The numerical results of our simulations are presented in Figures 2, 3, 4. Overall, we found that the general declining trend in the incidence of tuberculosis in the Province of Quebec will continue, especially as seen heuristically when the population is modelled as homogeneous. This will be particularly rapid among non-indigenous Canadians, while immigrant incidence rates will decrease at a slower rate. As the rate of TB in immigrants and the general population declines, the contribution to the total number of cases by the Inuit will increase.
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 et al.  from 2008: who predict that an increase in immigrant cases will cause an increase in the non-immigrant cases by 2012, although a re-analysis with recent data may delay that prediction. The slower decline and subsequent increase in immigrant cases is in part due to the proportion of exposed immigrants that arrives each year and in part due to the greater transmission factor compared to the Canadian-born population. The greater transmission factor may be due, in part, to the tendency of immigrant communities to live in closer proximity to one another as in Montreal, as described by Haase et al. . The low socioeconomic status of immigrants from high-incidence countries may also contribute to this. The results of our forecast show that if measures are not taken to screen immigrants for tuberculosis and to reduce the transmission of the disease between immigrants, the trend may reverse mid-century and TB may become more prevalent. Given that immigrant populations are a major contributor of tuberculosis cases, the future of the disease in Canada should be examined in context of future increased immigration.
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 et al. , who predict a plateau in the relative proportion of infections over long durations. Nevertheless, the number of new annual cases among the Inuit is expected to increase. Re-infection may play a significant role as suggested by our simulations, as the higher proportion of both recovered and infectious persons makes contact more likely. If these predictions are accurate, the costs of diagnosing and treating the disease, which are higher in the northern regions of the Province of Quebec due to low density populations and necessity of air travel, may increase.
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 et al. , for example, noted that the transmission was based on interactions between different villages in Nunavik. With small populations, stochasticity in transmission may play a role. Epidemiological models such as those taking into account stochastic processes  in disease progression may be more accurate in describing the Inuit population over long time spans.
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 [15, 16]. However, such analyses were deemed to be beyond the scope of this paper because they are not expected to lead to short-term changes in the forecasts. Long term trends can be examined (as appear on insets in Figures 2 and 3), but over many decades they will lose validity because the demographic models have constant birth, death, and immigration rates leading to exponential population growth. Over longer time periods, changes to these rates and the onset of logistic growth are expected to occur and more assumptions are required to model the demographics.
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  per patient, must be taken into account. If re-infection is included, as discussed previously, there is now an opportunity for latent individuals to become infectious.
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 . Among the non-indigenous Canadian-born population the burden will be minimal, as the disease slowly becomes less and less common. The results of these simulations, however, indicate that without intervention, tuberculosis among immigrant populations in the Province of Quebec will not vanish, and incidence of the disease will slowly increase among the Inuit. These may lead to greater expenditure required to control the disease, if public health policy is not modified to take into account tuberculosis within these two groups.
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.