A Mathematical Model of a Tuberculosis Transmission Dynamics Incorporating First and Second Line Treatment

This  paper  presents  a  new  mathematical  model  of  a  tuberculosis  transmission  dynamics incorporating first and second line treatment. We calculated a control reproduction number which plays a vital role in biomathematics. The model  consists  of  two equilibrium points namely disease  free  equilibrium and  endemic equilibrium  point, it has been shown that the disease free equilibrium point was locally asymptotically stable if the control  reproduction number  is  less  than one and also  the endemic equilibrium point was  locally asymptotically stable if the control reproduction number is greater than one. Numerical simulation was carried out which supported the analytical results. DOI: https://dx.doi.org/10.4314/jasem.v24i5.29 Copyright: Copyright © 2020 Andrawus et al. This is an open access article distributed under the Creative Commons Attribution License (CCL), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Dates: Received: 12 March 2020; Revised: 30 April 2020; Accepted: 26 May 2020

Tuberculosis (TB) is a protracted bacterial infectious disease caused by Mycobacterium tuberculosis which position a major health, social and economic burden globally, especially in low and middle income countries (WHO, 2013). The surge in HIV-TB coinfection and growing emergence of multidrugresistant TB (MDR-TB) and extensively drug resistant TB (XDR-TB) strains has further fuelled TB epidemic (WHO,2013;WHO, 2016). It is the second fatal disease due to a single infectious agent only after HIV/AIDS (WHO, 2013;WHO, 2016). TB usually affects the lungs but it can also affect other sites as well (extra-pulmonary TB). Tuberculosis is conveyed by tiny airborne droplets which are ejected into the air when a person with active pulmonary TB coughs or talks (Issarowa et al., 2015). According to the World Health Organization (WHO), in 2013, about 9million people were infected, worldwide, with TB and 1.5 million deaths from the disease were reported, 360,000 of whom were HIV-positive (Yang et al., 2014). Tuberculosis is seen to be declining slowly each year and an estimated 37 million lives were saved between 2000 and 2013 through effective diagnosis and treatment (Yang et al., 2014;Okuonghae and Ikhimwin, 2016). Numerous mathematical models have been developed and used to study the transmission dynamics of TB in a population (Aparicio and Castillo-Chavez, 2009;Castillo-Chavez and Song, 2004;Feng, 2000). For instance, Okuonghae (2013) worked on a deterministic TB model with genetic heterogeneity in susceptibility and disease progression. Zhang and Feng (2000) constructed and analyzed a dynamical model to investigate the spread of TB in a community with isolation and incomplete treatment. The purpose of this article is to formulate a mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment which provides insights into the drug resistant cases in first and second line treatment. The model considers human population N . The population at time t is divided into six ( The above assumptions result in the following system of nonlinear ordinary differential equations: Simplifying (2), we have

Boundedness of Solutions: Consider the region
It can be shown that the set is positively invariant and a global attractor of all positive solution of the system (1) Lemma 1 The region is positively invariant for the system (1). Proof: The rate of change of the total human population is give as using the integrating factor method is a positively invariant set under the flow described in (1). Hence, no solution path leaves through and boundary of . Also, since solution paths cannot leave , solutions remain non-negative for non-negative initial conditions. Solutions exist for all time t. In this region, the model (1) is said to be well posed mathematically and epidemiologically.

Positivity of Solution
Lemma 2 Let the initial data for the model (1) to solve the ODE using the integrating factor method Similarly, we can also show that, Now, it follows that the control reproduction number is given as The following result is established using Theorem 2 in (van den Driessche & Watmough, 2002).

Lemma 3: The DFE of the system (1) is locally asymptotically stable if < 1 and unstable if > 1.
The value is the humans effective reproduction number since there is the presence of control strategies. The threshold quantity is the control reproduction number for the model (1). By Theorem 1, biologically speaking, Tuberculosis is eliminated from the population when < 1 if the initial sizes of the populations of the model are in the region of attraction of 0 D . However, the disease free equilibrium may not be globally asymptotically stable even if < 1 in the case when a backward bifurcation occurs. That is, there is the presence of a stable EEP co-existing with the DFE.

Local Stability of Endemic Equilibrium
The Eigenvalues are giving as We claimed the following results Lemma 4: The positive endemic equilibrium state of the model (1)

RESULTS AND DISCUSSION
In this session we carried out the simulation of model (2), examining the effect of treating infectious individuals on different classes using Maple software. Table 1 contains all the values used in simulations.   Figure 2 is showing the effect of varying the treatment rate on Infectious individual, increasing treatment rate leads to reduction of Infectious individuals in a society. Epidemiologically, this will reduced the disease burden in a society. Figure 3 is showing the effect of varying a treatment rates on First line treatment (those that defaulted in treatment which will lead to antidrug resistant), increasing the treatments rates which will leads to fewer case of First line treatment. Figure 4 shows the effect of varying the treatment rates on Second line treatment (those that defaulted in treatment for the second time which will lead to antidrug resistant), increasing the treatment rates leads to a small number of such cases in a society that is the cases of second line treatment. Figure 5 shows the effect of varying treatment rates on recovered individuals, increasing the treatment rates which will leads to the recovered individual as years goes by.

Conclusion:
In this paper, a mathematical model of a tuberculosis transmission dynamics incorporating first and second line treatment was formulated and analyzed in order to gain better understanding of the disease dynamics. The DFE was shown to be locally asymptotically stable when the C R is less than one and the EEP was also shown to be locally asymptotically stable when the C R is greater than one. Numerical simulation shows that treating Infectious Individuals can leads to drastic reduction in second line treatment cases in a society which will also leads to reduction of Tuberculosis (TB) burden in a society.