A TB model: Is disease eradication possible in India?

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doi:10.3934/mbe.2018010

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2018, Volume 15, Issue 1: 233-254. Doi: 10.3934/mbe.2018010

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A TB model: Is disease eradication possible in India?

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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 28, 2016

Revised: March 14, 2017

Published: February 2018

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Abstract

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.

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Mathematics Subject Classification: 92D30, 92D25.

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Figure 1. Diagram showing the flow of population through $6$ different possible population classes

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Figure 2. 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.

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Figure 3. 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.

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Figure 4. 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.

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Figure 5. 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.

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Figure 6. 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.

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Figure 7. 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.

Download: Full-size image PowerPoint slide

Figure 8. 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.

Download: Full-size image PowerPoint slide

Figure 9. 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.

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Figure 10. 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 $~2900$.

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Figure 11. 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 $~2200$.

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Figure 12. 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.

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Figure 13. 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.

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Table 1. 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]

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