Spread and Control of Multi-drug Resistance Tuberculosis and Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis

In this work we considered nonlinear dynamical system to study the dynamics of two-strain Tuberculosis epidemic in Ethiopia. We proved that the solution of the considered dynamical system is positive and bounded. We found that the considered dynamical system has disease free and endemic equilibrium points. We proved that the local and global stability of disease free equilibrium point and endemic equilibrium point. We found the effective reproduction number of the dynamical system. Also, the effective reproduction number of the dynamical system which experience drug sensitive strain and the effective reproduction number of the dynamical system which experience multi drug resistance strain. Using real data collected from different health sectors from Ethiopia we found that the numerical value of the effective reproduction number of the drug sensitive tuberculosis is 1.03 and the effective reproduction number of the drug resistance tuberculosis is 4.78 and the effective reproduction number of the dynamical system max{1.03, 4.78}=4.78. So that MDR strain is spreads strongly than DS strain. Numerical simulation is also done to illustrate the influence of different parameters on the effective reproduction number. Using sensitive analysis we identify the most influential parameter to change the behavior of the solution of the considered dynamical system is the number of effective contacts of susceptible or vaccinated individuals make with an infectious individual.


Introduction
Tuberculosis is among the most ancient diseases. German Microbiologist Robert Koch discovered the causative organism Mycobacterium tuberculosis on 24 th March 1882. World Health Organization (WHO) declared tuberculosis as global epidemic in 1993 [1,2]. The lifetime risk of TB reactivation for a person with documented Latent Tuberculosis Infection (LTBI) is estimated to be 5%-10%, with the majority developing TB disease within the first five years after initial infection the risk of developing TB disease following infection depends on several factors [1,3]. There is a huge TB-latent human; this increased its average probability of re-activation due to the emergence and growth HIV and TB drug-resistant strains [1]. One of the biggest health challenges facing the world is tied in to the dramatic increases in the levels of drug resistance TB, particular in hospital settings [1,4]. In 2016, the World Health Organization (WHO) reports roughly 9.4 million new cases (incidence) per year, an active-TB prevalence of 14 million, and 1.6 to 1.9 million deaths per year, a number that includes 400,000 deaths coming from HIV positive individuals each year. Most active-TB cases are concentrated in South East Asia, African and Western Pacific regions [1,5]. In Ethiopia there were in average of 177 TB cases per 100,000 TB in 2016 [5,6,7,8].
Currently, WHO recommends that, the countries use three major categories of health interventions for TB prevention: treatment of LTBI; prevention of transmission of Mycobacterium tuberculosis through infection control; and vaccination of children with the Bacille Calmette-Guérin (BCG) vaccine. In 2016, 154 countries reported providing BCG vaccination as a standard part of these Programmes, of which 111 reported coverage above 90% [5]. Results of field trials of the BCG vaccine have differed widely, some indicating protection rates as high as 70% to 80%, others indicating the vaccine was completely ineffective in preventing TB [4].
Drug Susceptibility Sesting (DST) is very important to provide information about which drugs a person is resistant to. Treatment of tuberculosis disease is not simple and Drug Susceptible Tuberculosis (DS-TB) requires a multiple drug regimen taken for at least six months. But the treatment will only be successful if the drugs are taken exactly as required for the entire length of time [5]. The currently recommended treatment for cases of drug-susceptible TB must be faithfully carried out over 6-9 months regimen of four first-line drugs: isoniazid, rifampicin, ethambutol and pyrazinamide and is a source of concern due the fact that a number of TB-active individuals do not complete treatment giving rise to the emergence of drug resistance TB strains [1,5]. Treatment for rifampicin-resistant TB (RR-TB) and multidrugresistant tuberculosis (MDR-TB) is longer, and requires more expensive and more toxic drugs [5].
Multi-drug-resistant (MDR) tuberculosis is a form of tuberculosis caused by bacteria that do not respond to, at least, isoniazid and rifampicin, which are the two most powerful, standard anti-tuberculosis drugs [3,9,10]. According to the world health organization (WHO) global TB report in 2017, it is estimated that there will be 490,000 new cases of Multi-Drug Resistant Tuberculosis (MDR-TB) in 2016, in addition, 110,000 new patients who resistant to rifampicin meet the treatment conditions of multi drug resistance tuberculosis [5,11]. A combination of poor compliance and poor medical supervision or when the anti-TB drugs are mismanaged (incomplete course of treatment) or misused (wrong dose or time length to complete the drugs) can result multi-drug resistance. However some acquire MDR-TB by being infected with a multi-drug resistant strain. MDR-TB is transmitted in the same way as the normal drug sensitive strain [3,11,12]. Drug-resistant TB has a higher mortality rate, among them, multi-drug resistant tuberculosis (MDR-TB) is more prominent, and has become another new serious problem [11]. Multi-drugresistant tuberculosis (MDR-TB) treatment regimens are significantly longer, cause serious side effects and are very expensive. The latest data reported to WHO show a treatment success rate for multi-drug resistant tuberculosis (MDR-TB) of 54%, globally, reflecting high rates of loss to follow-up, unevaluated treatment outcomes and treatment failure [5,12] and TB treatment outcomes in Ethiopia have been assessed only in small and fragmented observational studies [13].
In this work we present a non-linear dynamical system to study the spread and control of the dynamics of tuberculosis in Ethiopia which describes the infectious disease of two strains tuberculosis. The interventions: vaccination, screening and treatments are incorporated in our model. Based on the Ethiopia context we introduced the model assumptions, construct flow chart of the model and develop the corresponding dynamical system. We investigated disease free equilibrium point, endemic equilibrium points and effective reproduction number. We showed that their local and global stability. Numerical simulations of the results are done by real data collected from different health sectors in Ethiopia. Finally we suggest our recommendation based on the finding of our work which is MDR strain is spreads strongly than DS strain.

The Initial Model
In the work [14], the total population ( ) was divided in to eight classes: Susceptible S (t),; Vaccinated V (t), early stage with high risk of developing active tuberculosis ( ) and Later (Long) stage with low risk of developing active disease ( ), individuals who screened and treating at early latent stage tuberculosis ( ) , Infectious individuals with tuberculosis ( ) , treating infectious ( ) and Recovered individuals ( ), recruitment of the population Λ with the proportions of which are vaccinated to protect them against tuberculosis infection and the remaining proportion are susceptible, force of infection rate = and is the probability that an individual is infected by one infectious individual, and c is the contact rate. The proportion of class have got a chance of screened. The proportion and (1 − ) of individuals of the early latent/exposed individuals for tuberculosis who do not get chance for screened will go to and respectively at the rate . Individual leaves class at the rate in which, the proportion goes to class ; the proportion (1 − ) recovers naturally and enter to recovered class R. The proportion of individuals in class goes for treatment in and the remaining proportion (1 − ) enters to class at the rate . Individuals leave the screened class and treating class at the rates , and ! respectively, and go to recovered class. " is the reduction in susceptibility of recovered individual. Natural death at the rate µ while infectious individuals in are die due to tuberculosis diseases at the rate $.
The corresponding dynamical system of the figure 1 is With the total population at a given time t is . ) , ) ) ) ) ) .
We extend the model in [26] by introducing additional assumptions as follows:

Model Assumptions
In this study, we introduce a deterministic TB model by disaggregating the mycobacterium tuberculosis in to two strains (DS-TB, MDR-TB). The total population is divided in to ten disjoint classes depending on the epidemiological status of individuals such as: Susceptible .
, who have never exposed to any strain of the Mycobacterium tuberculosis, Vaccinated , , who have taken BCG vaccine against mycobacterium tuberculosis, an early stage infected with high risk of developing active drug sensitive tuberculosisand Later (Long) stage infected with low risk of developing active drug sensitive tuberculosis -, Infectious individuals with drug sensitive tuberculosis -, Latently infected individuals with multi-drug resistant tuberculosis 4 , Infectious individuals with multi-drug resistant tuberculosis , Screened Early latently infected with drug sensitive tuberculosis -, Screened latently infected with multi-drug resistant tuberculosis and Recovered individuals . Assume that individuals are recruited into the population by a constant rate Λ with the proportions of which are vaccinated to protect them against tuberculosis infection. Furthermore, that the vaccine has a waning effect over time (after a time 5 6 vaccinated individuals become susceptible again) and reduces due to expiration of duration of vaccine efficacy. We assume that vaccinated individuals may infect with the rate of inefficacy of vaccine (7 [0, 1]. Susceptible population increases due to the coming in of new births not vaccinated against the infection and those who were vaccinated but lose their immunity. When some susceptible individuals come into contact with infectious individuals, they get infected and progress to latently infected classes of drug susceptible and multi-drug resistant tuberculosis at a force of infection ratesand respectively where 8 8 9 , ; <, = and 8 is the probability that an individual is infected by one infectious individual, and c is the number of effective contacts.
The proportion of the high risk latently drug susceptible TB infected developed active TB and the remaining proportion enters to long latent with drug sensitive TB at the rate . The proportion and 1 of individuals of the early latent/exposed individuals for drug susceptible tuberculosis who do not get chance for screened will go toand respectively at the rate . Thus, the proportion , 1 and 1 1 of individuals in the classis transferred to classes -, -andrespectively at a rate . Individual leaves classat the rate in which, the proportion > goes to classand; the remaining proportion 1 > recovers naturally and enter to recovered class R.
Individuals inand, -can also be infected by MDR TB (primary infection) if there is effective contact with individuals in class. Individuals leaveclass at the ratesthat the proportion of individuals in infectious classes of drug susceptible tuberculosis progress to the recovered class while the remaining 1 proportion of individuals with active drug sensitive TB may develop MDR TB because of improper treatment.
The proportion ? of the latently infected multi-drug resist tuberculosis are screened for treatment and the remaining proportion developed active drug resist tuberculosis and leaves E class at the rate of . Individual in class recovers at the rate and goes to R class. Individuals leave the screened classesand at the rates , and ! respectively, and go to recovered class, where @ !.
Due to the nature of the disease, the infection will only kill individuals whose TB progresses to the infectious stage. Moreover, individuals in the recovered class are temporarily recovered. Soon they revert back to the latently infected classesand 4, after been re-infected by either drug sensitive class and multi-drug resistant strain at the rate "and " respectively, where " is the reduction in susceptibility due to prior endogenous infection of tuberculosis. We assume that each class conforms to natural death at the rate + while infectious individuals inand are die due to TB diseases at the rate $ -and $ respectively. State variables and parameters in the dynamical system listed in the following table. The portion ofenter in to -?
The portion of 4 enter in to 8 ; <, = The recovery rate infectious individuals (< =DS strain, = =MDR strain). " Acquired immunity due to previous treatment.
Based on the above assumptions we do have the following flow chart: Based on the assumptions and the above flow chart we develop the following nonlinear dynamical system.

Positivity of Solutions of the Dynamical System (9) -(18)
Theorem-1: Let the initial data for the model (9) Therefore all of the state variables of the dynamical system (9)- (18) are positive for all t > 0, given any positive initial conditions.

Positively Invariant
Theorem-2: The dynamical system (9)   That is the set Ω is positively invariant.

Effective Reproduction Number ' -˜˜
We compute the effective reproduction number ™šš using next generation operator method. In the dynamical system (9) -(18) the rate of appearance of new infections ℱ and the transfer rate of individuals oe at the disease free equilibrium point , 0, 0, 0, 0, 0, 0, 0, 0 are given as: And the inverse of the matrix oe is: , , and ¥ ad = (5u®)- The spectral radius of ℱ, u5 is the required effective reproduction number ™šš and obtained as: .
Proof: We apply a matrix-theoretic method using the Perron eigenvector to prove the global stability of the disease-free equilibrium as in [15]. The dynamical system (9)-(18), the drug sensitive TB disease compartment of is R 5 = ( -, -, -) 7ℝ` Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis and non-disease (drug sensitive TB) compartment Ì 5 7ℝ f .

The Drug Resistant TB Strain only Equilibrium Point, ‡ ý
This equilibrium solution is obtained by setting -= 0 in equations (9)

Local stability of the Drug Resistant TB Strain only Equilibrium Point, ‡ ý
Now we apply the Gershgorin circle theorem, [16]

Endemic Equilibrium Point where Both TB Strains Co-exist, ‡
The endemic equilibrium where both TB strains co-exist is given as  Proof: The Jacobian matrix of the dynamical system (9)-(18) at endemic equilibrium where both TB strains co-exist 4` is given by:

Local Stability of Endemic Equilibrium Point where Both TB Strains Co-exist, ‡
Where, Drug-sensitive Tuberculosis in Ethiopia: A Mathematical Model Analysis The characteristic equation will be: Now we apply the Gershgorin circle theorem [16] to determine the sign of the eigenvalues of the characteristic equation |·(4`) − | = 0. The matrix ·(4`) is a strictly column diagonally dominant matrix. And also all diagonal elements of ·(4`) are negative. Hence, all eigenvalues of ·(4`) has negative real part if ™šš (DS) > 1 , ™šš (MDR) > 1 and * < å;x X n(ol% A )

Numerical Simulations
We perform some numerical experimentation on the tuberculosis model (9) We assumed that more than half of the population (62%) belongs to susceptible class .(0) = 62355690 and that a big percentage about 33% is infected with TB in latent stage that is -(0) = 3,000000, and -(0) = 30000000, . This is justified from the fact that "about one third of the world's population has latent TB", as it is indicated from the webpage of the World Health Organization (WHO, 2017).

Numerical Simulation for the Reproduction Number
We discussed on the relation between effective reproduction number and the parameters involved in it. Now we consider the parameters that involve in both ™šš (DS) and ™šš (MDR) and discuss on their impact on the transmission of DS-TB and/or MDR-TB strains. Here five parameters are involve in common for both effective reproduction numbers ™šš (DS and ™šš MDR .
Let us consider the parameter, the number of effective contacts c as a variable and keeping all other parameters constant and written the reproduction numbers as a function of c, ™šš          when @ 2.8and ™šš MDR @ 1 when q 2.8. In figure   10 the curve ™šš MDR ? 5.1225 1 ? and the line ™šš MDR 1 intersect at ? 0.8, ™šš MDR q 1 when ? @ 0.8 and ™šš MDR @ 1 when ? q 0.8.

Sensitivity Analysis
We apply the normalized forward sensitivity index of the effective reproduction number ™šš to a parameter is the ratio of the relative change in the variable to the relative change in the parameter [26]. If

Discussion
In this work we considered non-linear dynamical system to study the dynamics of a two strain Tuberculosis disease. The effective reproduction number is:  Figure 4 shows that both ™šš (DS) > 1 and ™šš (MDR) > 1 for every values of (, therefore both strains of the TB disease spread in the society what ever the value of ( is. Ofcourse the transmision of MDR-TB is higher than DS-TB.
In figure 5 shows that ™šš (DS) < 1 when * < 0.038 and ™šš (DS) > 1 when * > 0.038. But ™šš (MDR) > 1 for all values of * . This implies the MDR-TB spreads in the community for every value of * . And for the value of * < 0.038 the DS-TB does not spread in the society. That is, if * < 0.038 the only MDRTB spreads in the community and boths DSTB and MDRTB spreads in the community when * > 0.038 . From figure 6 we observe that ™šš (DS) < 1 when > 0.8 and ™šš (DS) > 1 when < 0.8 while, ™šš (MDR) > 1 for all values of . This shows that the MDR-TB spreads in the society for all values of , but DS-TB spreads for < 0.8. Implies both strain spreads in the community if < 0.8 and only MDRTB spreads in the community if > 0.8 . This indicates that giving BCG vaccine has no significant impact in the control of MDRTB; however we can reduce DSTB by BCG vaccine. From figures (3)(4)(5)(6)(7)(8)(9)(10) and the sensitivity index of effective reproduction number (Table 3) we observe that the parameters contact rate c, the rate of inefficacy of vaccine individuals (, the rate of vaccine waning *, the probability of transmissionand , the rate of progression of individuals from early latently infected with drug sensitive TB , the progression rate from Long latently infected DS-TB strain , the portion ofenter in to -, > and the progression rate from latency MDR-TB have positive contribution in the transmission of TB disease. While, the proportions new born vaccinated , natural death rate +, the proportion of individuals who do not get chance for screened atand will go toclass , the Proportion of latently infected drug sensitive TB at early stage for treatment , the recovery rates infectious individualsand , the induced death rates $ -and $ ; and of the portion of 4 enter in to , ? have negative impact on the transmission of TB disease. From table 3 for the parameters of which the sensitivity index of ™šš (DS) and ™šš (MDR) has positive sign the effective reproduction number increase as those parameters increase and vise verse, while for those parameters of which sensitivity index of ™šš (DS) or ™šš (MDR) has negative sign then the effective reproduction number increase as the parameters decrease and vise versa. The number of effective contact of susceptible or vaccinated individual with an infectious individual of both strains c, the probability of transmission followed by the recovery rates infectious individuals are the most influential parameters in the spread and control of tuberculosis disease, this is because of that magnitude of the sensitivity indices Π { 3 êëë , Π A 3 êëë , Π « A 3 êëë , Π 0 3 êëë are, Π « 0 3 êëë are maximum compared to others. The result of this study indicates that reducing the number of effective contact and increasing recovery rate have great role to control tuberculosis disease.

Conclusion
In this study we have presented and analyzed the two strain TB model with interventions: vaccination of newly born babies, screening of latently infected and treatments of infectious individuals for both strains of tuberculosis (drug sensitive and multi-drug resistance tuberculosis). We found that ™šš (DS) = - ¶-ol(6l(5u-)o)

6lo
‰¦ §(5u¨)©ªl¦(©lo)(5u §)(5u¨)" the effective reproduction numbers of drug sensitive and multi-drug resistance tuberculosis respectively. And, thus ™šš = åÂR° ™šš (DS), ™šš (MDR)µ is the effective reproduction number of the system (9)- (18). We have discussed on the existence of disease free equilibrium point, endemic equilibrium (drug-sensitive TB only endemic equilibrium, drug-resistance TB only endemic equilibrium and endemic equilibrium when both strains exist) points and presented the conditions that the local and global stability of those equilibrium points. We evaluated the numerical value of the reproduction numbers. Consequently, ™šš (DS) = 1.03 and ™šš (MDR) = 4.78, which show that the disease of both strain tuberculosis spread in the community and MDRTB spreads vastly in the society. The sensitivity analysis shows that the number of effective contact of susceptible or vaccinated individual with an infectious individual of both strains is the most influential parameter to change the reproduction number respectively.

Recommendation
In order to decrease the spread of both strains in the society, we recommend that the number of contact of susceptible individuals, c should be less than two. The second influential parameter to reduce the transmission of tuberculosis in Ethiopia is the recovery rate infectious individuals, . In order to control the spreads of MDR-TB it needs to raise the value of over 2.8. The result of this study indicates that reducing the number of contact of susceptible or vaccinated individuals with an infectious and increasing recovery rate have great role to control tuberculosis disease.