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Conceived and designed the experiments: EBP TC MM. Performed the experiments: EBP. Analyzed the data: EBP. Contributed reagents/materials/analysis tools: EBP. Wrote the paper: EBP TC MM.

The authors have declared that no competing interests exist.

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.

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.

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.

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.

Mathematical models of tuberculosis (TB) are important tools for investigating the dynamics of epidemics and identifying strategies for disease control

Transmission of TB occurs when people with infectious pulmonary disease cough, sneeze, talk or otherwise aerosolize tubercle bacilli which may be inhaled by others. Among those infected, approximately 5% develop active disease within the first few years but the majority remain asymptomatically infected. Of these, 5–10% are expected to reactivate their latent infection and develop TB disease later in life

Most mathematical models of TB transmission classify individuals as either susceptible (S), exposed (E), infectious, (I) or recovered (R)

Compartmental

In this paper, we explore the impact of age structure on TB model dynamics. We investigate the role of survivorship by comparing model outcomes using constant, exponentially distributed and Gompertz-like lifespans. Of these approaches, the Gompertz function produces the most realistic patterns of human mortality by allowing the risk of death to increase exponentially with age

We begin by describing 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. Finally, we use an individual-based model to examine how age structure may affect the projected impact control interventions. Full details of the models and equations are contained in

We capture the natural history of TB by classifying individuals based on age and infection status using an

Mortality in infectious disease models is commonly included as a constant age-independent mortality rate,

(A) The age-specific mortality rate in Ukraine in 1959 (green solid line) and 2006 (black solid line), arising from constant (dashed red line), exponential (solid blue line) and Gompertz-like (dot-dashed grey line) lifespans, fitted to Ukraine 2006 data. (B) The probability of surviving until a given age for the same three types of survival functions: constant, exponential and Gompertz-like lifespans. The probability of survival for Ukraine in 2006 is taken from the life tables in

Human survivorship is better characterized by a mortality rate that increases exponentially with age (

We use mortality data from Ukraine in 2006 as a country with detailed population data

TB models are usually calibrated to an observed prevalence of disease by varying the transmission parameters

Natural mortality affects steady state dynamics in two ways: by reducing the length of time an individual is infectious and by decreasing the probability of a secondary infection progressing to active disease during their lifetime. Assuming constant lifetimes increases the lifespan for the majority of individuals and because the population size is fixed, there is slower population turnover. This affects the number of individuals in each infection state: the numbers of susceptible persons and infectious cases are increased, but the number of latently infected persons is reduced.

For given TB parameters, the proportion of the population not exposed to disease is up to 50% greater with constant lifetimes. New cases of active disease are modelled as resulting from a recent primary infection (

Rather than fixing parameters, TB models are generally fitted to an observed prevalence. We show that the type of survivorship directly affects estimates of epidemiological parameters by absorbing the difference in life-years. For a fixed prevalence, constant lifespans are consistent with lower parameter estimates than exponential lifespans. For example, if we require equal numbers of latently infected individuals to progress to disease in both models, then we must assume a smaller progression rate for the constant lifetime model. This can be seen analytically by rearranging the equations for prevalence of disease so that

We use the basic reproductive ratio (

A model with constant lifespans results in a smaller effective infectious period, although for a mean infectious period of less than 15 years the estimates differ by less than 5%. A larger difference arises when we consider the probability of progressing from latency to active disease in each of the models. In a model with exponential lifespans, the probability of progressing from latency to active disease is up to twice as likely (

Red dashed lines: all individuals live for a constant lifetime; blue solid lines: individuals experience a constant mortality rate and have exponentially distributed lifetimes. (A) The basic reproductive ratio (

Our analysis indicates that the main effects of including realistic age structure are to alter our estimates of the proportion of people with latent infections and the proportion that are expected to progress from latency to active disease. A further manifestation is that a model with exponential lifespans will predict at least 25% more disease due to reactivation and reinfection than a model with constant lifespans (

To illustrate our results further, we fit three individual-based models, differing only in survivorship function, to the same observed data. In each case we explore the effect of control interventions by examining the level of interventions required for disease eradication. The age structure and natural history parameters for each model are determined by the type of survivorship.

We fit each model separately to Ukraine prevalence of 350/100,000

Red dashed lines: all individuals live for a constant lifetime; blue solid lines: individuals experience a constant mortality rate and have exponentially distributed lifespans; grey dot-dashed lines: mortality increases exponentially with age (Gompertz mortality). Each model is fitted to an equilibrium prevalence rate of 350/100,000 by allowing the transmission rate to vary. (Baseline parameters given in Table S1 of

The basic reproductive ratio, calculated using equations derived in

The higher value of

Using the above parameters, we consider the necessary reduction in transmission to eradicate disease in each of the models. All three models predict similar reductions in prevalence for decreases in transmission rate of less than 10% (

Red dashed lines: constant lifespan; solid blue lines: exponential lifespan and black dot-dashed lines Gompertz survival. In each case, we determined the necessary decrease in parameters for the prevalence to fall to zero. (A) Reduction in transmission rate,

For reductions in the average infectious period, the model with constant lifespans requires the smallest reduction in order to achieve disease eradication. An average infectious period of 16.2 months is needed for eradication in the constant lifespan model, whereas the exponential lifespan model requires a further reduction to 12.2 months. The estimate of 15.4 months from the Gompertz model is 25 days less than the constant lifespan model estimate and 100 days greater than the exponential model estimate (

Our analysis illustrates that natural (non-disease related) mortality and age structure play an important role in the dynamics of chronic infectious diseases such as TB. We explored three common characterisations of survivorship: constant or fixed lifetimes, exponentially-distributed lifetimes and lifetimes resulting from a Gompertz mortality function (mortality that increases exponentially with age). Our results indicate that the choice of survivorship strongly affects steady-state dynamics, parameter estimation and predictions about the effectiveness of control interventions.

Our findings are consistent with previous investigations into the effects of realistic infectious periods of acute diseases

In addition to exploring the effect of mortality on steady-state dynamics, we also investigated the public health implications of mortality assumptions in mathematical models that are used to inform recommendations for disease control. We find that models that assume exponential lifetimes result in larger parameter estimates and overestimate the effect of treating TB disease and underestimate the effect of treating latent TB infection when compared to the same interventions in models with more realistic lifetimes. Quantifying the importance of particular routes to disease is important for balancing individual- and population-level control interventions that will lead to a reduction in disease burden.

Age-structure and natural mortality have not been a focus of TB models. Assumptions about survivorship are often hidden; our results suggest that these modelling assumptions should be more explicitly stated given their impact on model dynamics and results. Two types of mortality patterns are commonly used in models of TB epidemics. The majority of non-age structured TB models implicitly assume that lifetimes are exponentially distributed

The aim of our analysis was to quantify the differences resulting from commonly used modelling assumptions about natural mortality. Additional complexity resulting from altered living standards, treatment rates and co-morbidity with other diseases are important features that could potentially confound the effects of altered survivorship. We focused on TB natural history in otherwise healthy populations; the interaction between the HIV and TB affects both TB natural history and the underlying demography

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We are grateful to Leon Danon, Andrew Conlan and the anonymous reviewers for comments and suggestions on the manuscript. Thanks to the members of the Murray lab for discussing an early version of this work.