Mathematical analysis and numerical simulation of co-infection of TB-HIV

Abstract In this article, we consider the model having co-infection of TB and HIV for numerical analysis and modelling. For this purpose, used non-standard finite difference scheme with Mickens approach ϕ(h)=h+O(h2) rather than h to control this disease. Furthermore, we analyse the well-posedness and stability of the system. Also, check the sensitivity analysis of the verse condition R0t and R0h as well as analyzed qualitatively. Finally represents the numerical solution of the model which supports the theoretical results.


Introduction
Human immunodeficiency virus is a global health problem which is a lentic virus and causes HIV infection.It is one of the most studied infectious diseases in the world.If an infected individual does not use medicines against HIV, then the total survival time of that individual is 9-11 years (Lawn & Zumla, 2011;Tahir, Inayat, & Zaman, 2019).After discovery of the virus, the models for HIV infection was discovered at late 1980s.In the 1900, SIR and SIS model by Kendrick and McCormack as well as bread-and-butter models of mathematical epidemiologists were discovered and this model for HIV was inspired from these models.These models were used to investigate infections between persons and population (Anderson & May, 1991).Viruses can investigate in person's body by 'viral dynamics' model (Nowak & May, 2000).These models have since been used to describe many other human virus infections, such as Hepatitis B and C, influenza, dengue and herpes simplex virus (Alison, 2018).A system of nonlinear ordinary differential equation is used to made viral dynamic model which further used to investigate clinical parameters of HIV patient (Huang, Wu, & Acosta, 2010).
The outbreak of HIV was considered to wellbeing debacle at current times.HIV considered to most exasperating disease to different parts of the world, this made scientists to study about infection and its affects on human body during second pandemic of disease.This decrease in overspread of disease.But spread of the disease remains continue and treatment remains inaccessible to the mind-boggling standard of the individuals who require it (Sadegh & Miehran, 2015).Accordingly, various mathematical modelling have been arranged in the past time frames for HIV/AIDS spread elements.Global stability of equilibrium point for scientific models of HIV/AIDS spread elements have been pondered by various creators.It is normal that the HIV scourge feasts both through horizontal and vertical transmission (Silva & Torres, 2018).
In this article, brief introduction of the model is given in Section 2, also discussed the stability, wellposedness and qualitative analysis in Section 3. Furthermore, the system is analyzed by local asymptotically stable.Also, check the sensitivity analysis of the verse condition R t 0 and R h 0 as well as analyzed qualitatively.We proposed the NSFD scheme for co-infection of TB-HIV model.Numerical simulations are carried out to support the analytical results in Sections 4 and 5.

Mathematical model
In this analysis of co-infection of HIV and TB, we suppose that total population is homogeneous and closed.Let N denotes the total number of population.Next, we divide this N into six different classes and parameters used in model, where the state space parameters and associate parameters and their description is presented in Tables 1 and 2 respectively.
The mathematical model of co-infection HIV-TB can be represented as follows, along with initial condition Sð0Þ > 0, I t ð0Þ > 0, I h ð0Þ > 0, A h ð0Þ > 0, A ht ð0Þ > 0: Here S(t) is susceptible human, I t ðtÞ is infectious human with TB, I h ðtÞ is infectious human with HIV, I ht ðtÞ is I ¼ infectious human with co-infection of HIV-TB, A h ðtÞ is infectious human with only HIV and susceptible to TB, A ht ðtÞ is Infectious human with both HIV and TB in time (Fatmawati, 2016).While area of biological interest of model ( 1) is 3. Well-Posedness of the model Theorem 3.1.Assumed that the model (1) enclosing all possible results with respect to non-negative initial solution then it is non-negative over the whole time.
Proof.Assume that we have non-native initial solution of the model (1), that is Sð0Þ !0, I h ð0Þ !0, I t ð0Þ !0, I ht ð0Þ !0, A h ð0Þ !0, A ht ð0Þ !0 (2) By model (1), the principal condition is evaluated as follows, The solution of S can be evaluated by following expression, where B ¼ Ð t 0 ÀBðsÞds and AðuÞ ¼ Ð u 0 ÀBðwÞdw: Therefore, Sð0Þ !0 for all time t !0: The non-negativity of rest parameters, the model ( 1) can be demonstrated as follows, The above system (5) can be demonstrated in the form of matrix as follows, and It is clear that the matrix M is a Matzler matrix (a matrix whose all diagonal entries are strictly negative and non-diagonal entries are non-negative).From this fact, we investigate R 5 þ is invariant along with the stream of model ( 6), which completes the proof.w Theorem 3.2.Suppose that the underlying circumstances for considered model (1), the following conditions holds, Here, Then, when the zeros of considered model exists on an interval J, it holds for subsequent a priori limits Proof.The result can be proved by three cases, Case: 1 Subsequently, I t ðtÞ !0 we have the following transformation by using first and second equation of model ( 1), Since, I h !0, I t !0: So, we have, Applying Gronwall inequality, we get This implies that, Case: 2 Subsequently, I h ðtÞ !0 we have the following transformation by using first and third equation of model ( 1), Since, I h !0, I t !0, I ht !0: So, we have, Applying Gronwall inequality, we get This implies that, Case: 3 Subsequently, I ht ðtÞ !0 we have the following transformation by using first and second equation of model ( 1), Since, I h !0, I t !0, I ht !0: So, we have, Applying Gronwall inequality, we get This implies that, The boundedness of A ht is proved similarly, which completes the proof.w

Qualitative analysis
In this section we present the stability analysis of considered model presented in Equation (1).Furthermore, we investigate the model ( 1) is locally stable at disease free and endemic equilibrium points.The disease free equilibrium points of model ( 1) is computed as follows, Let E 0 ¼ ðS 0 , I 0 t , I 0 h , I 0 ht , A 0 h , A 0 ht Þ be represent the disease free equilibrium points then by solving Equation (1) in a well-known mathematical software MATHEMATICA, the solution is computed as follows, Next, we find existence and evaluate the equilibrium points of considered model (1).Assume that EðS, I t , I h , I ht , A h , A ht Þ be an equilibrium point such that, we have following TB-HIV endemic equilibrium point Next we substitute the value of S, I h , I ht in 3rd equation of system (1), we have the term I t satisfy the following quadratic equation, where ARAB JOURNAL OF BASIC AND APPLIED SCIENCES where are represented the reproductive number.
Theorem 3.3.For disease free equilibrium point E 0 if real part of eigenvalues of Jacobian matrix J of considered system (1), then the system (1) is locally asymptotically stable otherwise it is unstable.
Proof.Let J be a Jacobian matrix of system (1) is and is computed by following expression, where The eigenvalue of matrix J 0 is computed by finding the solution of detðJ 0 À kIÞ ¼ 0, where I is an 6 Â 6 identity matrix.Take jJ 0 À kIj ¼ 0, Thus the eigenvalues of J 0 are follows, All of the eigenvalues of considered Jacobian matrix J are strictly negative, therefore the system is locally asymptotically stable at disease free equilibrium.w

Sensitivity analysis
The sensitivity analysis of reproduction number R t 0 , with respect to each parameters is, From above sensitivity analysis, we analyze that the R t 0 become sensitive by a very slight variation in parameters.The reproduction number R t 0 increase with increment of K, b t and decrease with decrease with parameters d, a 1 , l 1 : The sensitivity of reproduction number R h 0 , with respect to each parameters is,

Proposed NSFD scheme
The fundamental hypothesis of nonstandard finite difference (NSFD) displaying was set up by Micken's with a portion of his articles from the eighties.An essential reference is the book (Mickens, 2000) which he altered, and where he gave the first section (Mickens, 2000).The discretized transformation of model ( 1) in the form of NSFD scheme using first order forward method is discriminated as follows, This implies that Also, we have This implies, Also, we have Also, we have This implies, Also, we have   Also, we have

Results and discussion
In this section, we present some numerical simulations of susceptible human S, infectious human with TB I t , infectious human with HIV I h , infectious human with both of HIV-TB I ht , infectious human with only HIV and susceptible to TB A h and infectious human with both HIV and TB A ht for both case of R h 0 < 1, R t 0 < 1 and R h 0 > 1, R t 0 > 1 using values of associated parameters taken in Tables 1 and 2

Conclusion
In this article, we present a mathematical model for the co-infection of TB and HIV transmission along with variable aggregate population size.The wellposedness of the co-infection TB-HIV model is presented.Furthermore, we analyze the existence of a disease-free and endemic equilibrium for the coinfection of TB-HIV.Also, we present the basic reproduction number R t 0 and R h 0 for Tb and HIV infection respectively.The formulation and sensitivity analysis of fundamental reproduction numbers for Tb and HIV are analyzed.The formulation of the proposed co-infection TB-HIV model in terms of non-standard finite difference schemes is presented.Finally, we present the graphical representation of the solution using the Non-Standard Finite Difference scheme which provides the solution according to our steadystate for both disease-free and endemic equilibrium.

Figure 1 .
Figure 1.Susceptible population S(t) in time t at different step size for DFE.

Figure 2 .
Figure 2. Infected population with TB I t ðtÞ in time t at different step size for DFE.

Figure 3 .
Figure 3. Infected population with HIV only I h ðtÞ in time t at different step size for DFE.
respectively.By analyzing the graphical results assure us to justify the theoretical literature review of stability and instability of co-infection of TB-HIV.The following graphical results are plotted by a well-known

Figure 4 .
Figure 4. Infected population with both TB and HIV I ht ðtÞ in time t at different step size for DFE.

Figure 5 .
Figure 5. Infected with AIDS only and susceptible to TB class A h ðtÞ in time t at different step size for DFE.

Figure 6 .
Figure 6.Infected population with TB and AIDS both class A ht ðtÞ in time t at different step size for DFE.

Figure 7 .
Figure 7. Susceptible population S(t) in time t for EEP at different step size.

Figure 8 .
Figure 8. Infected population with TB I t ðtÞ in time t for EEP at different step size.

Figure 9 .
Figure 9. Infected population with HIV only I h ðtÞ in time t for EEP at different step size.

Figure 10 .
Figure 10.Infected population with both TB and HIV I ht ðtÞ in time t for EEP at different step size.

Figure 11 .
Figure 11.Infected with AIDS only and susceptible to TB class A h ðtÞ in time t for EEP at different step size.

Figure 12 .
Figure 12.Infected population with TB and AIDS both class A ht ðtÞ in time t for EEP at different step size.

Table 1 .
Description of associate parameters with their values for disease free equilibrium point.

Table 2 .
Description of associate parameters with their values for endemic equilibrium point.
Consequently, I h ðtÞ H m : Replacing this in fifth equation of model, we have From above sensitivity analysis, we analyze that the R h 0 become sensitive by a very slight variation in parameters.The reproduction number R h 0 increase with increment of K, b h and decrease with decrease with parameters d, c 1 , l 2 :