Compartmental Model

Compartmental Model of Population Pools, Flows and Transmission in Economic Evaluation

A compartmental model sorts a population into a few states, counts how many people are in each, and moves them between states at defined rates, usually written as differential or difference equations. In health economics the best-known examples are the SIR family of infectious disease models, where the rate of new infection depends on how many people are infectious, so an evaluation can count the protection a vaccination or screening programme gives to people it never reaches. The SIR model, SEIR model, dynamic transmission model and basic reproduction number have their own pages.

Population pools linked by flows

Chhatwal and He describe compartments as population pools, with movement between them represented by differential equations, and note that compartmental models typically treat everyone within a compartment as identical. The ISPOR-SMDM conceptualisation report (Roberts and colleagues, 2012) groups them with decision trees and Markov processes as models homogeneous within each state, unlike agent-based models and microsimulation.

$$\frac{dx_j(t)}{dt}=\sum_{k\neq j} f_{kj}(t)-\sum_{k\neq j} f_{jk}(t)+b_j(t)-d_j(t)$$

where $x_j(t)$ is the number of people in compartment $j$ at time $t$, $f_{jk}(t)$ is the flow from compartment $j$ to compartment $k$ per unit time, $b_j(t)$ is inflow from outside the model, such as births, and $d_j(t)$ is outflow that leaves it, such as deaths. Each internal flow leaves one compartment and enters another, so summing over compartments cancels them and the total changes only through external flows. Chhatwal and He note that compartmental models can be open, admitting births and migration, or closed, whereas Markov models generally start with a closed cohort.

Natural history fixes the compartments

The compartments should mirror what infection does to a host. Jit and Brisson set out the common choices. SIS models assume no natural immunity, which suits gonorrhoea; SIR models assume near-lifelong immunity, as for measles; SIRS models let immunity wane, as for influenza; and SEIR models add a latent period in which a person is infected but not yet infectious, which slows an epidemic. With frequency-dependent transmission the SIR equations are as follows.

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

where $S$, $I$ and $R$ are the numbers susceptible, infectious and recovered, $N=S+I+R$, $\beta$ is the transmission rate per infectious person per unit time and $\gamma$ is the recovery rate, so the mean infectious period is $\gamma^{-1}$ and the basic reproduction number is $\beta/\gamma$.

Jit and Brisson report that SIS, SIR and SIRS structures can all be fitted to the same outcome data, yet a long-lasting vaccine looks most cost-effective under SIS and least under SIR. A European consensus framework for vaccine evaluation (Ultsch and colleagues, 2016) shows the variation in practice. A study of seasonal influenza vaccination in Spain used SIR, whereas one in England used SEIR.

The force of infection makes the model dynamic

The force of infection is the instantaneous rate at which susceptible people become infected; in the SIR equations it is $\lambda(t)=\beta I(t)/N(t)$. Jit and Brisson explain that it is constant in a static model but a function of the proportion infected in a dynamic one, so protecting some people lowers the risk faced by everyone else. A compartment diagram is therefore not dynamic by itself. Compartments with fixed transition rates form a static cohort, and the feedback term is what changes the economics. The ISPOR-SMDM dynamic transmission report (Pitman and colleagues, 2012) cites a review in which only 11% of 208 cost-effectiveness studies of vaccination programmes used an approach that could capture indirect effects.

Jit and Brisson give two reasons why, against no intervention, a dynamic model usually makes a preventive programme look more cost-effective than a static one. The first is that it predicts more infections prevented, including through herd immunity, and the second is that it predicts a faster fall in infections, which counts for more under positive time preference. Pitman and colleagues recommend a dynamic model when an intervention changes transmission in the target population or alters the distribution of strains.

Worked example of one vaccination programme counted two ways

All numbers are illustrative. A closed population of 10,000 has 9,900 susceptible and 100 infectious people, with $\beta=0.3$ and $\gamma=0.1$ per day. The model advances in one-day steps (the Euler method), holding each day's flows constant. The flows on day 1 without vaccination are as follows.

$$\beta\frac{SI}{N}=0.3\times\frac{9900\times100}{10000}=29.7$$

where 29.7 is the expected number of new infections. Recoveries are $\gamma I=0.1\times100=10$, so the compartments become 9,870.3 susceptible, 119.7 infectious and 10 recovered, still summing to 10,000.

A programme instead gives a perfect vaccine to 2,000 susceptible people at the start, moving them to the immune compartment. Day 1 infections fall to 23.7 (0.3 × 7,900 × 100 / 10,000), so 6.0 are averted. On day 2 the static and dynamic counts part company.

Over two days the static model counts 13.16 infections averted (6.0 + 7.16) and the dynamic model 14.58 (6.0 + 8.58). The difference of 1.42, about a tenth of the dynamic total, is the indirect effect, because fewer people were infected on day 1, so fewer were infectious on day 2. At an illustrative GBP 40 per infection treated, averted treatment costs are GBP 526 under the static count and GBP 583 under the dynamic count. The gap compounds with each generation of transmission, which is why the static estimate is usually conservative against a no-intervention comparator.

Deterministic, stochastic and discrete-time versions

Pitman and colleagues describe deterministic models as approximating average behaviour, most appropriate when all subgroups are large and easier to fit to data. Stochastic models matter when a population or subgroup is small, as at the start or end of an epidemic, when local extinction is likely; population-based models can be either, while individual-based models are invariably stochastic. The WHO guide on economic evaluation of immunisation programmes (2019) notes that dynamic models are typically solved as differential equations in continuous time, with discrete time or age steps often used for practical reasons. A discrete step holds the rate constant, and under a constant hazard the day-1 probability of infection in the example is as follows.

$$p=1-e^{-\lambda\Delta t}=1-e^{-0.003}=0.0029955$$

where $p$ is the per-person probability, $\lambda=0.003$ per day is the force of infection and $\Delta t$ is the one-day step. Applied to 9,900 people this gives 29.66 infections rather than 29.7, a gap that grows with longer steps or higher rates (see rate-to-probability conversion). Siebert and colleagues advise converting probabilities between time units through rates.

Mixing, heterogeneity and state expansion

Chhatwal and He point out that a compartmental model captures heterogeneity only by adding compartments, which can greatly enlarge it, and usually assumes homogeneous mixing; in return, it allows analytical insight that agent-based models do not. Roberts and colleagues warn of state expansion. Age or risk groups are added by stratifying each compartment, as shown below.

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

where $\lambda_a(t)$ is the force of infection on susceptible people in group $a$, $\beta_{ab}$ is the transmission rate from an infectious person in group $b$ to a person in group $a$, and $I_b$ and $N_b$ are the infectious and total numbers in group $b$. Pitman and colleagues ask published models to give their differential equations, all initial values and parameters including the mixing matrix, and the type of mixing assumed. With several infected compartments the basic reproduction number becomes the dominant eigenvalue of a next-generation matrix, whose construction for compartmental models Diekmann and colleagues set out.

Costs, QALYs and the time horizon

Economic outcomes attach to compartments and flows. Costs accrue for each person-day in a compartment. Event costs attach to flows such as new infections, as follows.

$$C=\int_0^T e^{-rt}\Big[\sum_j c_j\,x_j(t)+\sum_{j,k} e_{jk}\,f_{jk}(t)\Big]dt$$

where $C$ is the discounted total cost over horizon $T$, $r$ is the continuous discount rate, $c_j$ is the cost per person per unit time in compartment $j$ and $e_{jk}$ is the cost per transition from $j$ to $k$. QALYs follow the same form with utility weights in place of $c_j$, and the totals are for the whole modelled population.

Pitman and colleagues observe that preventive programmes are often sensitive to the discount rate and time horizon, and that a lifetime horizon is poorly defined for a population that turns over; they recommend sensitivity analysis on both. Jit and Brisson give varicella vaccination as a case where a short horizon makes the programme look less cost-effective, because zoster rises in older unvaccinated adults, and a longer horizon reverses this. Pitman and colleagues also ask reports to split infections prevented into direct and indirect where appropriate. Incremental costs and QALYs then feed the ICER.

Choosing a compartmental, Markov cohort or agent-based structure

Model choice turns on whether interaction matters and how large the population is. The WHO guide advocates the model that minimally meets the analytical requirements, and Jit and Brisson advise using transmission dynamic models only when an intervention can change the force of infection substantially, because they need more data and computing.

Calibration, identifiability and validation

Jit and Brisson note that infectious disease models are first calibrated to pre-intervention epidemiology by choosing parameters for infectivity and immunity. Pitman and colleagues note the importance of identifiability and state that difficulty in calibrating across several domains suggests the structure or assumptions are wrong. Alarid-Escudero and colleagues show the decision stakes. In a three-state Markov model calibrated to relative survival alone, two equally good-fitting parameter sets gave treatment benefits of 0.67 and 0.31 years, and a second target made the model identifiable.

For model validation, the ISPOR-SMDM report by Eddy and colleagues (2012) distinguishes face validity, verification, cross validity, external validity and predictive validity. Verification of a compartmental model includes mass balance at every step, non-negative compartments, no new infections when transmission is set to zero, and identical results for all strategies when the intervention effect is zero.

Where compartmental models are misread

The same boxes and arrows can hide very different mathematics. Three errors recur.

1. A compartment diagram implies dynamics. Without a force of infection that responds to prevalence, the model is static and omits herd effects.
2. Fractional people behave like whole people. Deterministic compartments hold expected values; with small numbers the chance of fade-out needs a stochastic model.
3. Indirect effects are always gains. Pitman and colleagues note that a higher age at infection raises complications for infections such as hepatitis A and varicella, and that strain replacement can limit health gains.

Sources

• Alarid-Escudero F, MacLehose RF, Peralta Y, Kuntz KM, Enns EA. Nonidentifiability in model calibration and implications for medical decision making. Medical Decision Making. 2018;38(7):810-821. https://doi.org/10.1177/0272989X18792283
• Chhatwal J, He T. Economic evaluations with agent-based modelling: an introduction. PharmacoEconomics. 2015;33(5):423-433. https://doi.org/10.1007/s40273-015-0254-2
• 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. https://doi.org/10.1098/rsif.2009.0386
• Eddy DM, Hollingworth W, Caro JJ, Tsevat J, McDonald KM, Wong JB. Model transparency and validation: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-7. Value in Health. 2012;15(6):843-850. https://doi.org/10.1016/j.jval.2012.04.012
• 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
• Pitman R, Fisman D, Zaric GS, Postma M, Kretzschmar M, Edmunds J, Brisson M. 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
• Roberts M, Russell LB, Paltiel AD, Chambers M, McEwan P, Krahn M. Conceptualizing a model: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-2. Value in Health. 2012;15(6):804-811. https://doi.org/10.1016/j.jval.2012.06.016
• Siebert U, Alagoz O, Bayoumi AM, Jahn B, Owens DK, Cohen DJ, Kuntz KM. State-transition modeling: a report of the ISPOR-SMDM Modeling Good Research Practices Task Force-3. Value in Health. 2012;15(6):812-820. https://doi.org/10.1016/j.jval.2012.06.014
• Ultsch B, Damm O, Beutels P, Bilcke J, Brüggenjürgen B, Gerber-Grote A, et al. Methods for health economic evaluation of vaccines and immunization decision frameworks: a consensus framework from a European vaccine economics community. PharmacoEconomics. 2016;34(3):227-244. https://doi.org/10.1007/s40273-015-0335-2
• World Health Organization. WHO guide for standardization of economic evaluations of immunization programmes. 2nd edition (WHO/IVB/19.10). Geneva: WHO; 2019. Section 6.3.