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February  2018, 15(1): 233-254. doi: 10.3934/mbe.2018010

## A TB model: Is disease eradication possible in India?

 1 Public Health Foundation of India, Plot No. 47, Sector-44, Gurgaon-122002, Haryana, India 2 Dipartimento di Matematica "Giuseppe Peano", Università di Torino, via Carlo Alberto 10, 10123 Torino, Italy

Received  October 29, 2016 Revised  March 15, 2017 Published  May 2017

Tuberculosis (TB) is returning to be a worldwide global public health threat. It is estimated that 9.6 million cases occurred in 2014, of which just two-thirds notified to public health authorities. The "missing cases" constitute a severe challenge for TB transmission control. TB is a severe disease in India, while, worldwide, the WHO estimates that one third of the entire world population is infected.

Nowadays, incidence estimation relies increasingly more on notifications of new cases from routine surveillance. There is an urgent need for better estimates of the load of TB, in high-burden settings. We developed a simple model of TB transmission dynamics, using a dynamical system model, consisting of six classes of individuals. It contains the current medical epidemiologists' understanding of the spread of the Mycobacterium tuberculosis in humans, which is substantiated by field observations at the district level in India. The model incorporates the treatment options provided by the public and private sectors in India. Mathematically, an interesting feature of the system is that it exhibits a backward, or subcritical, bifurcation.

One of the results of the investigation shows that the discrepancy between the diagnosis rates of the public and private sector does not seem to be the cause of the endemicity of the disease, and, unfortunately, even if they reached 100% of correct diagnosis, this would not be enough to achieve disease eradication.

Several other approaches have been attempted on the basis of this model to indicate possible strategies that may lead to disease eradication, but the rather sad conclusion is that they unfortunately do not appear viable in practice.

Citation: Surabhi Pandey, Ezio Venturino. A TB model: Is disease eradication possible in India?. Mathematical Biosciences & Engineering, 2018, 15 (1) : 233-254. doi: 10.3934/mbe.2018010
##### References:
 [1] M. A. Behr, S. A. Warren, H. Salomon, P. C. Hopewell, A. P. de Leon, C. L. Daley and P. M. Small, Transmission of mycobacterium tuberculosis from patients smear-negative for acid-fast bacilli, Lancet, 353 (1999), 444-449.  doi: 10.1016/S0140-6736(98)03406-0. [2] S. Bernardi and E. Venturino, Viral epidemiology of the adult Apis Mellifera infested by the Varroa destructor mite, Heliyon, 2. [3] V. Chadha, S. Majhi, S. Nanda and S. Pandey, Prediction of prevalence and incidence of tuberculosis in a district in India, submitted. [4] V. K. Chadha, P. Kumar, S. M. Anjinappa, S. Singh, S. Narasimhaiah, M. V. Joshi, J. Gupta, Lakshminarayana, J. Ramchandra, M. Velu, S. Papkianathan, S. Babu and H. Krishna, Prevalence of pulmonary tuberculosis among adults in a rural sub-district of South India, PLoS ONE, 7. [5] V. K. Chadha, R. Sarin, P. Narang, K. R. John, K. K. Chopra, R. J. D.K Mendiratta, V. Vohra, A. N. hashidhara, G. Muniraj, P. G. Gopi and P. Kumar, Trends in the annual risk of tuberculous infection in India, International Journal of Tuberculosis and Lung Disease, 17 (2013), 312-319. [6] G. W. Comstock, V. T. Livesay and S. F. Woolpert, The prognosis of a positive tuberculin reaction in childhood and adolescence, American J of Epidemiology, 99 (1974), 131-138.  doi: 10.1093/oxfordjournals.aje.a121593. [7] E. L. Corbett, C. J. Watt, N. Walker, D. Maher, B. G. Williams, M. C. Raviglione and C. Dye, The growing burden of tuberculosis: Global trends and interactions with the HIV epidemic, JAMA Internal Medicine, Formerly known as Archives of Internal Medicine, 163 (2003), 1009-1021.  doi: 10.1001/archinte.163.9.1009. [8] D. W. Dowdy and R. E. Chaisson, The persistence of tuberculosis in the age of dots: Reassessing the effect of case detection, Bulletin World Health Organisation, 87 (2009), 296-304.  doi: 10.2471/BLT.08.054510. [9] S. H. Fercbee, Controlled chemoprophylaxis trials in tuberculosis. a general review, Bibliotheca Tuberculosea, 26 (1970), 28-106. [10] K. Floyd, V. K. Arora, K. J. R. Murthy, K. Lonnroth, N. Singla, Y. Akbar, M. Zignol and M. Uplekar, Cost-effectiveness of PPM-DOTS in India, Bulletin of the World Health Organization, 84 (2006), 437-439. [11] P. G. Gopi, R. Subramani, K. Sadacharam and P. R. Narayanan, Yield of pulmonary tuberculosis cases by employing two screening methods in a community survey, International Jounral of Tuberculosis and Lung Disease, 10 (2006), 343-345. [12] M. Martcheva, An Introduction to Mathematical Epidemiology, Springer-Verlag, New York, 2015.  doi: 10.1007/978-1-4899-7612-3. [13] S. Martorano Raimundo, E. Venturino, Drug resistant impact on tuberculosis transmission, Wseas Transactions on Biology and Biomedicine, v. 5, 85-95, ISSN 1109-9518,2008. [14] S. Martorano Raimundo, H. M. Yang and E. Venturino, Theoretical assessment of the relative incidences of sensitive and resistant Tuberculosis epidemic in presence of drug treatment, Mathematical Biosciences and Engineering, 11 (2014), 971-993. [15] R. Naresh, S. Pandey and J. B. Shukla, Modeling the cumulative cffect of ecological factors in the habitat on the spread of tuberculosis, International Journal of Biomathematics, 2 (2009), 339-355.  doi: 10.1142/S1793524509000728. [16] W. H. Organization, World health organization global TB report 2016,2016, URL http://www.who.int/tb/publications/global_report/gtbr2016_executive_summary.pdf?ua=1, Accessed online 20-January-2017. [17] W. H. Organization, World health organization tb report 2015,2016, URL http://www.who.int/tb/publications/global_report/gtbr15_main_text.pdf, Accessed online 20-January-2017. [18] S. Pandey, V. K. Chadha, R. Laxminarayan and N. Arinaminpathy, Estimating tuberculosis incidence from primary survey data: A mathematical modeling approach, International Journal of TB and Lung Disease, 21 (2017), 366-374. [19] S. Pandey, S. Nanda and P. S. Datti, Mathematical analysis of TB model pertaining to India, Submitted. [20] A. Perasso, An introduction to the basic reproduction number in mathematical biology, private communication. [21] I. Registrar General, Sample registration survey bulletin, 2011, December 2011, URL http://censusindia.gov.in/vital_statistics/SRS_Bulletins/Bulletins.aspx. [22] S. Satyanarayana, S. A. Nair, S. S. Chadha, R. Shivashankar, G. Sharma, S. Yadav, S. Mohanty, V. Kamineni, C. Wilson, A. D. Harries and P. K. Dewan, From where are tuberculosis patients accessing treatment in India? results from a cross-sectional community based survey of 30 districts, PLoS One, 16. [23] V. Sophia, V. H. Balasangameswara, P. S. Jagannatha, V. N. Saroja and P. Kumar, Treatment outcome and two and half years follow-up status of new smear positive patients treated under RNTCP, Indian Journal of Tuberculosis, 51 (2004), 199-208. [24] K. Styblo, The relationship between the risk of tuberculous infection and the risk of developing infectious tuberculosis, Bulletin of the International Union Against Tuberculosis and Lung Disease, 60 (1985), 117-129. [25] R. Subramani, S. Radhakrishna, T. R. Frieden, C. K. P. G. Gopi, T. Santha, F. Wares, N. Selvakumar and P. R. Narayanan, Rapid decline in prevalence of pulmonary tuberculosis after dots implementation in a rural area of South India, International Jounral of Tuberculosis and Lung Disease, 12 (2008), 916-920. [26] A. Thomas, P. G. Gopi, T. Santha, V. Chandrasekaran, R. Subramani, N. Selvakumar, S. I. Eusuff, K. Sadacharam and P. R. Narayanan, Predictors of relapse among pulmonary tuberculosis patients treated in a dots programme in South India, International Journal of Tuberculosis and Lung Disease, 9 (2005), 556-561. [27] M. Uplekar, S. Juvekar, S. Morankar, S. Rangan and P. Nunn, Tuberculosis patients and practitioners in private clinics in India, International Journal of Tuberculosis and Lung Disease, 6 (1998), 324-329. [28] M. W. Uplekar, S. K. Jvekar and D. B. Parande, Tuberculosis management in private practice and its implication, Indian Journal of Tuberculosis, 43 (1996), 19-22. [29] P. van den Driessche and J. Watmough, A simple sis epidemic model with a backward bifurcation, J. of Mathematical Biology, 40 (2000), 525-540.  doi: 10.1007/s002850000032. [30] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [31] F. van Leth, M. J. van der Werf and M. W. Borgdorff, Prevalence of tuberculous infection and incidence of tuberculosis: A re-assessment of the styblo rule, Bulletin of the World Health Organization, 86 (2008), 20-26.

show all references

##### References:
 [1] M. A. Behr, S. A. Warren, H. Salomon, P. C. Hopewell, A. P. de Leon, C. L. Daley and P. M. Small, Transmission of mycobacterium tuberculosis from patients smear-negative for acid-fast bacilli, Lancet, 353 (1999), 444-449.  doi: 10.1016/S0140-6736(98)03406-0. [2] S. Bernardi and E. Venturino, Viral epidemiology of the adult Apis Mellifera infested by the Varroa destructor mite, Heliyon, 2. [3] V. Chadha, S. Majhi, S. Nanda and S. Pandey, Prediction of prevalence and incidence of tuberculosis in a district in India, submitted. [4] V. K. Chadha, P. Kumar, S. M. Anjinappa, S. Singh, S. Narasimhaiah, M. V. Joshi, J. Gupta, Lakshminarayana, J. Ramchandra, M. Velu, S. Papkianathan, S. Babu and H. Krishna, Prevalence of pulmonary tuberculosis among adults in a rural sub-district of South India, PLoS ONE, 7. [5] V. K. Chadha, R. Sarin, P. Narang, K. R. John, K. K. Chopra, R. J. D.K Mendiratta, V. Vohra, A. N. hashidhara, G. Muniraj, P. G. Gopi and P. Kumar, Trends in the annual risk of tuberculous infection in India, International Journal of Tuberculosis and Lung Disease, 17 (2013), 312-319. [6] G. W. Comstock, V. T. Livesay and S. F. Woolpert, The prognosis of a positive tuberculin reaction in childhood and adolescence, American J of Epidemiology, 99 (1974), 131-138.  doi: 10.1093/oxfordjournals.aje.a121593. [7] E. L. Corbett, C. J. Watt, N. Walker, D. Maher, B. G. Williams, M. C. Raviglione and C. Dye, The growing burden of tuberculosis: Global trends and interactions with the HIV epidemic, JAMA Internal Medicine, Formerly known as Archives of Internal Medicine, 163 (2003), 1009-1021.  doi: 10.1001/archinte.163.9.1009. [8] D. W. Dowdy and R. E. Chaisson, The persistence of tuberculosis in the age of dots: Reassessing the effect of case detection, Bulletin World Health Organisation, 87 (2009), 296-304.  doi: 10.2471/BLT.08.054510. [9] S. H. Fercbee, Controlled chemoprophylaxis trials in tuberculosis. a general review, Bibliotheca Tuberculosea, 26 (1970), 28-106. [10] K. Floyd, V. K. Arora, K. J. R. Murthy, K. Lonnroth, N. Singla, Y. Akbar, M. Zignol and M. Uplekar, Cost-effectiveness of PPM-DOTS in India, Bulletin of the World Health Organization, 84 (2006), 437-439. [11] P. G. Gopi, R. Subramani, K. Sadacharam and P. R. Narayanan, Yield of pulmonary tuberculosis cases by employing two screening methods in a community survey, International Jounral of Tuberculosis and Lung Disease, 10 (2006), 343-345. [12] M. Martcheva, An Introduction to Mathematical Epidemiology, Springer-Verlag, New York, 2015.  doi: 10.1007/978-1-4899-7612-3. [13] S. Martorano Raimundo, E. Venturino, Drug resistant impact on tuberculosis transmission, Wseas Transactions on Biology and Biomedicine, v. 5, 85-95, ISSN 1109-9518,2008. [14] S. Martorano Raimundo, H. M. Yang and E. Venturino, Theoretical assessment of the relative incidences of sensitive and resistant Tuberculosis epidemic in presence of drug treatment, Mathematical Biosciences and Engineering, 11 (2014), 971-993. [15] R. Naresh, S. Pandey and J. B. Shukla, Modeling the cumulative cffect of ecological factors in the habitat on the spread of tuberculosis, International Journal of Biomathematics, 2 (2009), 339-355.  doi: 10.1142/S1793524509000728. [16] W. H. Organization, World health organization global TB report 2016,2016, URL http://www.who.int/tb/publications/global_report/gtbr2016_executive_summary.pdf?ua=1, Accessed online 20-January-2017. [17] W. H. Organization, World health organization tb report 2015,2016, URL http://www.who.int/tb/publications/global_report/gtbr15_main_text.pdf, Accessed online 20-January-2017. [18] S. Pandey, V. K. Chadha, R. Laxminarayan and N. Arinaminpathy, Estimating tuberculosis incidence from primary survey data: A mathematical modeling approach, International Journal of TB and Lung Disease, 21 (2017), 366-374. [19] S. Pandey, S. Nanda and P. S. Datti, Mathematical analysis of TB model pertaining to India, Submitted. [20] A. Perasso, An introduction to the basic reproduction number in mathematical biology, private communication. [21] I. Registrar General, Sample registration survey bulletin, 2011, December 2011, URL http://censusindia.gov.in/vital_statistics/SRS_Bulletins/Bulletins.aspx. [22] S. Satyanarayana, S. A. Nair, S. S. Chadha, R. Shivashankar, G. Sharma, S. Yadav, S. Mohanty, V. Kamineni, C. Wilson, A. D. Harries and P. K. Dewan, From where are tuberculosis patients accessing treatment in India? results from a cross-sectional community based survey of 30 districts, PLoS One, 16. [23] V. Sophia, V. H. Balasangameswara, P. S. Jagannatha, V. N. Saroja and P. Kumar, Treatment outcome and two and half years follow-up status of new smear positive patients treated under RNTCP, Indian Journal of Tuberculosis, 51 (2004), 199-208. [24] K. Styblo, The relationship between the risk of tuberculous infection and the risk of developing infectious tuberculosis, Bulletin of the International Union Against Tuberculosis and Lung Disease, 60 (1985), 117-129. [25] R. Subramani, S. Radhakrishna, T. R. Frieden, C. K. P. G. Gopi, T. Santha, F. Wares, N. Selvakumar and P. R. Narayanan, Rapid decline in prevalence of pulmonary tuberculosis after dots implementation in a rural area of South India, International Jounral of Tuberculosis and Lung Disease, 12 (2008), 916-920. [26] A. Thomas, P. G. Gopi, T. Santha, V. Chandrasekaran, R. Subramani, N. Selvakumar, S. I. Eusuff, K. Sadacharam and P. R. Narayanan, Predictors of relapse among pulmonary tuberculosis patients treated in a dots programme in South India, International Journal of Tuberculosis and Lung Disease, 9 (2005), 556-561. [27] M. Uplekar, S. Juvekar, S. Morankar, S. Rangan and P. Nunn, Tuberculosis patients and practitioners in private clinics in India, International Journal of Tuberculosis and Lung Disease, 6 (1998), 324-329. [28] M. W. Uplekar, S. K. Jvekar and D. B. Parande, Tuberculosis management in private practice and its implication, Indian Journal of Tuberculosis, 43 (1996), 19-22. [29] P. van den Driessche and J. Watmough, A simple sis epidemic model with a backward bifurcation, J. of Mathematical Biology, 40 (2000), 525-540.  doi: 10.1007/s002850000032. [30] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6. [31] F. van Leth, M. J. van der Werf and M. W. Borgdorff, Prevalence of tuberculous infection and incidence of tuberculosis: A re-assessment of the styblo rule, Bulletin of the World Health Organization, 86 (2008), 20-26.
Diagram showing the flow of population through $6$ different possible population classes
With the remaining parameter values taken from the Table, the plot of the $R_0$ surface as function of $(\beta,\sigma )\in \{[0,6]\times [0,1]\}$ is shown in the left frame. The countour line indicating the domain in which $R_0$ is larger than 1 is shown in the corresponding right frame. Therefore the disease is endemic on the right portion of the parameter space plot. The star denotes the situation with these parameters as given originally in the Table.
With the remaining parameter values taken from the Table, the plot of the $R_0$ surface as function of $(\beta,\phi_1 )\in \{[0,6]\times [0,1]\}$ is shown in the left frame. The countour line indicating the domain in which $R_0$ is larger than 1 is blown-up and shown for $(\beta,\phi_1 )\in \{[0,6]\times [0,0.1]\}$ in the corresponding right frame. Therefore the disease is endemic in the upper right corner of the plot. The star denotes the situation with these parameters as given originally in the Table. The star denotes the situation with these parameters as given originally in the Table.
With the remaining parameter values taken from the Table, the plot of the $R_0$ surface as function of $(\beta,\nu_1 )\in \{[0,6]\times [0,1]\}$ is shown in the left frame. The countour line indicating the domain in which $R_0$ is larger than 1 is shown in the corresponding right frame. Therefore the disease is endemic on the right portion of the parameter space plot. The star denotes the situation with these parameters as given originally in the Table.
With the remaining parameter values taken from the Table, the plot of the $R_0$ surface as function of $(\beta,\nu_2 )\in \{[0,6]\times [0,1]\}$ is shown in the left frame. The countour line indicating the domain in which $R_0$ is larger than 1 is shown in the corresponding right frame. Therefore the disease is endemic on the right portion of the parameter space plot. The star denotes the situation with these parameters as given originally in the Table.
With the remaining parameter values taken from the Table, the plot of the $R_0$ surface as function of $(\beta,\alpha_0 )\in \{[0,6]\times [0,1]\}$ is shown in the left frame. The countour line indicating the domain in which $R_0$ is larger than 1 is shown blown-up, for $(\beta,\alpha_0 )\in \{[0,6]\times [0,0.3]\}$, in the corresponding right frame. Therefore the disease is endemic in the very thin strip at the bottom right corner of the plot. The star denotes the situation with these parameters as given originally in the Table.
With the remaining parameter values taken from the Table, the plot of the $R_0$ surface as function of $(\beta,\alpha_2 )\in \{[0,6]\times [0,1]\}$ is shown in the left frame. The countour line indicating the domain in which $R_0$ is larger than 1 is shown in the corresponding right frame. Therefore the disease is endemic on the right portion of the parameter space plot. The star denotes the situation with these parameters as given originally in the Table.
With the remaining parameter values taken from the Table, the plot of the $R_0$ surface as function of $(\beta,\phi_2 )\in \{[0,6]\times [0,1]\}$ is shown in the left frame. The countour line indicating the domain in which $R_0$ is larger than 1 is shown in the corresponding right frame. Therefore the disease is endemic on the right portion of the parameter space plot. The star denotes the situation with these parameters as given originally in the Table.
With the remaining parameter values taken from the Table, the plot of the $R_0$ surface as function of $\nu_1,\nu_2\in \{[0,1]\times [0,1]\}$ is shown. It is always above the level 1. Therefore the disease remains endemic independently of the performance of the two hospitalization sectors.
Endemic equilibrium for $A=100$, $\beta=1.5$ and with the other parameter values taken from the Table. Left frame: susceptibles $S$ at steady level $~2100$; Center frame: treated but latently infected $T_1+T_2+L_2$ at steady level $~200$; Right frame: infected in the active stage of the disease $D+L_1$ at steady level $~2900$.
Endemic equilibrium for $A=100$, $\beta=0.9$ and with the parameter values taken from the Table. Left frame: susceptibles $S$ at steady level $~5000$; Center frame: treated but latently infected $T_1+T_2+L_2$ at steady level $~200$; Right frame: infected in the active stage of the disease $D+L_1$ at steady level $~2200$.
Endemic equilibrium for $\phi_2=0$. The top frame shows the 2 intersections of the straight line $\Phi_s$ (red) with the hyperbola $\Psi_s$ (blue); note that the vertical line on the left represents the vertical asymptote. The center frame is a blow up of the 2 intersections closest to the vertical axis, while the bottom one shows the intersection farther on the right.
Top frame: disease eradication for $\phi_1=0.001$, $\phi_2=0$, $\sigma=0$, $\mu_{12}=0$, $\mu_{22}=0$, $\nu_1=1$, $\nu_2=1$. The top frame shows the plot over the whole relevant range of the straight line $\Phi_s$ (red) and the hyperbola $\Psi_s$ (blue), again with the vertical line on the left representing the vertical asymptote of the latter. The other frames are blow ups of the former. The second one from top shows the range $[2000,3000]$ with no intersections, the third one the range $[3000,4220]$ again with no intersections, the bottom one contains the range $[4220,4230]$, with a much lower vertical scale, where again no intersections occur. Bottom frame: disease eradication for $\phi_1=0.001$, $\phi_2=0$, $\sigma=0.01$, $\mu_{12}=0.001$, $\mu_{22}=0.001$, $\nu_1=.999$, $\nu_2=.999$ for which $\Omega = -2.316$. The frames contain similar information as for the left column.
Model parameters
 Description Symbol Value Unit Reference Immigration rate $A$ $30$ [21] transmission rate $\beta$ $5.31$ [1,4,5,7,8][11,24,25,31] Proportion of infectious rapidly progressing to active disease $\sigma$ $0.015$ pure number [6] Progression from latent to diseased class $\phi_1$ $0.02284$ year$^{-1}$ [9] Diagnosis and treatment rate in the public sector $\nu_1$ $0.49$ year$^{-1}$ [16,17,22] Diagnosis and treatment rate in the private sector $\nu_2$ $0.41$ year$^{-1}$ [16,17,22] Recovery (cure) rate after treatment in the public sector $\mu_{11}$ $0.89$ year$^{-1}$ [16] Recovery (cure) rate after treatment in the public sector $\mu_{21}$ $0.51$ year$^{-1}$ [10,27,28] Failure rate after treatment in the private sector $\mu_{12}$ $0.064$ year$^{-1}$ [16] Failure rate after treatment in the private sector $\mu_{22}$ $0.32$ year$^{-1}$ [27] Relapse from treatment $\phi_2$ $0.11$ year$^{-1}$ [23,26] Natural death rate $\alpha_0$ $0.0071$ person$^{-1}$ year$^{-1}$ [21] Latently infected population $L_2$ death rate $\alpha_1$ $0.016$ year$^{-1}$ [23] Diseased population death rate (Case fatality rate in untreated) $\alpha_2$ $0.32$ year$^{-1}$ [17] Population under treatment death rate in public sector $\alpha_3$ $0.074$ year$^{-1}$ [16] Population under treatment death rate in private sector $\alpha_4$ $0.32$ year$^{-1}$ [27]
 Description Symbol Value Unit Reference Immigration rate $A$ $30$ [21] transmission rate $\beta$ $5.31$ [1,4,5,7,8][11,24,25,31] Proportion of infectious rapidly progressing to active disease $\sigma$ $0.015$ pure number [6] Progression from latent to diseased class $\phi_1$ $0.02284$ year$^{-1}$ [9] Diagnosis and treatment rate in the public sector $\nu_1$ $0.49$ year$^{-1}$ [16,17,22] Diagnosis and treatment rate in the private sector $\nu_2$ $0.41$ year$^{-1}$ [16,17,22] Recovery (cure) rate after treatment in the public sector $\mu_{11}$ $0.89$ year$^{-1}$ [16] Recovery (cure) rate after treatment in the public sector $\mu_{21}$ $0.51$ year$^{-1}$ [10,27,28] Failure rate after treatment in the private sector $\mu_{12}$ $0.064$ year$^{-1}$ [16] Failure rate after treatment in the private sector $\mu_{22}$ $0.32$ year$^{-1}$ [27] Relapse from treatment $\phi_2$ $0.11$ year$^{-1}$ [23,26] Natural death rate $\alpha_0$ $0.0071$ person$^{-1}$ year$^{-1}$ [21] Latently infected population $L_2$ death rate $\alpha_1$ $0.016$ year$^{-1}$ [23] Diseased population death rate (Case fatality rate in untreated) $\alpha_2$ $0.32$ year$^{-1}$ [17] Population under treatment death rate in public sector $\alpha_3$ $0.074$ year$^{-1}$ [16] Population under treatment death rate in private sector $\alpha_4$ $0.32$ year$^{-1}$ [27]
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