Medmetrics Inc, Suite # 215, 925 De Maisonneuve Avenue West, Montreal, QC H3A 0A5, Canada

Adjunct Professor (Health Economics), McGill University, Montreal, QC, Canada

Abstract

Background

While the overall population prevalence of tuberculosis in Quebec has been declining for many years, tuberculosis is still disproportionately more prevalent among the immigrant and Inuit communities. As such, the aim of this study was to forecast the incidence of tuberculosis in the Province of Quebec over time in order to examine the possible impact of future preventative and treatment programs geared to reducing such disparities.

Methods

A compartmental differential equation based on a **S**usceptible **E**xposed **L**atent **I**nfectious **R**ecovered (SELIR) model was simulated using the Euler method using Visual Basic for Applications in Excel. Demographic parameters were obtained from census data for the Province of Quebec and the model was fitted to past epidemiological data to extrapolate future values over the period 2015 to 2030.

Results

The trend of declining tuberculosis rates will continue in the general population, falling by 42% by 2030. The incidence among immigrants will decrease but never vanish, and may increase in the future. Among the Inuit, the incidence is expected to increase, reaching a maximum and then stabilizing, although if re-infection is taken into account it may continue to increase. Tuberculosis among non-indigenous Canadian born persons will continue to decline, with the disease almost eradicated in that group in the mid 21st century.

Conclusions

While the incidence of tuberculosis in the Province of Quebec is expected to decrease overall, certain populations will remain at risk.

Background

Tuberculosis (TB), which is caused by the bacteria

The rate of tuberculosis is approximately ten times higher in immigrants compared to Canadian-born persons, in part due to immigration from countries with high levels of TB such as Haiti and India

While historical data can be used to gauge the present situation and observe trends, it does not provide immediate knowledge of the future of the disease. To predict the epidemiology of a disease in the future, mathematical models must be employed. Epidemiological modelling of disease, including tuberculosis, is an established practice. In this regard, the first mathematical model of tuberculosis was developed by Waaler in 1962 **s**usceptible, **i**nfectious, and **r**ecovered (SIR). There are demographic parameters such as birth and death rates as well as the natural history of infection parameters such as the rate of infection that determine the relative behaviour of these groups. A paper by Garcia

Methods

An ordinary differential equation was used to model the spread of tuberculosis throughout the population of the Province of Quebec. A compartmental model was developed with five categories: **S**usceptible, **E**xposed, **L**atent, **I**nfectious, **R**ecovered, known as a SELIR model (Figure _{2} and γ_{1} of recovering, respectively. These recovery parameters are based on current epidemiological data and thus take into account current treatment methods. The coupled differential equations that make up the model are thus:

Flow chart of the model

**Flow chart of the model.**

Values for parameters were selected based on various estimates and published values and are described in Table

**Symbol**

**Meaning**

**Typical value**

**Source**

Greek letter parameters were subject to a sensitivity analysis varying them by up to 25%.

b

Birth/immigration rate

0.03/person/year

μ

Death rate

0.007/person/year

β

Transmission Parameter

5/person/year

Fitting

q

Proportion of exposed immigrants

0.01

α

Probability of infectiousness

0.1

K

Rate of progression

0.05/person/year

γ_{1}

Rate of recovery (latent)

0.22/person/year

γ_{2}

Rate of recovery (infectious)

0.87/person/year

The model was modified to account for immigration. Instead of a birth rate adding to the population of susceptibles, a constant immigration rate added the majority of the new population to the susceptible group, but a small fraction q to those exposed. This is designed to mirror the fact that each year there is a sub-population of new immigrants who have been exposed to TB. Because most cases last for less than a year, the number of cases per capita can serve as a proxy for the order of magnitude of the incidence of the disease. Thus, the fraction of exposed immigrants was estimated as ten times the incidence per capita of immigrants in the Province of Quebec using 2009 data. This is based on the fact that, according to Colijn

If the initial numbers of the various populations are not consistent with what would emerge naturally from this model, there is a rapid increase or decrease of their ratios in the first few years of simulation. To guarantee smoothness, the fits were pre-seeded, running for six years before being fit to the data, such that the initial ratios were consistent with the dynamics of the model. This ensured that the ratio of exposed to infectious persons was stable over long time scales, and that any discontinuities occur before the fitting and forecasting began. In order to propagate the uncertainties in the fits to historical data, two Gaussian random number distributions were generated with means and standard deviations taken from the regression of the infection parameter and the exposed:infectious ratios. The distributions were used to forecast the disease into the future, using 200 iterations of the model to generate a range for the prediction, establishing upper and lower bounds for the spread of disease. We performed a sensitivity analysis using Latin hypercube sampling and found that the model is robust under variations of the parameters, with the model being most sensitive to the infectiousness ratio α.

The parameter that is used as an output from the simulations is the number of new cases per year, as this is the common form of the available epidemiological data and serves as the best comparison between model and reality. It can also be used to examine the financial burden of the disease in an economic extension of the model.

The model was simulated using Euler’s method with a time-step of one month (the results were compared to those of the Runge–Kutta method and found to be identical). This was implemented using MATLAB for quick development as well as Visual Basic for Applications (VBA), interfaced through Microsoft Excel. The latter implementation, while not the fastest, was chosen to allow the program to be used by public health professionals who are not familiar with computer programming. This is part of a longer-term project

Results

Data from Canada

There were 586 cases in the Province of Quebec in 1985 and 195 in 2009. Overall, the fit of the model to the data was found to be high (R^{2}=0.92). Fitting to the data, the background exposure level was found to be 168 times the infectious level, and a transmission parameter of 4.0±0.4 was found. This is consistent with a similar model by Castillo-Chavez

Although not the primary target of tuberculosis interventions, the model was used to forecast the data in non-indigenous people of Canadian birth. Tuberculosis among this group has been decreasing over the past several years, and is in fact consistent with exponential decay (R^{2}=0.85). Fitting the model to the data, we found that the decay of new infections was consistent with a lower bound of zero transmissions (R^{2}=0.56). Forecasting into the future using these results, we expect that there will be fewer than 30 new cases yearly by 2030 (Figure

Retroactive and future predictions of the model for the Canadian-born non-indigenous Quebec population

**Retroactive and future predictions of the model for the Canadian-born non-indigenous Quebec population**. Error bars represent standard deviations of forecast distributions. Inset: Comparison of mean long term trend treating the Canadian-born and immigrant populations separately (blue), and with full interaction (green).

The fitting for the immigrant model begins with the 2001 census for the Province of Quebec, which reported an immigrant population of 706,975 and 37,604 new immigrants that year, while 150 new cases among immigrants were reported. An unknown, but important parameter is the percentage of exposed immigrants. We estimate this to be 130 exposed per 100,000 new immigrants yearly, based on 2009 epidemiology data. Using the 2001 through 2009 data as a basis, we found a best-fit (R^{2}=0.68) transmission parameter of 2.72 ± 1.98 and an initial exposed to infectious ratio of 167, both consistent with our estimates for the homogeneous population. The model predicts a slow decline in the disease, with the median prediction dropping below 100 cases within several years (Figure

Retroactive and future predictions of the model for the immigrant population

**Retroactive and future predictions of the model for the immigrant population.** Error bars represent standard deviations of forecast distributions. Inset: Long-term prediction, showing the eventual rise in the late 21st century.

The Inuit population of the Province of Quebec is largely isolated

The Inuit population can be difficult to model epidemiologically because of its low population: minor variations in cases can have large effects on the overall trend. The Inuit population of the northern region of the Province of Quebec was 7,765 in 1996 according to the Circumpolar Statistics Database ^{2}=0.64). The transmission parameter was found to be much higher than among immigrants: 61.90 ±16.3. The exposed to infectious ratio was found to be 216. The validation and forecast for the Inuit population can be seen in Figure

Retroactive and future predictions of the model for the Inuit Quebec population

**Retroactive and future predictions of the model for the Inuit Quebec population.** Error bars represent standard deviations of forecast distributions. The red curve represents the prediction when re-infection (a transfer of recovered persons to the exposed category) is taken into account, and the error bars are suppressed at early times as a visual aide.

Defining upper and lower bounds on the predictions, we found that the disease will increase to a plateau of about 30 cases in the 2020’s.

Epidemiological reports indicate that re-infections were responsible for 6% of total cases in Quebec in 2000

**Year**

**Canadian born**

**Immigrants**

**Inuit**

**Total**

**Drop from 2009**

Parentheses represent bounds on predictions.

2015

48 (16)

102 (22)

30 (7)

181 (28)

16%

2020

38 (15)

86 (22)

33 (6)

158 (27)

26%

2025

24 (15)

76 (22)

35 (5)

141 (28)

34%

2030

19 (11)

64 (22)

36 (3)

124 (24)

42%

Discussion

The numerical results of our simulations are presented in Figures

The predictions for immigrant cases show that the disease will not vanish. The upper bound on the transmission rate predicts that the number of immigrant cases may start to increase in the 2040s. This is consistent with the predictions of Zhou

The high transmission parameter and predicted increase for the Inuit may appear alarming, but the growth is not expected to continue indefinitely and in the long term the per capita incidence disease will be present but stable. This is similar to the findings of Blower

More data may be needed to understand the long-term trends in this population and a more complicated model than a homogeneous differential equation may be necessary to take into account such fluctuations. Nguyen

It may be of interest for future researchers to study in greater depth the dynamics of this particular model, beyond short- and medium-term predictions for Quebec. Future work can examine it in the context of broader analyses of tuberculosis dynamics in low incidence-high immigration countries

It is of note that the latent category in this model does not explicitly have an effect on the dynamics: it merely serves as another compartment for exposed individuals. When re-infection is taken into account there is the possibility of people in the latent category becoming infectious after recovery. Although the model may be simplified by removing the transition to latency, it is necessary for future work in which the financial cost of latent tuberculosis, on the order of several hundred dollars

Certain occurrences may reduce the validity of these results, for example, if at some time in the future a new treatment for TB is discovered and implemented, or a new deadlier or more drug-resistant strain breaks out. The predictions of this model would no longer adequately describe those situations, although modifications to the recovery parameters may account for such developments. To incorporate such an effect in the model, a parameter can be modified at a specific time (for example, the recovery parameter would be increased in the case of a better drug) which would manifest itself as a cusp on the forecast curve. However, predicting such changes are beyond the scope of this paper.

The simulations described in this paper are part of a larger study that serves to estimate the future financial burden of tuberculosis

While it may appear that these results are specific to the Province of Quebec, they highlight the challenges of disease management in the 21st century. The Province of Quebec is an example of an affluent industrialized society with a large immigrant population and a disadvantaged indigenous population. In this respect, the epidemiology of tuberculosis between these three groups is similar in other Canadian provinces as well as other nations with similar population structures including the United States, Australia, New Zealand and South Africa as well as European nations with high rates of immigration. The forecasts presented herein may serve as a case study for a situation that may arise worldwide in the coming decades.

Conclusions

A compartmental differential equation model was used to forecast the yearly incidence of tuberculosis for three at risk populations in the Province of Quebec. Tuberculosis has been declining in the Province of Quebec for many years, but is more prevalent among the immigrant and Inuit communities. Historical epidemiological data was used to verify the model, which was then used to forecast the rates for three population groups. It was found that the disease is vanishing among non-indigenous Canadian born persons, that the incidence is expected to decrease among immigrants but not vanish, and that among the Inuit a short term increase and subsequent leveling is predicted. Overall, the number of new annual cases will decrease.

Competing interests

Alexander Klotz, Abdoulaye Harouna and Andrew F. Smith were employees of Medmetrics Inc at the time this research was conducted. The authors have no other competing interests.

Authors’ contributions

AK: AK provided data analysis and literature searching skills. AK worked on the statistical analysis and interpretation of data flowing from the review of the literature and analyzed data on the trends of tuberculosis in the Province of Quebec. Lastly, AK participated in the drafting of the article reporting on the findings of this research. AH: AH contribution to this paper was in the form of providing support with the literature searches, extracting the data for analysis, assisting with the analysis of underlying trends in the data and in assisting with the drafting the manuscript reporting on the findings. AFS: AFS was responsible overall for obtaining funding for this research, the original conceptualization of the research aims, underlying methodological approach, devising the literature search strategy, analyzing the data, as well as drafting, editing and critically evaluating the intellectual content of the final manuscript. All authors read and approved the final manuscript.

Acknowledgements

The authors also wish to thank Drs. John White and Marcel Behr, both of McGill University and Dr Suneil Malik of the Infectious Disease Program in the Office of Biotechnology, Genomics and Population Health at the Public Health Agency of Canada for comments and suggestions on earlier drafts of this manuscript.

Funding

Funding for this research was provided to Medmetrics Inc, by McGill University, Montreal, Quebec, Génome Québec and the Ministère de Développement économique, innovation et Exportation du gouvernement du Québec.

Pre-publication history

The pre-publication history for this paper can be accessed here: