Dynamic Transmission Model

What a dynamic transmission model captures

An intervention can therefore protect treated or vaccinated people directly and change transmission risk for others indirectly. This page explains the feedback mechanism, core compartment equations, mixing and immunity assumptions, calibration, uncertainty and use in economic evaluation.

The defining feature is endogenous transmission: the force of infection is generated by the model's evolving population state. A model that applies a fixed infection risk unaffected by prevalence is not dynamically transmitting infection, even if it follows people over time.

Transmission creates feedback between people

Infectious disease risk depends on contact between susceptible and infectious people. As incidence rises, more people may become infectious and create additional exposure; as immunity rises, transmission opportunities can decline. This feedback allows a dynamic model to estimate indirect effects, epidemic peaks, recurrent waves and changes in the timing of infection.

A basic force of infection is:

$$\lambda(t)=\beta\frac{I(t)}{N(t)}$$

where:

• (\lambda(t)) is the instantaneous infection hazard for a susceptible person.
• (\beta) is the effective transmission parameter.
• (I(t)) is the number of infectious people.
• (N(t)) is the relevant population size.

The expected number of new infections over a short interval (dt) is approximately:

$$dC(t)=\lambda(t)S(t)dt$$

where (S(t)) is the number susceptible. The equation assumes homogeneous mixing; structured populations require group-specific contact and infectiousness.

A simple SIR model shows the core mechanism

The susceptible-infectious-recovered, or SIR, model divides the population into mutually exclusive compartments. Susceptible people can become infectious, and infectious people can recover and enter a recovered or immune compartment. Ordinary differential equations describe the continuous-time flows.

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

where (1/\gamma) is the mean infectious duration under an exponential waiting-time assumption. If births, deaths, waning immunity or migration are absent:

$$S(t)+I(t)+R(t)=N$$

The SIR structure is a teaching model rather than a universal disease model. Latency, asymptomatic infection, repeated infection, vaccination, disease severity and treatment may require additional compartments or an individual-based structure.

A worked early-growth example

Suppose (\beta=0.30) per day, (\gamma=0.10) per day, (S=9{,}900), (I=100) and (N=10{,}000). The equations show the instantaneous expected flow into and out of the infectious compartment at that moment. This local calculation does not by itself describe the full epidemic, which requires integration over time.

New infections per day are:

$$\beta\frac{SI}{N} =0.30\times\frac{9{,}900\times100}{10{,}000} =29.7$$

Recoveries per day are:

$$\gamma I=0.10\times100=10$$

The instantaneous net change in infectious people is:

$$\frac{dI}{dt}=29.7-10=19.7$$

At this state, infections are increasing. As the susceptible proportion falls or interventions reduce transmission, the balance can reverse.

The reproduction number summarises transmission potential

The basic reproduction number (R_0) is the expected number of secondary infections generated by a typical infectious person in a wholly susceptible population under specified conditions. In the basic homogeneous SIR model:

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

The effective reproduction number changes as susceptibility and interventions change:

$$R_t=R_0\frac{S(t)}{N(t)}$$

In the worked example, (R_0=3) and (R_t=2.97). An (R_t) above one indicates growth under the model at that time, while an (R_t) below one indicates decline.

The reproduction number is context dependent. Contact patterns, immunity, behaviour, season, variant characteristics and interventions can change it, so a value should always be tied to a population, period and model definition.

Indirect effects distinguish dynamic from static models

An intervention that prevents infection can also reduce onward transmission. These indirect effects may protect unvaccinated people, shift infections to older ages or alter future epidemic timing. A dynamic model updates infection pressure as intervention coverage changes and can therefore represent these population-level consequences.

A static cohort model may approximate direct effects under a fixed force of infection but generally cannot reproduce transmission feedback. Using a static model for a transmission-changing intervention can understate or mischaracterise population impact.

Vaccination can affect several transmission pathways

Vaccines may reduce susceptibility to infection, probability of symptoms, infectiousness after breakthrough infection, duration of infection or severity conditional on disease. These effects enter different parts of the model and should not be collapsed into one efficacy parameter without justification. Waning, dosing schedules and delayed onset of protection may also matter.

If vaccine efficacy against infection is (VE_{inf}), the vaccinated susceptible hazard may be:

$$\lambda_V(t)=\left(1-VE_{inf}\right)\lambda(t)$$

If efficacy against onward infectiousness is (VE_{trans}), infectious vaccinated people may contribute a factor (1-VE_{trans}) to transmission. The model should distinguish these mechanisms because they can yield similar trial endpoints but different indirect effects.

Under simplifying assumptions of a perfect, lifelong vaccine in a homogeneous population, the critical immunity proportion is:

$$p_c=1-\frac{1}{R_0}$$

If vaccine efficacy against infection is less than one, a simplified coverage threshold is:

$$c^*=\frac{1-1/R_0}{VE_{inf}}$$

These expressions are not general operational targets. Heterogeneous mixing, waning immunity, imperfect uptake, importation and multiple vaccine effects can substantially change the threshold or make elimination infeasible.

Population structure changes who infects whom

Real populations do not mix uniformly. Age, household, school, workplace, geography and behaviour influence contact patterns and susceptibility. Stratified dynamic models represent these differences using group-specific states and a mixing matrix.

For susceptible group (a), a structured force of infection may be:

$$\lambda_a(t)= q_a\sum_b \beta_{ab}c_{ab} \frac{I_b(t)}{N_b(t)}$$

where (q_a) is susceptibility in group (a), (c_{ab}) is contact from group (a) with group (b), and (\beta_{ab}) includes per-contact transmission. The exact decomposition varies, but contact rates and transmission probabilities should not be counted twice.

Mixing matrices should be checked for population reciprocity when contacts are measured from both groups' perspectives:

$$N_a c_{ab}\approx N_b c_{ba}$$

Survey sampling, smoothing and demographic change can violate this relationship, requiring documented adjustment.

The next-generation matrix generalises reproduction numbers

In a structured model, one infectious person can generate infections across several groups or infected compartments. The next-generation matrix records expected new infections by recipient and source type near the disease-free state. Its dominant eigenvalue is the model's basic reproduction number under the defined conditions.

If (K) is the next-generation matrix:

$$R_0=\rho(K)$$

where (\rho(K)) is the spectral radius, or largest absolute eigenvalue. The matrix formulation helps identify which groups contribute to transmission, but its interpretation depends on how infected states, durations and new-infection terms are defined.

Natural history determines the compartment structure

Compartments should represent distinctions that materially affect transmission, outcomes or intervention impact. Adding compartments can improve realism but also increases data requirements and identifiability problems. The structure should be justified by the disease pathway and the decision rather than by a desire for detail.

Common extensions include:

• Exposed compartments represent infection before infectiousness.
• Asymptomatic and symptomatic compartments represent different detection, severity or infectiousness.
• Severity compartments represent outpatient care, hospitalisation, intensive care and death.
• Vaccination compartments represent dose, protection level or time since vaccination.
• Waning compartments allow recovered or vaccinated people to return toward susceptibility.
• Treatment compartments allow therapy to change duration, infectiousness or progression.
• Strain compartments represent variants with different transmissibility, immune escape or severity.

Each transition requires a rate, probability or duration assumption. Several parameter combinations can generate similar incidence, so external evidence and identifiability analysis are essential.

Deterministic and stochastic models answer related questions

A deterministic model calculates expected flows and produces the same trajectory for the same inputs. A stochastic model represents random infection, recovery and other events, producing a distribution of epidemic paths. Stochasticity is especially important in small populations, early invasion, elimination and rare-event settings.

In a stochastic compartment model over a short interval (\Delta t), new infections may be sampled as:

$$X_{inf}\sim \operatorname{Binomial} \left(S,1-e^{-\lambda(t)\Delta t}\right)$$

Recoveries may be sampled as:

$$X_{rec}\sim \operatorname{Binomial} \left(I,1-e^{-\gamma\Delta t}\right)$$

The time step must be small enough that the approximation is accurate, or an exact event-simulation algorithm should be used. Parameter uncertainty and stochastic epidemic variability should be reported separately because increasing simulation replications reduces only Monte Carlo error.

Calibration links unobserved transmission processes to data

Many transmission parameters are not observed directly. Calibration selects or weights parameter values so model outputs reproduce relevant surveillance, serology, hospitalisation, mortality or other targets. A good fit to one time series does not establish that every parameter or mechanism is identified correctly.

A weighted least-squares objective may be:

$$J(\theta)= \sum_{k=1}^{K}w_k \left[y_k-m_k(\theta)\right]^2$$

where (y_k) is target (k), (m_k(\theta)) is the corresponding model output and (w_k) is a prespecified weight. A likelihood-based calibration instead defines a probability model for observing the data given the simulated latent process.

Calibration should document:

• Which data were used as targets and which were reserved for validation.
• How under-ascertainment, delays and reporting changes were represented.
• Which parameters were fixed, estimated or constrained by prior evidence.
• The fit measure, weights, likelihood or posterior framework.
• Parameter correlations, non-identifiability and alternative good-fitting solutions.
• The criterion for an acceptable fit.

Observation models separate infections from reported data

The transmission model represents latent infections and disease states, while surveillance records only a filtered and delayed subset of those events. An observation model connects latent states to cases, admissions, deaths or test results. Without it, changes in testing or reporting may be misinterpreted as changes in transmission.

If latent incident infections at time (t) are (C_t), expected reported cases after an ascertainment probability (p_t) and delay distribution (g_d) may be:

$$E[Y_t]= \sum_{d=0}^{D}p_{t-d}C_{t-d}g_d$$

The model may also include overdispersion or day-of-week effects. The observation process should be validated independently where possible because it can absorb errors in the transmission structure.

Economic outcomes follow infections through clinical pathways

A dynamic model generates infections, cases and clinical events over time. Economic evaluation attaches costs and health outcomes to these events while preserving their timing and population distribution. The analysis should include intervention costs, disease-management costs, adverse effects and downstream consequences relevant to the perspective.

Discounted total cost under strategy (s) can be represented as:

$$C_s=\int_0^T c_s(t)d_C(t)\,dt$$

and discounted health loss as:

$$DALY_s=\int_0^T b_s(t)d_H(t)\,dt$$

where (c_s(t)) is the cost rate, (b_s(t)) is the DALY rate and (d_C(t)) and (d_H(t)) are the applicable discount functions. Incremental outcomes compare strategies simulated under otherwise consistent assumptions.

If QALYs are used instead, utilities must be mapped to time in health states without double counting morbidity already represented in disease events. The economic module should reconcile case counts and outcomes with the epidemiological module.

The time horizon must capture delayed and recurrent effects

Transmission-changing interventions can alter the timing of epidemics and future immunity. A short horizon may count postponed infections as permanently prevented or miss later replacement waves. The horizon should capture material direct, indirect and downstream effects, subject to increasing uncertainty in long-term projections.

Analysts should examine:

• Whether cumulative infections converge across strategies after temporary disruption.
• Whether immunity wanes or demographic turnover replenishes susceptibility.
• Whether intervention programmes continue, stop or change over time.
• Whether strain replacement or behavioural adaptation changes long-run outcomes.
• How much incremental health and cost occurs after observed data end.

Longer horizons do not automatically improve validity. They make structural and behavioural assumptions more influential, so scenario analysis is essential.

Uncertainty has several distinct sources

Dynamic models combine parameter, structural, calibration and stochastic uncertainty. These sources can interact because several parameter sets or structures may fit historical data but produce different intervention forecasts. A single best-fitting trajectory understates decision uncertainty.

Important uncertainty layers include:

Probabilistic draws should preserve correlation created by calibration. Sampling calibrated parameters independently can generate combinations inconsistent with the data.

Dynamic models differ from static infection models

A static model applies infection risk that is fixed or externally specified rather than generated by the model population. It may be adequate when the intervention does not materially affect transmission or when indirect effects are deliberately excluded. A dynamic model is required when feedback in infection pressure is central to the decision.

The added complexity should be justified by expected impact on the decision. A dynamic structure that cannot be calibrated or validated may be less informative than a transparent approximation with clearly stated limitations.

Common modelling errors

Dynamic transmission models can reproduce observed data for the wrong mechanistic reasons. Careful checks must therefore cover conservation, event timing, force-of-infection construction, calibration and economic linkage. The following errors can materially distort intervention value.

• Treating prevalence and incidence as interchangeable breaks the connection between infectious stock and new infection flow.
• Double counting contact and transmission coefficients inflates the force of infection.
• Using a static infection risk after an intervention changes prevalence removes indirect effects despite a dynamic label.
• Ignoring population reciprocity in contact matrices can create impossible contact totals between groups.
• Applying efficacy against disease as efficacy against infection overstates transmission reduction when the vaccine acts only on symptoms or severity.
• Fitting cumulative outcomes only can hide incorrect epidemic timing and transient behaviour.
• Using reported cases as infections without an observation model ignores ascertainment and delay.
• Sampling calibrated parameters independently destroys posterior or accepted-set correlation.
• Choosing a horizon that captures the first decline but not later rebound can overstate cases averted.
• Adding economic consequences that do not reconcile with simulated cases and states disconnects the epidemiological and economic modules.

A practical validation sequence

Validation should proceed from mathematical conservation and limiting cases to calibration, out-of-sample prediction and intervention behaviour. Epidemiological and economic outputs should be reconciled at each stage. The following sequence provides an auditable minimum.

1. Define the population groups, infection states, interventions and time scale in a conceptual model.
2. Write every flow equation and force-of-infection term with consistent units and denominators.
3. Verify population conservation or reconcile births, deaths and migration at every time point.
4. Test disease-free, no-intervention and zero-transmission limits for predictable behaviour.
5. Reproduce analytically available quantities, including early growth, mean durations and (R_0) in simplified cases.
6. Check contact-matrix orientation and reciprocity before calibration.
7. Validate vaccine or treatment effects at the mechanism they actually change, including susceptibility, infectiousness, duration and severity.
8. Calibrate to multiple epidemiological targets and retain parameter correlation and alternative good-fitting sets.
9. Validate against data not used for calibration when available.
10. Reconcile simulated infections, cases, severe outcomes, costs and health losses across the epidemiological and economic modules.
11. Test longer horizons, alternative structures and future-context scenarios for rebound, waning and replacement effects.
12. Quantify parameter, calibration, structural and stochastic uncertainty separately before combining them for decision analysis.

A dynamic transmission model is valuable when population feedback changes the expected effect of an intervention. Its credibility depends on transparent mechanisms, calibrated but identifiable parameters, and validation that connects transmission dynamics to health and economic outcomes.