The Impact of Realistic Age Structure in Simple Models of Tuberculosis Transmission

Background Mathematical models of tuberculosis (TB) transmission have been used to characterize disease dynamics, investigate the potential effects of public health interventions, and prioritize control measures. While previous work has addressed the mathematical description of TB natural history, the impact of demography on the behaviour of TB models has not been assessed. Methods A simple model of TB transmission, with alternative assumptions about survivorship, is used to explore the effect of age structure on the prevalence of infection, disease, basic reproductive ratio and the projected impact of control interventions. We focus our analytic arguments on the differences between constant and exponentially distributed lifespans and use an individual-based model to investigate the range of behaviour arising from realistic distributions of survivorship. Results The choice of age structure and natural (non-disease related) mortality strongly affects steady-state dynamics, parameter estimation and predictions about the effectiveness of control interventions. Since most individuals infected with TB develop an asymptomatic latent infection and never progress to active disease, we find that assuming a constant mortality rate results in a larger reproductive ratio and an overestimation of the effort required for disease control in comparison to using more realistic age-specific mortality rates. Conclusions Demographic modelling assumptions should be considered in the interpretation of models of chronic infectious diseases such as TB. For simple models, we find that assuming constant lifetimes, rather than exponential lifetimes, produces dynamics more representative of models with realistic age structure.

Technical Appendix S1: The impact of realistic age structure in simple models of tuberculosis transmission Ellen Brooks-Pollock * , Ted Cohen, Megan Murray * E-mail: ebrooks@hsph.harvard.edu We describe a simple age-structured model of TB transmission with alternative assumptions about survivorship. We use analytic approximations of the prevalence of infection and disease and the basic reproductive ratio as the basis for investigating the effect of changing survivorship in models with otherwise identical natural history assumptions.

A basic model for TB transmission
TB dynamics can be captured by classifying individuals based on age and infection status [1][2][3]. Susceptibles become infected at a time varying rate which reflects the prevalence of infectious cases of all ages a at a given time and the probability of transmission, β, given an encounter between an infectious case and a susceptible individual, The equations for the proportion of the population in each infection state are: We take the total population size asN =S +Ē +Ī +R = 1. In this system, the number of births and deaths are balanced and the number of births is equal to the number of people aged zero, N (0) = S(0, t), The other boundary conditions are E(0, t) = I(0, t) = R(0, t) = 0. For the static population structures used in this paper, equations (1)-(4) reduce to a system of ordinary differential equations (ODEs) [2]. We investigate the behaviour of the model for parameter ranges taken from the literature (table S1, [4,5]).

Models of survivorship
The survivorship function (l(a)) and mortality rate (µ(a)) for the three types of survival used in this paper can be described in terms of age a and life expectancy, L: Exponential lifespan : Gompertz : where l(a) = exp(− a 0 µ(t)dt) gives the probability of surviving until age a. A constant lifespan underestimates early mortality, although produces a reasonable approximation of a realistic survivorship curve, while an exponential lifespan overestimates mortality in younger ages and underestimates mortality in older age groups, resulting in an underestimation of early survival (Fig. ??). We note that in populations with high infant mortality, fitting a constant mortality rate to mortality data yields a mortality rate that is greater than the reciprocal of the life expectancy. For our example of the Ukraine in 2006, infant mortality is relatively low therefore the optimal fitted mortality rate is 1/66.5 years −1 , which is similar to the reciprocal of the life expectancy of 1/69 years −1 .
The Gompertz function contains positive parameters α and ζ, which we calculated as α = 3 × 10 −4 and ζ = 7.35 × 10 −2 using Ukraine mortality data [6]. The intercept parameter α affects mortality for all age groups; the slope parameter ζ differentially affects mortality for older ages. The Gompertz function can be fit to mortality data by linear regression after taking the logarithm, log(µ(a)) = log(α) + ζa.
There are a number of approaches for improving the fit of the Gompertz mortality function such as the Gompertz-Makeham equation which includes an additional constant mortality rate for all ages groups, µ(a) = c + αe ζa [7].

Prevalence of disease
The prevalence of disease is often used as a measure of a TB epidemic, with models calibrated so that the equilibrium prevalence is consistent observed TB notification rates [4,8,9]. The number of susceptibles at equilibrium for each type of survival by integrating equation (1) twice, once with respect to time to obtain the proportion of individuals aged a that are susceptible, S(a), and then with respect to age to find the total proportion of susceptibles,S [10]: These formulae indicate thatS const >S expo , although when exp(−λL) is small the difference is negligible.
However for TB, λL can be less than one, therefore the proportion of susceptibles estimated using constant lifespans can be 50% larger than the proportion estimated using exponential lifespans.
To examine contribution of reinfection and reactivation to disease, we consider active cases produced via the latent state. Very approximately, we assume the recovery rate from active TB is on the order of years −1 , the force of infection is on the order of decades −1 and the average latent period, including individuals that do not progress from latency, occurs over centuries. If we let d ∼ ε, then λ ∼ ε 2 and σ ∼ ε 3 . The equations for the number of infectious people produced via latency, estimated by constant and exponential lifespans, simplify tō where σ′ = σ + ελ Accordingly, due to both the higher proportion of susceptibles, and the difference in lifespans,Ī const > I expo .
The remaining individuals in the population have been infected but do not currently have active disease, i.e. individuals with a positive skin test, in the exposed (E) and recovered (R) stages of disease.
A model with constant lifespans will predict a smaller proportion of the population with latent infection than a model with exponential lifespans, i.e.Ē const +R const <Ē expo +R expo . In the exponential lifetime model, the proportion of individuals that progresses from latency to disease, either due to reactivation or reinfection, is α = σ′/(σ′ + µ). In contrast, the proportion of individuals that progress from latency to disease in a model with constant lifetimes is 1 − exp(−α/ (1 − α)). The proportion of individuals that progress in each model is equal for the extreme values where either noone or everyone progresses to disease. For all intermediate values, fewer individuals will progress from latency to disease in an exponential lifetimes model. Therefore if we used the same parameters in each model, any interventions aimed at reducing reactivation or reinfection would have a greater impact in the constant lifetimes model.

The basic reproductive ratio
The basic reproductive ratio is the average number of secondary cases produced by a single infectious case and can be calculated using the survival function, l(a) [11,12], Equation (11) is the product of the transmission rate, β, the effective infectious period, (1−exp(−da))l(a)da, and the probability of progressing to active disease, (1 − exp(−σa))l(a)da. Thus, background mortality affects R 0 in two ways, by reducing the length of time an individual is infectious, and by increasing the probability of a secondary infective dying while latently infected. Applying the survivorship functions above, yields The equation for R expo 0 can be recognised as the standard form of R 0 [1,13]. The difference between these estimates of R 0 can be approximated by considering the latent period less than the life expectancy, σ ≤ 1/L, an infectious period less than the average latent period and life expectancy, d >> σ and exp(−dL) ≈ 0 and comparing equations (12) and (13):

Model generality
Our analysis uses the simple scenario of a chronic infectious disease at equilibrium to illustrate the potential effects of ignoring the importance of the birth-death process. The effects that we report are due to slow disease natural history (so that the majority of secondary infections never progress to disease) and a low prevalence of disease (small λ affecting the average age of infection). The model and parameter specification are such that these results will be potentially relevant for diseases where the average age of infection is in adulthood, such as TB in other mammals, rabies [14] or dengue [15] in low prevalence settings.
Further difficulties in capturing realistic TB dynamics are the variety of risk factors that have been found to be important for infection and disease. Notably, our model does not include age-related risks of disease and for increased realism in ageing populations, we would have to include complexities such as declining incidence over time by a mechanism that reduces the number of contacts with transmission potential. Relaxing the assumption of a constant population size is likely to reveal more interesting features of TB in diverse demographic landscapes.