Representing tuberculosis transmission with complex contagion: an agent-based simulation modeling approach

Abstract

Objective:

A recent study reported a tuberculosis (TB) outbreak where, among newly infected individuals, exposure to additional active infections was associated with higher probability of developing active disease. Referred to as complex contagion, multiple re-exposures to TB within a short period after initial infection is hypothesized to confer greater likelihood of developing active infection in one year. The purpose of this paper is to develop and validate an agent-based simulation model (ABM) to study the effect of complex contagion on population-level TB transmission dynamics.

Methods:

We built an ABM of a TB epidemic using data from a series of outbreaks recorded in the 20^th century in Saskatchewan, Canada. We fit three dynamical schemes: base, with no complex contagion; additive, where each re-exposure confers independent risk of activated infection; and threshold where a small number of re-exposures confers low risk and a high number of re-exposures confers high risk of activation.

Results:

We find that the base model fits mortality and incidence output targets best, followed by the threshold then additive models. The threshold model fits incidence better than the base model, but overestimates mortality. All three models produce qualitatively realistic epidemic curves.

Conclusion:

We find that complex contagion qualitatively changes the trajectory of a TB epidemic, though data from a high incidence setting are reproduced better with the base model. Results from this model demonstrate the feasibility of using ABM to capture nuances in TB transmission.

INTRODUCTION

Tuberculosis (TB) is the leading cause of death due to single infection worldwide ¹. With a global goal of eliminating TB by 2050 ^2,3 and rising threats of multi-drug resistant and extremely-drug resistant TB infection ^1,4, it is more urgent than ever to elucidate why TB persists as a threat to global health.

Complex transmission dynamics make TB difficult to control. One complexity of transmission, referred to as complex contagion, is the phenomenon by which multiple re-exposures to active TB disease during early latency increase the risk of progression to active infection ⁵ (Figure 1). While previous work has looked at the effect of reinfection occurring five or more years after initial infection ^6–8, the dynamical impact of numerous exposures within one year of initial infection is not known.

Figure 1/.

Model schematics 1A Data structure for one neighborhood in model

Model schematics 1B Disease state transition diagram

The population-level implications of complex contagion are difficult to study using traditional epidemiological methods. Previous studies using comprehensive contact tracing and diagnostic testing reveal complex contagion in TB epidemiological data, ^5,9. It is not feasible to apply these techniques at a large scale, and the population-level impacts of complex contagion are not well understood. Disease transmission models are a cost-effective way to understand complex transmission dynamics when relevant population-level data are difficult to collect. In particular, agent-based simulation models (ABMs) are flexible tools for inferring population-level effects from individual-level dynamics ^10,11.

One study found that within the first year following infection, each additional exposure to active TB increased the odds of progressing to active disease by a factor 1.11 (95% CI, 1.06–1.16)⁵. Results from a stochastic model fit to a small TB outbreak suggest complex contagion worsens an epidemic ¹². They find that a model with complex contagion fit better than a model without¹². Building on this work, we develop an ABM to represent a TB epidemic in a large population over a longer time period while incorporating complex contagion transmission dynamics. The purpose of this study is to demonstrate the feasibility of modeling a TB epidemic with complex contagion dynamics by fitting versions of the model with and without complex contagion to data.

We use data from the First Nations and Métis populations of Saskatchewan, Canada collected during a series of severe TB epidemics during the 20^th century. This study population is of particular interest because TB was a leading contributor to mortality in Canada ¹³, with disproportionate effects on these populations^14–16. The epidemics studied here are well documented, with data comprising decades-long time series, surveys, and medical records ^14–16. Modeling data from these epidemics can help elucidate the role of complex contagion in a high incidence setting during a prolonged outbreak.

METHODS

Study Population

Our ABM is developed with archival epidemiological and census data from First Nations and Métis communities of Saskatchewan, Canada. Population and data sources are described in Pepperell et al.¹⁷.

Overview of the model

We developed an agent-based model (ABM) that simulates TB transmission on a network of connected communities. An ABM is an extension of traditional discrete-event simulation where agents in the model have unique attributes, interact with each other and the environment, and updates occur in discrete time steps. TB infections can be protracted ^18–20, taking years or even decades after an initial infection for a person to develop active disease. Additionally, untreated active disease on average lasts 2 – 3 years until death or self-cure²⁰. To capture this wide range of latent and active disease times, we have modeled minimum time steps of one month. Model geography is comprised of neighborhoods representing one community each from our study population and agents represent individuals. Agents move between grids within each neighborhood and contacts occur between agents within a grid space and every surrounding grid (Figure 1). Agents move once per time step and can move between neighborhoods, returning to their home neighborhood at the start of each time step. The model allows for population growth, which we refer to as recruitment. Recruitment can occur through migration or birth, though for the purposes of modeling the study data, recruitment is entirely due to new births because migration in the study population was minimal during the study period¹⁴.

We use this ABM to evaluate three different TB transmission dynamics: the standard transmission model where additional exposure to active TB during early latency does not confer additional risk of activation (base model), additive complex contagion where each additional exposure during early latency confers an independent risk of activation (additive model), and threshold complex contagion where each additional exposure during early latency confers a small additional risk of activation up to some threshold of exposures, after which the risk jumps significantly (threshold model). See Figure 2 for a schematic of the relationship between exposure and risk in all three models.

Figure 2/.

Diagrams of complex contagion 2A: Schematic of the relationship between exposures to active TB during latency and the increasing risk of developing active infection.

Diagrams of complex contagion 2B Comparison of risk of activation during latency as a function of exposures during the first year of latency in the base (blue), additive (orange), and threshold (pink) models.

Agents & the environment

Agents are assigned a probability of changing neighborhoods during initialization. Moving probability is drawn from an exponential distribution such that on average, agents will visit another neighborhood at least once per year of model time. Within the neighborhood, agents have a uniform probability of ending up on any grid space. Once assigned a grid space, agents mix homogeneously with each other within the grid space they are on as well as on any surrounding grid space (up to nine grids total). Grids are designated as contaminated at time step t if an infectious agent visits that grid. Agents on the contaminated grid and surrounding grids are then assigned exposures equivalent to the number of infectious agents on those grid spaces. For this model, each neighborhood is initialized with enough grids such that, on average, each grid holds four agents at the end of the warmup period.

At the beginning of each time step, all agents return to their home neighborhood. Agents then move throughout the geography in a two-step process. First, a Bernoulli trial determines if an agent will move outside of their neighborhood. If the agent moves, the new neighborhood is determined with uniform probability from non-home neighborhoods. Then, the agent is assigned a grid with uniform probability from the neighborhood they are staying in or traveling to. At the beginning of each year, defined as a time step that is a multiple of 12, each neighborhood shrinks due to non-TB deaths then grows through new births. The non-TB death rate is set to the inverse of life expectancy (Table 1: model parameters). The neighborhood grows according to birth rates for the study population. All newly recruited agents are uninfected.

Table 1:

Input parameters

Symbol	Parameter	Mean value	Distribution	Range	Source
β	Transmission rate per contact per month	0.013B 0.006A 0.005T	Uniform	[0.001, 0.06]	Calibrated
ptransmissible	Probability of developing transmissible TB for active cases	0.54B 0.74A 0.68T	Uniform	[0.25, 0.85]	Calibrated
pfast	Probability of developing active TB within one year of initial infection	0.3	--	--	1,2
vfast	Progression rate to TB, fast ( < 1 year)	0.384	Exponential	--	2
vslow	Progression rate to TB, slow ( > 1 year)	0.00438 (corresponds to 5-10% progression in 20 years)	Exponential	--	1,2
c	Time to self-cure	36 months	Exponential	--	3
μTB	Time to TB death	24 months	Exponential	--	3
ω	Relapse rate	3% per year	Exponential	--	1
trate	Treatment rate	--	Piecewise linear function (see supplement)	[0.005, 0.52]	4,5
tcompletion	Treatment completion probability	0.6	--	--	4,5
tlength	Treatment length	12 months	Gamma(12,1)	--	4
pprophylaxis	Probability of receiving treatment during latency each month	0.27B 0.08A 0.08T	Uniform	--	6
Λ	Recruitment rate per year per 1,000 individuals	50	--	--	5,7
1/μ	Life expectancy (1/death rate)	60 years	--	--	5
m	Move probability	0.15	Exponential	--	7,8

^1.Blower SM, Mclean AR, Porco TC, et al. The intrinsic transmission dynamics of tuberculosis epidemics. Nature Medicine. 1995;1(8):815-821. doi:10.1038/nm0895-815

^2.Ackley SF, Liu F, Porco TC, Pepperell CS. Modeling historical tuberculosis epidemics among Canadian First Nations: effects of malnutrition and genetic variation. PeerJ. 2015;3:e1237. doi:10.7717/peerj.1237

^3.Tiemersma EW, van der Werf MJ, Borgdorff MW, Williams BG, Nagelkerke NJD. Natural History of Tuberculosis: Duration and Fatality of Untreated Pulmonary Tuberculosis in HIV Negative Patients: A Systematic Review. PLoS One. 2011;6(4). doi:10.1371/journal.pone.0017601

^4.Tuberculosis Morbidity and Mortality 1938-1963, National Health and Welfare Volume 1224.

^5.Canada L and A. Indian Affairs Annual Reports, 1864-1990. Published March 19, 2013. Accessed March 5, 2020. http://www.bac-lac.gc.ca/eng/discover/aboriginal-heritage/first-nations/indian-affairs-annual-reports/Pages/introduction.aspx

^6.Wherrett GJ. Miracle of the Empty Beds: History of Tuberculosis in Canada. University of Toronto Press; 1978.

^7.Lux MK. Medicine That Walks. University of Toronto Press; 2001.

^8.Waldram JB, Herring DA, Young TK. Aboriginal Health in Canada. 2nd ed. University of Toronto Press; 2006.

Disease state transitions

Each agent has a dynamic TB disease state modeled as a discrete-time Markov chain with transitions drawn from exponential distributions based on previous models^12,21 (Figure 1). Agents can be in Uninfected (U), Latent (L), Active—transmissible (AT), Active—non-transmissible (AN), Recovered (R), Relapsed (RL), TB death (TD), or Natural death (ND) states. Agents are only infectious in the Active—transmissible (AT) state and susceptible to infection in the Uninfected (U) and Recovered (R) states. Agents in the Latent (L) state have been exposed but are not infectious. Once infected, the agent is subjected to a Bernoulli trial to determine if they will experience a fast or slow progression to active infection with probability of fast progression (p_fast). Progression rate is drawn from an exponential distribution for both fast and slow activations. For fast activations, progression takes at most 12 months (v_fast) and for slow activations, it takes more than 12 months, with only 5 – 10% of infections activating within 20 years of initial infection (v_slow). Activation time for each latently infected agent is stored in an activation clock and decreases at each time step. Once the activation clock reaches 0, the agent will move to either AT or AN state. See Table 1 for further explanation of parameter values. Agents in either active state (AT or AN) can self-cure or receive treatment and enter the (R) state or experience death due to TB (TD). Recovered agents can experience Relapse (RL) or become independently re-infected, reentering either active state (AT or AN). Agents in any alive state can die of non-TB causes.

Exposures

We evaluated three types of exposure models: 1) base model with no complex contagion, 2) additive complex contagion, and 3) threshold complex contagion. In all three models, uninfected agents’ exposures are counted at each time step. An agent is considered exposed if an actively infected, transmissible agent is on the same Grid or an adjacent Grid at that time step. At the subsequent time step, the agent is subjected to a Bernoulli trial with probability β that exposure leads to latent infection. The trial is repeated once for each exposure that agent accrued over the previous time step.

For the complex contagion models, exposures during latency are also counted. In the additive model, exposures during latency are counted at each time step and each results in a Bernoulli trial to determine if a latent exposure will become an active infection with probability β. In the threshold model, exposures during latency are counted over the first 12 months since initial infection, the time period determined by Ackley et al. to be of most importance in complex contagion.¹² Exposures are treated as in the additive model until the threshold exposure level is reached. At that time, the agent becomes twice as likely to have a successful exposure¹² (i.e. a Bernoulli trial with probability 2* β) and any additional exposures over the threshold are subjected to Bernoulli trials with probability β. If one of these exposures is successful, that new infection is determined to be a fast activation (i.e. < 12 months in latency) or slow activation (> 12 months) by a Bernoulli trial with probability p_fast of becoming a fast activation. Then a new length of latency is drawn from an exponential distribution with rate v_fast for fast activations and v_slow for slow activations. If this length of latency is less than the current time left on the agent’s activation clock, the time on the clock is reduced to the lower value. See Supplemental Figure 1 for a flow diagram of how latent exposures are modeled under complex contagion.

Treatment

We model three treatment introductions coinciding with changes in treatment policy in the study population over the modeled time period. The first two introductions reflect the discovery of TB-treating drugs: streptomycin and para-aminosalicylic acid (PAS) (1944–1946) and isoniazid (1952)¹⁴. Streptomycin was not very effective until combined with PAS ²², so we average the years of discovery to introduce treatment in 1945. The second introduction occurs in 1952, with the discovery of isoniazid and more effective therapy ²³. Finally, we introduce treatment for latent infections likely to become active, in 1966¹⁵. Infections likely to become active were typically identified in the study population by mass chest x-ray screenings and risk level was determined based on the results of the chest x-ray ^14,15. In the model, we do not include mass screenings so any latently infected agent ‘likely to become active’ is eligible to start receiving treatment. We have used ‘within 24 months of activation’ (i.e. 24 time steps or fewer on the activation clock) as a proxy for an infection likely to become active based on TB treatment guidelines^24,25. Any latently infected agent with 24 or fewer time steps remaining on their activation clock has probability p_prophylaxis of starting treatment at each time step.

Treatment rate (t_rate) was calibrated during the testing phase of model building. Length of treatment and probability of successfully completing treatment (t_completion) are based on previous analyses of the study data²⁶. After treatment is introduced in the model, any active agent has some probability of entering treatment, determined by a Bernoulli trial. If the agent is selected for treatment, a Bernoulli trial determines if the agent will be successfully complete treatment. If treatment will be completed, duration is a Gamma random variable with a mean of 12 months (shape = 12, scale = 1). If the agent ceases treatment before cure, duration is a Gamma random variable with a mean of 9 months (shape = 9, scale = 1). Once an agent completes treatment, s/he moves to the Recovered (R) state. If the agent fails treatment, s/he returns to the Active (AT or AN) state s/he came from for the remainder of the duration of the original time in active state. Agents are assigned a new treatment completion probability so that, if selected again, they might successfully complete treatment. Latently infected agents that successfully complete treatment return to Uninfected (U) state.

Input parameters

The model was parameterized from three sources. The first data source includes archival epidemiological and census data as described above. Second, published clinical and epidemiological literature pertaining to TB infection was used for clinical parameters. Third, other TB transmission models were used for parameters pertaining to disease state transitions (Table 1: model parameters).

The archival dataset includes TB sanatorium admission and discharge records (described in Zwick and Pepperell 2020²⁶ and Pepperell et al. 2010¹⁷), incidence time series (1952 – 1975) ²⁷, and mortality time series (1935 – 1958) ²⁷. We parameterized length of treatment and proportion of successfully completed treatments based on analyses in Zwick and Pepperell 2020²⁶. We used mortality time series for calibration and testing and incidence time series for validation output targets. Census data were used to estimate birth and non-TB death rates²⁸. Clinical and epidemiological literature was used to estimate the proportion of agents that activated within one year, TB-specific mortality rate, self-cure rate, and relapse rate. Disease state transitions are assumed to be exponentially distributed ^8,21,29,30.

Model fitting

Target outputs for model fitting were determined by dividing study data into three subsets for calibration, testing, and validation based on the time period. Calibration was performed using mortality data from 1934 – 1944, testing was performed using mortality data from 1945 – 1959, and validation was performed using incidence data from 1965 – 1975. None of the parameters that were estimated during calibration were changed during the testing and validation phases except for the following: treatment rate was increased during the testing phase; and treatment rate and treatment completion probability were increased during the validation phase. Additionally, we introduced treatment for latent infections starting in 1966 during the validation phase. Treatment rates were determined from previous analyses of medical records²⁶. See Supplemental Table S1 for the changes in parameter values through these different phases.

Calibration

Because of the novel contact structure in this model, there are no data available to directly estimate the probability of infection per exposure (β). We also lacked data to directly estimate the proportion of active infections that are transmissible (p_{transmissible}). To fit these parameters, we developed a calibration process similar to Codella et al ³¹. For β, we tested values ranging from 0.001 – 0.06 incremented by 0.001 and for p_{transmissible}, we tested values ranging from 0.25 – 0.85 incremented by 0.05, generating 780 parameter combinations for the base and additive models. For the threshold model, we also calibrated threshold number of exposures, at 5, 10, and 15 exposures per year for a total of 2,340 calibration sets (Table 2: fitting procedure). We simulated each parameter set using 1000 replications to sufficiently account for the stochasticity of the model (Supplemental Figure 2).

Table 2:

Fitting procedure

Phase	Variables fitted	Target output	Sets tested
Calibration	Transmission rate (β) Transmissible TB probability (f) Threshold value	Mortality rate (1934-1944)	780B 780A 2340T
Testing	Treatment rate (trate) Treatment completion (tcompletion)	Mortality rate (1945-1958) Treatment rate (1945-1955)	672B 96A 216T
Validation	--	Incidence rate (1965-1975)	28B 4A 9T

We determined best fitting parameter sets by comparing simulated mortality rate to observed mortality rate from 1934-1944. We calculated the 95% confidence interval around the observed mortality rate, generating an envelope around our target output ³². We counted the number of times simulated mortality rate violated those confidence bounds (i.e. envelope violations). Then, we calculated the mean square error (MSE) to determine the distance between simulated mortality rate and observed mortality rate. We define MSE for parameter set P as:

$$MS{E}_P={\underset{t=1}{\overset{n}{\Sigma }}\left(y_t-\frac{{\sum }_{s=1}^{1000}{\widehat{y}}_{t,P,s}}{S}\right)}^2$$

for each time point t and each simulation run s. We retained all parameter sets p that fell within the lowest 10% MSE and had two or fewer envelope violations. The set of best fitting parameter sets, P^TOP was used for all subsequent phases of fitting and analysis.

Testing

We tested three different treatment models against mortality rates from 1945 – 1958²⁷ using the distance measurement defined above. For the testing phase, we fit treatment rate per month (t_rate) and treatment completion probability (t_completion). We tested three treatment structures: one introduction in 1952, two introductions in 1945 and 1952, and two introductions in 1945 and 1952 with linearly increasing treatment intensity over time. For each model, we tested treatment probabilities ranging from 0.04% to 50% per agent per month and treatment completion probabilities of 50%, 55%, and 60% based on data from literature²⁷. See Table 2 for fitting sets from each model and supplemental material for treatment rate functions.

Verification and validation

We followed verification and validation techniques similar to Codella et al. ³¹ and Kleijnen ³³. Our model is written in C++ using Microsoft Visual Studio IDE 2017. The model has agent-level and time step-level debugging output which was used to determine whether agent behavior matched the intended agent logic (Supplemental Material).

We validated our model in five phases ³¹: face validation, calibration, testing, validation against withheld data, and sensitivity analysis. Face validation, an ongoing process, involved checking that the agent logic, environment, agent interactions, disease state transitions, and parameters produced a sufficiently accurate representation of a real TB epidemic given the simplifying assumptions necessary to create the model. One of the authors, an infectious disease physician and TB researcher, provided expertise on TB transmission and control, helping ensure that agents interactions and TB transmission occurred with sufficient accuracy.

Validation against withheld data was performed by comparing modeled incidence rates against observed incidence in the study population between 1965 and 1975. Incidence was not used as a performance metric at any other point in the development of our model. Sensitivity analysis was performed to examine the effect of changes in input parameters on model results.

Validation against withheld data

We withheld all data from 1959 – 1975 during the building, calibration, and testing phases of model fitting. This includes incidence time series data starting in 1952 through 1975, a subset of which was used as a validation set. The only structural changes to the model after testing are capping the treatment probability at 80%, increasing treatment completion probability to 90% after 1958, and introducing treatment for agents with latent infections. We take these values from medical histories of the study population ¹⁵. Treatment for latent infections is introduced in the model at the time step corresponding to January 1965. Treatment probability per agent per time step for latently infected agents likely to become active (p_prophylaxis) is 0.27 in the base model and 0.08 in the additive and threshold models (Table 1: model parameters).

Sensitivity analysis

We performed a deterministic one-way sensitivity analysis to look for significant change in MSE in modeled output. We vary each parameter within 50% of its fitted value and run the model holding the rest of the parameter values constant. We considered any MSE from the sensitivity run that exceeded the maximal accepted MSE from P^TOP to be a significant change in MSE.

RESULTS

Initial conditions

We initialize with 3,786 agents across 37 neighborhoods with five active transmissible infected agents randomly distributed across geographic space. We find that the model enters a steady state of infections just after the 600^th time step (50 years of model time), so we set a warmup period of 660 time steps (55 years). Each parameter set is run for 1,000 replications to sufficiently account for the stochasticity of the model (Supplemental Figure 2).

Calibration

We retained 28 sets for P^TOP from the base, 4 from the additive, and 9 from the threshold models (Table 2: fitting procedure and Supplemental Figure 3). The base model calibration resulted in an average probability of infection per exposure per month (β) of 1.3% and an average proportion of transmissible active infections (p_{transmissible}) of 54%. Both additive and threshold model calibrations tended towards lower β, with averages of 0.6% and 0.5% respectively. The additive model calibration resulted in an average p_{transmissible} of 74% while the threshold model averaged 68%. The threshold model calibrated to an average threshold of 10 exposures per year. All parameter results are in Tables 1 and 2.

Testing and treatment structure

We find that two treatment introductions for active infections (1945 and 1952) with increasing intensity result in the best fit to mortality data from 1945–1958. We fit a piecewise linear function such that the rate of treatment administered to active cases per month (t_rate) increases from 0.04% to 3.5% between 1945 and 1951 then jumps to 7.2% in 1952, increasing to 36% in 1958. This treatment function was preferred in all three versions of the model. The base model has a 55% treatment completion probability (t_completion), while the additive and threshold models had 50% t_completion.

Treatment probability per latently infected agent per month (p_prophylaxis) was determined to be 0.27 in the base model and 0.08 in both the additive and threshold models to match the incidence rates during the validation period (Figure 5, left side). We find that significantly more latently infected agents need treatment to achieve the same incidence level in the base model compared to both complex contagion models (Figure 5, right side).

Figure 5.

(L) The effect of treatment of latent TB infections on incidence rates for the Base (B), Additive (A), and Threshold (T) models under varying treatment probabilities. (R) Latent infection treatment rate per year across treatment probabilities. Treatment probability is per latently infected agent per month. The modeled values are bolded in black: 0.27 for the base model and 0.08 for the additive and threshold models.

Validation against withheld data

We find that the threshold model has the lowest MSE, followed by the additive then the base model, fit to 1965–1975 incidence data. However, this seems to be an artifact of how high the modeled 1965 incidence is compared to observed incidence (Figure 4). The threshold model underestimates true incidence for most of the validation period, while both the base and additive models seem to provide better fit after 1965. Indeed, when we recalculate MSE excluding 1965, we find that base has the best fit, followed by the additive, then the threshold models. We also find the base model fits mortality targets from calibration and testing the best (Figure 3). Additionally, we find that the additive and threshold models produce fewer latent infections and require less treatment of latent infections to achieve similar epidemic decline than the base model.

Figure 4.

Modeled TB incidence rate for Base (blue), Additive (orange), and Threshold (pink) models averaged over all best fitting sets. Points represent observed incidence rates.

Figure 3.

Modeled TB-specific mortality rate for Base (blue), Additive (orange), and Threshold (pink) models averaged over all best fitting sets. Points represent observed TB mortality rates.

Sensitivity analysis

We find that all three models were most sensitive to changes in birth rate and the proportion of fast activations. The threshold model was particularly sensitive to movement between neighborhoods while the base and additive were not. We find that the models are fairly robust to changes in fast activation rate, recovery rate, relapse rate, slow activation rate, and treatment length. Life expectancy and TB death rate had variable effects on incidence and TB mortality across the three models. Full sensitivity results are presented in Supplemental Figures 4 and 5.

DISCUSSION

In this study, we develop and fit an ABM that models TB transmission and control under three dynamical schemes. We also estimate transmission rate and the proportion of transmissible TB cases in a high incidence setting. These together provide a framework for future models of TB transmission that may help epidemiologists and infectious disease experts understand the effect of underlying transmission dynamics on the scope of a TB epidemic.

We showed that all three versions of the model produce qualitatively realistic epidemic curves (Figure 4). Each version of the model reproduces the relationship between mortality and incidence in settings before/without and after/with antibiotic treatment from canonical works by Styblo³⁴ and Grigg¹⁹. While the base model provides the best fit to mortality data, we find that the threshold model fits incidence better from 1952 onwards (Figure 4) despite over estimating mortality in previous years. This is because more infections activate sooner in the complex contagion versions of the model, resulting in more deaths early on and a smaller pool of latent infections remain after the initial epidemic subsides. The burden of latent infections affects the reemerging epidemic: once the initial epidemic is brought under control, incidence starts climbing again around 1959 (Figure 4). This secondary epidemic is most apparent in the base model, which produced a steep uptick in incidence starting in 1959, while both complex contagion models produced incidence curves that appear to plateau rather than start a secondary epidemic (Figure 4).. We also found that all three models fit a high proportion of fast activations compared to other TB models^12,21,35, though our estimates did not exceed the reported values in the medical literature ¹⁸. We expect a high proportion of fast activations under complex contagion^5,12 and hypothesize that some of the dynamical impacts of complex contagion are captured in this parameter.

There are few ABMs of TB transmission dynamics fit to data. There are models using theoretical populations to test the effect of close versus casual contacts¹⁰, determine the role of age structure in transmission³⁶, or predict the emergence of drug resistance³⁷. However, none of these are fit to data nor do they consider complex transmission dynamics. Tuite et al.³⁸ fit a model to data in a similar high incidence setting in Northern Canada, however their model only captures a 10-year period. TB epidemics have slow transmission dynamics and long doubling times²¹, so to fully capture a TB epidemic, we modeled almost 100 years and fit over 40 years to data. Only one previous transmission model that we know of has been used to study complex contagion: the study by Ackley et al.¹², which wasused as a starting point for our model. Like Ackley et al., we find that of the two complex contagion schemes, threshold fits better than additive and base models to incidence, however the base model had better fit to mortality during calibration and testing.

One of the most striking differences in the complex contagion models compared to the base model is the burden of latent infections. This is evident both in the reemerging epidemic, around 1959, as slow progressing latent infections start to activate and increase incidence rates (Figure 4) and in the differential effects of treatment for latent infections across models (Figure 5). To achieve similar incidence rates in the base model compared to either complex contagion model, the probability that a latently infected agent starts treatment each month (p_prophylaxis) is over three times higher in the base model at 0.27 compared to 0.08 in both complex contagion models. Additionally, latent agents receive treatment in the base model at or over four times the rate as the complex contagion models in 1966, the peak of latent treatment in the model (600 per 100,000 population in the base model versus 125–150 per 100,000 population in the additive and threshold models, respectively) (Figure 5). At the same p_prophylaxis value, the base model has significantly higher incidence while still treating more latently infected agents than either complex contagion model (Figure 5). This suggests that depending on dynamical context (complex contagion or no complex contagion), we can expect widely different burdens of latent TB infection in a population with widely different latent treatment rates required to prevent a secondary epidemic.

In addition to wide variation in the amount of underlying latent infections across models, we find that complex contagion produces a steeper epidemic curve (Figure 4). Both model behaviors have implications for treatment interventions. Because the epidemics take off more steeply under complex contagion, early active case detection and intervention would be crucial to bringing the epidemic under control in complex contagion conditions. Fewer latent infections would reduce the amount of latent detection and treatment under the same circumstances, which is apparent in the lower latent treatment rates from the complex contagion models (Table 1). Whereas in the base model, we observe a longer epidemic start up and many more latent infections, as evidenced by the increasing incidence rate starting around 1959 (Figure 4). This suggests that in the absence of complex contagion, focusing on treatment for latently infected individuals is important for completely stopping the outbreak.

We acknowledge that the study data are not granular enough to fully distinguish all three exposure models. For example, though we note a major difference in the burden of latent TB infection in both complex contagion models compared to the base model, we do not have study data to compare these model results with the true burden of latent TB infection. Similarly, the active TB incidence and TB mortality time series used for fitting are not detailed enough to fully distinguish the three exposure models as in Ackley et al.¹², which used contact tracing data. However, successful fit of all three models to data demonstrates the feasibility of incorporating complex contagion in dynamical models of TB transmission and suggests the potential importance of incorporating complex contagion transmission dynamics into future TB models.

Our model does have some additional limitations. Due to data limitations, we do not introduce treatment for latent infections until 1966, though medical histories ^14,15 and model behavior suggest that latent treatment may have been introduced earlier. We also assume that treatment has a binary outcome: successful completion or failure to complete treatment. If an agent fails treatment, they return to their previous active disease state. In reality, individuals who do not complete treatment, receive only partial treatment, or have their treatment interrupted and later resumed have a higher risk of developing difficult-to-treat multi-drug resistant TB^4,39, which we do not model. However, while this is an area for further exploration, we do not find evidence that changes to treatment (Supplemental Figures 4 and 5) affected the output of our model significantly. The model could be easily adapted to include different treatment outcomes. Finally, though we include structure at the community level, we do not account for household structure or network movement within a neighborhood. This simplifying assumption was deemed appropriate to study the direct effects of complex contagion, though we acknowledge that contact structure plays an important role in TB transmission ^10,40.

We develop this model with an appropriate level of detail to capture individual-level interactions, while still being generalized enough to avoid extreme assumptions or limitations due to data scarcity. To maintain focus on population-level effects of transmission dynamics, we do not consider different treatment efficacy or effectiveness in this model. Further, we collapsed non-infectious pulmonary TB and extra-pulmonary TB into one category, capturing differences in infectious TB only to reduce the number of parameters we had to fit.

All three versions of this model serve as a basis for future, more complex models of TB transmission and control. Future applications of the model may explore different formulations for complex contagion, including more complex risk functions or investigate the applicability of complex contagion in lower incidence settings.