MATHEMATICAL MODELLING OF PRIMARY INFECTION TUBERCULOSIS WITH ASTHMA AS A SECOND INFECTION

ABSTRACT


INTRODUCTION
Tuberculosis infects more than 9 million people and causes the death of 1.5 million people every year worldwide.Treatment of tuberculosis takes at least 6 months, causing drug resistance worldwide to increase and threaten the effectiveness of tuberculosis treatment [1].Tuberculosis is a respiratory disease associated with asthma.Asthma become a common infection in patients with a history of tuberculosis who was successfully treated in the past [2], [3].Asthma is a disease caused by inflammation of the bronchi which causes the bronchi to narrow, so that air flow in the bronchi to the lungs is limited [4].Current asthma treatment including long-term control care and environmental control measures can reduce asthma exacerbations due to airborne allergens [5], [6].
Some mathematical modelling on tuberculosis has been done such as in [7] which purposes optimal control strategies to reduce the number of tuberculosis patient in Philippines.Another researchers in [8] purpose mathematical model which include diagnosis and treatment on their model.Mathematical modelling on asthma currently not an epidemiological models because asthma is not an infectious disease, such as in [9].They present a computational model that describe mechanism in the lug.
Secondary infection mathematical model has been done by [10] but they were taking influenza as a primary infection and Bronchitis as a secondary infection.In their model, they gave treatment for influenzas but didn't put any treatment for the second infection.Based on this, we will do a mathematical modelling of the tuberculosis transmission with asthma as a secondary infection with each infection undergoing a different treatment.Next, an analysis of the equilibrium point and its local stability will be carried out on the mathematical model that has been formed.As a representation, a numerical simulation will be carried out using Maple software.Maple is a computer-based mathematical application for analytical and numerical mathematical calculations [11].

RESEARCH METHODS
To conduct this research first we collect the medical literature about tuberculosis and asthma.We also look up the mathematical model of secondary infection.Base on those literatures we build some assumption that proper to the medical phenomena then construct the mathematical models.After the mathematical models formed, we analyse the equilibrium points, the basic reproduction number, and their local stability.Finally, we do some simulation by choosing some parameters values.Some are based on data, but the others are by assumption.We describe the methods on Figure 1.

Mathematical Model
An infectious disease in a population will divide the population into several classes [11].Mathematical modelling in this study was divided into 6 classes, which are healthy individuals are susceptible to tuberculosis infection (), individuals infected with tuberculosis ( 1 ), individuals infected with tuberculosis undergoing treatment ( 1 ), individuals infected with asthma ( 2 ), individuals infected with asthma undergoing treatment ( 2 ), and individuals recovering from tuberculosis infection ().The parameters used in the model are presented in the following Table 1.The transmission model for tuberculosis disease with asthma as a secondary infection is illustrated in the modelling diagram is show in Figure 2 below The equation of the mathematical model is obtained as follows 1), the equilibrium point can be found by making zeros on the right side of the equation.As a result, two equilibrium points are obtained, namely 1. Disease-free equilibrium point ( 0 ) = ( * ,  1 The development of transmission of tuberculosis disease with asthma as a secondary infection is determined by the basic reproduction number ( 0 ) by finding the largest positive eigen values with the next generation matrix involving compartments that cause infection [12], namely  1 ,  1 ,  2 , and  2 .By some computation obtained the value of  0 =  ( 1 ++ 1 ) . The existence of the equilibrium points given in Theorem 1 as follows.Theorem 1.Given  0 =  ( 1 ++ 1 ) .
1.If  0 ≤ 1, then the system of Equation (1) has one equilibrium point, which is the disease-free equilibrium point ( 0 ).

2.
If  0 > 1, then the system of Equation (1) has two equilibrium points, which are the disease-free equilibrium point ( 0 ) and the endemic equilibrium point ( 1 ).
Proof.To find the equilibrium we solve the system equal to zero, so we will have From the second equation we got And we can conclude that While  1 = 0 it is easy to substitute and get the free disease equilibrium point  0 = ( * ,  1 * ,  1 * ,  2 * ,  2 * ,  * ) = ( Λ  , 0,0,0,0,0).
. Substituting this  into system (1) then we will get the second equilibrium point  1 = ( * * ,  1 * * ,  1 * * ,  2 * * ,  2 * * ,  * * ) as mentioned above.But we need a positive equilibrium point, and by define  0 =  ( 1 ++ 1 ) the existence of the second equilibrium point will be guaranteed while  0 > 1 .∎Then stability analysis will be carried out using the linearization method.So that the eigen values of the Jacobian matrix are obtained.The Jacobian matrix is as follows.
Next, substitute the value of the equilibrium point into the Jacobian matrix and look for the eigen values from matrix () using the formula ( − ()) = 0. Equality this called equality characteristics from () [13], [14] , [8], [9].The results were analyzed using the Routh-Hurwitz criteria.Theorem 2 was obtained as follows.
Theorem 2. Given  0 by the system of Equation (1).Based on  0 this obtained 1.The disease-free equilibrium point is ( 0 ) locally asymptotically stable if R 0 < 1.
So that we will have six eigen value which are Because all parameters are non-negative we only need to check  5 .From the previous result, we got  * =  0 ( 1 ++ 1 ) , and substitute it to  5 , we will have Here we can conclude that if  0 < 1 then all of the eigen values are negative, or in other word  0 is locally asymptotically stable.
Using Ruth-Hurwitz criterion can be seen that if  0 > 1 then all real part of the eigen values are negative, or in other word  1 is locally asymptotically stable.∎

Simulation Analysis at the Endemic Equilibrium Point(𝑬 𝟏 )
While we set parameter value  1 = 0,3 then obtained the basic reproduction number R 0 = 1,20438976.Because  0 > 1there will be endemic conditions.Obtained values  0 = ( * * ,  1 * * ,  1 * * ,  2 * * ,  2 * * ,  * * ) = (1.245444,0.122633866, 0.02830012062, 0.01540700398, 0.05016006932, 0.02867745556).The graph is shown in Figure 3 below Based on Figure 3 above, it is found that healthy individuals who are susceptible to tuberculosis infection () have decreased in their initial conditions.Then the size of susceptible individuals increases to a certain  and there is no change or constant at a point 1.245444 at a certain .Individuals infected with tuberculosis  1 () experienced an increase in initial conditions.Then up to a certain point when the tindividual infected with tuberculosis decreased and there was no change or constant at the point 0.12263 at a certain .Individuals infected with tuberculosis who underwent tuberculosis treatment  1 () experienced an increase in baseline conditions.Then up to a certain point when tindividuals infected with tuberculosis undergoing treatment experience a decrease and there is no change or constant at a the point 0.02830 at a certain .Individuals infected with asthma  2 () have an increase in initial conditions.Then up to a certain point when tan individual is infected with asthma there is a decrease and there is no change or constant at the point 0.01541 at a certain .Individuals infected with asthma undergoing asthma treatment  2 () experienced an increase in initial conditions.Then up to a certain point when tan individual is infected with tuberculosis, there is a decrease and there is no change or constant at the point of 0.05106 at a certain .Individuals recovering from tuberculosis () experienced an increase in initial conditions.Then up to a certain point when the tindividual recovers from tuberculosis, there is a decrease and there is no change or constant at the point 0.02868 at a certain .

Changing on Parameter Value 𝜸 𝟏
To determine the effect of tuberculosis treatment rate, we varying the value of the parameters γ 1 presented in Table 3 below: (1.5, 0, 0, 0, 0, 0) 0.7 0.7333371141 (1.5, 0, 0, 0, 0, 0) 0.9 0.6133855447 (1.5, 0, 0, 0, 0, 0) Obtained a graph as shown in Figure 4 below.Based on Figure 4 (a), and Figure 4 (b) it can be seen that the lower the rate of tuberculosis treatment, the higher the number of individuals in the () and  1 ().While in Figure 4 (c) it can be seen that the lower the rate of tuberculosis treatment, the lower the number of individuals in the sub-population  1 ().The subpopulation  2 (),  2 (), and () has the same graph as in Figure 4 (c).It can be seen that the lower the rate of tuberculosis treatment results in the lower number of individuals infected with asthma, asthma-infected individuals undergoing asthma treatment, and individuals recovering from tuberculosis.The effect of tuberculosis treatment rate on individuals in each population is the higher the effect of tuberculosis treatment, the faster the development of individuals in each population will reach a stable point in time  increased rate of tuberculosis treatment.

Changing on Parameter Value 𝒅 𝟏
To find out the effect of the death rate of an individual infected with tuberculosis, it will be done by varying the value of the parameter value  1 presented in Table 4 below:   it can be seen that the lower the death rate of individuals infected with tuberculosis, the higher the number of individuals in each population.Population  2 (),  2 (), and () have the same graph as in Figure 5 (c).It can be seen that the lower the death rate of individuals infected with tuberculosis, the higher the number of individuals infected with asthma, asthmainfected individuals undergoing asthma treatment, and individuals recovering from tuberculosis.The effect of the death of an individual infected with tuberculosis on an individual in each population is the higher the effect of the death of an individual infected with tuberculosis, the faster the development of individuals in each population will reach a stable point in time .So, to achieve a disease-free condition or a condition where tuberculosis disease with asthma as a secondary infection will disappear, it is necessary to increase the proportion of deaths of individuals infected with tuberculosis.

CONCLUSIONS
Through this research, it is found that the mathematical model of the spread of tuberculosis with asthma as a secondary infection has two equilibrium points, which are the disease-free equilibrium point ( 0 ) and the endemic equilibrium point ( 1 ).The spread of tuberculosis with asthma as a secondary infection is indicated by the basic reproduction number ( 0 ) that influenced by recruitment rate, infection rate, the natural death rate, tuberculosis treatment rate and the death rate of tuberculosis patient.The disease-free equilibrium point is ( 0 ) locally asymptotically stable if  0 < 1.Meanwhile, the endemic equilibrium point is ( 1 ) locally asymptotically stable if  0 > 1.Based on the numerical simulations performed, it was found that the change in the proportion value  1 had  1 a significant effect on the basic reproduction value ( 0 ) and the equilibrium point obtained.By giving a value  1 and  1 the higher it will decrease the value  0 and the development of individuals in each population is getting faster towards a stable point.

Figure 2 .
Figure 2. Simulation of Disease-Free Equilibrium Point

Figure 4 .
Figure 4. Simulation of the effect of tuberculosis treatment rate for each sub-population, (a) susceptible subpopulation, (b) tuberculosis patients sub-population, (c) tuberculosis patients undergoing treatment subpopulation.

Figure 5 .
Figure 5. Simulation by Varying the Death Ratescaused of Tuberculosis on (a)susceptible sub-population, (b)tuberculosis patients sub-population, (c)tuberculosis patients undergoing treatment sub-population.

Table 2 . Parameter Value
below