ResearchParameter estimation of tuberculosis transmission model using Ensemble Kalman filter across Indian states and union territories
Introduction
Tuberculosis (TB) is a well known infectious disease caused by bacterium M. tuberculosis which generally spreads through air. In 2011, 2.0–2.5 million new TB cases were estimated in India out of the global annual incidence of 9.4 million cases [1], [2]. A large number of factors contribute to the spread of TB; such as high prevalence of HIV/AIDS and diabetes, poor hygiene, crowding, illiteracy and lack of awareness make the TB situation critical in Indian context. All these factors directly contribute to high infection rates among the population. In 1997, the Government of India, with the help of World Bank, initiated RNTCP based on the internationally recommended Directly Observed Treatment Short-course (DOTS) strategy [2], [3]. RNTCP is the largest TB control program in terms of treatment of patients with full nationwide coverage.
Mathematical models and statistical techniques play a significant role in understanding the transmission dynamics of TB. In simple deterministic model of infectious disease, the number of susceptible persons who are infected by an infectious individual per unit of time is proportional to the total number of susceptible persons. This proportional coefficient is defined as infection rate. Estimation of parameters of mathematical model, for instance, infection rate contributes to better quantify the spread of disease.
Generally, inference of these parameters is a difficult task because of poor compatibility between observed data and models. Simulations and epidemiological data have been used to estimate the key parameters of deterministic models. Different techniques have been introduced and applied to estimate the parameters of TB models. Approximate Bayesian computation approach has been used to estimate TB transmission rate parameters for United States [4]. A synchronisation based method has been implemented to infer the parameters such as treatment rate, disease induced mortality rate and infection rate of a TB model. In particular, the infection rate in the study is estimated to be 2.04 for the quarterly data during 2003–2007 for Cameroon [5]. Liu et al. estimated the reactivation and infection rate of a TB model for China by assuming these rates as sinusoidal functions and infection rate is estimated to be 2.23 person per month for the period 2005–2009 [6]. A qualitative analysis of the TB model for Nigeria has been performed to analyse the effect of DOTS strategy [7]. Mandal et al. estimated parameters such as infection rate and treatment rate using annual prevalence and incidence data of TB [8]. In particular, the infection rate of TB have been estimated to be 11.03 per year for India [8]. Mishra et al. integrated quarantine compartment into TB model, which incorporates the multidrug-resistant TB patients. The model is further analysed and simulated in using TB data of Jharkhand, India [9].
In the present paper, we use Ensemble Kalman filter (EnKf) approach to estimate the parameters of a deterministic model of TB. Kalman filter has been extensively used to infer the parameters of models of various infectious diseases [10], [11], [12], [13], [14]. Parameters of an HIV/AIDS model has been estimated using Kalman filter approach [10]. An extension of Kalman filter has been implemented to analyse the spatio-temporal behaviour of measles outbreak using count data for the period 1960–1970 in Landon [12]. Influenza data of different cities within the United States has been analysed and demonstrated that ensemble filters were found to be more accurate than other filters in predicting the peaks of the influenza [13].
Section snippets
Dynamic model
In this paper, we use a variation of SIR (Susceptible-Infected-Recovered) model defined as SLIS (Susceptible-Latent-Infected-Susceptible). There are three exclusive groups of individuals; namely, susceptible, S, latently infected, L (infected with M. tuberculosis but not infectious), and actively infected with M. tuberculosis, I (infected and infectious). The model does not take into account genetic and demographic heterogeneity. The following are the governing differential equations for the
Results
The SLIS-EnKf framework explained in Section Methods and data is implemented in MATLAB and results are post processed and plotted in R-statistical programming language with the “spplot” package. In Fig. 1, reported smear positive cases of TB are plotted for Manipur state for the time period 2006 to 2011. It is observed that there is a declining trend in the number of cases. A seasonal pattern is observed in the number of cases over the years, with peak infection rate occurring in the mid of the
Discussion
We have implemented EnKf in conjunction with a deterministic model of TB. There are many alternate assimilation approaches that may be tested further [18]. The parameter estimation framework presented here captures seasonality well in the data which could not be expected from standard-likelihood methods. The technique is also computationally inexpensive when compared to Monte Carlo methods.
India's DOTS programme is the fastest expanding programme, placing more than 100,000 patients on treatment
Conclusion
To make RNTCP program more effective a shift in strategies is needed for states with higher transmission rates as compared with states with higher fraction of smear positive cases. For example, states with higher transmission rate require more focus on educational and awareness programs to mitigate the transmission of TB bacteria from infected to susceptible. Human behaviour plays an important role in the spread of infectious agent; transmission of the disease can be reduced by increasing the
Ethics
No ethical permission is required to conduct the present study.
Authorship contribution
SA, AB and PL designed the study. PN and VP carried out the analysis. All authors critically reviewed the manuscript.
Conflicts of interest
The authors of this paper have no conflicts of interest to declare.
Funding sources
The author(s) received no financial support for the research, authorship, and/or publication of this article.
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