing between system dynamics, discrete event simulation and agent-based modelling based on the structure and complexity of the health care delivery problem.
Journal ArticleView source →Methods for Health Economic Evaluation of Vaccines and Immunization Decision Frameworks: A Consensus Framework from a European Vaccine Economics Community — Ultsch, Damm, Beutels, Bilcke, et al., Vol. 34, No. 3 ed., 2016 (PharmacoEconomics)
A consensus framework on the immunisation-specific methodological issues in economic evaluation of vaccines — herd/indirect effects, discounting, dynamic transmission modelling — developed to support national vaccine HTA guidelines in Europe.
Journal ArticleView source →
Media
1
Infectious Disease Modelling Specialization — Imperial College London, 3-course specialization ed., 2023 (Coursera)
An Imperial College London specialization introducing mathematical modelling of infectious disease in R — compartmental and dynamic transmission models — foundational for the economic evaluation of vaccines and control programmes.
Online CourseView source →
Frequently Asked Questions (6)
What is the force of infection?
The rate at which susceptible individuals become infected, determined jointly by disease prevalence and the rate and probability of transmission per contact.
Source: Kermack & McKendrick 1927
How does the force of infection differ from a fixed infection risk?
A fixed infection risk treats the chance of catching a disease as a constant, whereas the force of infection lets it vary with how much disease is currently circulating. As more people become infectious, a susceptible person faces a greater rate of exposure, so the force rises, and as prevalence falls it declines. This dependence on the current state of the epidemic is what makes it dynamic, and it is why transmission models, rather than static ones, are needed to represent it. Keeling and Rohani (2008) define this quantity.
Source: Keeling & Rohani 2008
What determines the force of infection?
The force of infection is determined by the number or proportion of infectious individuals, the contact rate between individuals, and the probability of transmission per contact. As more people are infectious, the force of infection rises, and as susceptibles are depleted and individuals recover, it falls. It thus depends on the current state of the epidemic, making it dynamic. This dependence on prevalence is what distinguishes transmission models from static ones and produces effects such as herd immunity.
Source: Kermack & McKendrick 1927
Why does the force of infection change over time?
The force of infection changes over time because it depends on the current prevalence of infection, which rises and falls over the course of an epidemic. As infection spreads and the number of infectious individuals grows, the force of infection increases, accelerating new infections, and as susceptibles are used up and people recover, it declines. This feedback between prevalence and the force of infection drives the characteristic rise and fall of an epidemic and underlies the indirect protection of herd immunity.
Source: Kermack & McKendrick 1927
How does the force of infection relate to herd immunity?
The force of infection relates to herd immunity because reducing the number of infectious individuals, for instance through vaccination, lowers the force of infection for everyone, including the unvaccinated. As immunity rises in the population, the prevalence of infection falls, so the per-susceptible risk of infection drops, protecting susceptible individuals indirectly. This is herd immunity, and it arises because the force of infection depends on prevalence, so measures that cut transmission reduce the risk faced by all, which transmission models capture.
Source: Kermack & McKendrick 1927
How is the force of infection used in modelling?
In modelling, the force of infection is the quantity that governs the flow of individuals from susceptible to infected, calculated each time step from the current prevalence, contact rate, and transmission probability. It links the model's state to the rate of new infections, driving the epidemic's dynamics. By making the force of infection depend on prevalence, transmission models capture the indirect effects of interventions, so representing it correctly is central to evaluating measures such as vaccination that work through reducing transmission.
Source: Kermack & McKendrick 1927
Trust Record
Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 9 Oct 2025
Content version: 1.0.0
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