## Modelling of HIV-TB Co-infection Transmission Dynamics

**Nita H. Shah**^{1,}, **Jyoti Gupta**^{1}

^{1}Department of Mathematics, Gujarat University, Ahmedabad, Gujarat

### Abstract

In this paper, we have formulated a model for HIV-TB co-infection using differential equations in order to understand the dynamics of disease spread. The model is analysed for all the parameters responsible for the disease spread in order to find the most sensitive parameters out of all. Steady state conditions are derived. A threshold parameter *R*_{0} is defined and is shown that the disease will spread only if its value exceeds 1. Numerical simulation is done for the model using MATLAB which shows the population dynamics in different compartments.

### At a glance: Figures

**Keywords:** mathematical modelling, differential equations, numerical simulation, HIV/AIDS, tuberculosis

*American Journal of Epidemiology and Infectious Disease*, 2014 2 (1),
pp 1-7.

DOI: 10.12691/ajeid-2-1-1

Received November 26, 2013; Revised December 12, 2013; Accepted December 26, 2013

**Copyright**© 2013 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Shah, Nita H., and Jyoti Gupta. "Modelling of HIV-TB Co-infection Transmission Dynamics."
*American Journal of Epidemiology and Infectious Disease*2.1 (2014): 1-7.

- Shah, N. H. , & Gupta, J. (2014). Modelling of HIV-TB Co-infection Transmission Dynamics.
*American Journal of Epidemiology and Infectious Disease*,*2*(1), 1-7.

- Shah, Nita H., and Jyoti Gupta. "Modelling of HIV-TB Co-infection Transmission Dynamics."
*American Journal of Epidemiology and Infectious Disease*2, no. 1 (2014): 1-7.

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### 1. Introduction

Tuberculosis is the most common HIV-related opportunistic infection in world, and caring for patients with both diseases is a major public health challenge. India has about 1.8 million new cases of tuberculosis annually, accounting for a fifth of new cases in the world - a greater number than in any other country. Patients with latent *Mycobacterium tuberculosis *infection are at higher risk for progression into active TB if they are co-infected with HIV. In recent decades, the dramatic spread of the HIV epidemic in sub-Saharan Africa has resulted in notification rates of TB increasing up to 10 - times. The incidence of TB is also increasing in other high HIV prevalence countries where the population with HIV infection and TB overlap. This is the underlying factor that suggests that TB control will not make much head way in HIV prevalent settings unless HIV control is also achieved.

The related spread of two or more infections has always been a cause of concern for human beings. HIV-TB co-infection is the largest cause among them. Tuberculosis (TB) is the most common opportunistic disease affecting HIV positive people and the leading cause of death in patients with AIDS.

As per the World Health Organisation (WHO) estimation, one third of the world’s population is infected with Mycobacterium Tuberculosis. All over the world, approximately 15% of TB patients have HIV co-infection. HIV patients are at higher risk for catching primary TB infection as well as reactivation of latent TB infection. HIV infection primarily affects those components of host immune system responsible for cell mediated immunity. These cells help us fight against various infections. Once they are destroyed our body’s resistance to fight infections goes down. Thus if a person with latent TB infection catches HIV infection, then the host’s immunity reduces resulting in active tuberculosis. Moreover the infection is poorly contained following reactivation, resulting in widespread dissemination causing extra-pulmonary disease.

The interaction between HIV and tuberculosis in patients who are co-infected is bidirectional. HIV infection accelerates the activation of tuberculosis and tuberculosis accelerates the HIV infection developing into AIDS. Also, the relative risk of death and development of other opportunistic infections is higher in HIV-TB co-infected patients as compared to those having only one disease out of two.

Research work is already in progress for understanding the dynamics of this co-epidemic. Krischner modelled the CD4+ cell counts and viral load in the case of two infections together ^{[1]}. Morris formed basic models and derived steady states for them ^{[2]}. Pawlowski *et al* reviewed the available literature in order to highlight immunological events responsible for developing the one infection in the presence of the other ^{[3]}. Escombe *et al* analysed how infectiousness of a co-infected person differs from the one having only TB ^{[4]}. Sharomi *et al* formulated the mathematical model and also included the treatment factor in it ^{[5]}. Baur *et al* examined that how the latent TB patient moves to active TB class if it catches HIV-1 infection too ^{[7]}. Shah *et al* developed a model for tuberculosis to explain the dynamics of disease taking pulmonary and extra-pulmonary TB separately ^{[8]}. Shah *et al* also developed a model to predict future trends of HIV/AIDS ^{[9]}.

### 2. Mathematical Model

The entire population is divided into twelve compartments which are (1) Susceptible (*S*) (i.e. no disease), (2) Latent TB – No HIV(*E*_{1}), (3) Latent TB with HIV(*E*_{2}), (4) Active TB – No HIV(*I*_{1}), (5) Active TB with HIV (*I*_{2}), (6) No TB but HIV (*H*), (7) TB treated – No HIV (*T*_{1}), (8) TB treated with HIV (*T*_{2}), (9) AIDS – No TB (*A*_{1}), (10) AIDS with Latent TB (*A*_{2}), (11) AIDS with Active TB (*A*_{3}), (12) AIDS Treated TB (*A*_{4}).

Here, it is assumed that a person when moves any of AIDS class, does not spread disease any more. The population dynamics among these compartments is shown in Figure 1:

**Fig**

**ure**

**1.**Flow of Population in Various compartments

The state variables and the parameters used in model formulation are as follows:

• *S*: Number of susceptible (i.e. no disease)

• *E*_{1}: Number of persons with latent TB infection and no HIV

• *E*_{2}: Number of persons with latent TB infection and HIV positive

• *H*: Number of persons with HIV infection and no TB

• *I*_{1}: Number of persons with active TB infection and no HIV

• *I*_{2}: Number of persons with active TB infection and HIV positive

• *T*_{1}: Number of persons with treated TB and no HIV

• *T*_{2}: Number of persons with treated TB and HIV positive

• *A*_{1}: Number of persons with AIDS and no TB

• *A*_{2}: Number of persons with AIDS with latent TB infection

• *A*_{3}: Number of persons with AIDS with active TB infection

• *A*_{4}: Number of persons with AIDS with treated TB

• B: Recruitment rate in susceptible class

• *μ*: Natural death rate

• *δ*_{A}: AIDS induced death rate

• *δ*_{T}* *: Tuberculosis induced death rate

• *β*_{1}: Probability of transmission of TB infection from an infective to a susceptible per contact per unit time

• *β*_{2}: Probability of transmission of HIV infection from an infective to a susceptible per contact per unit time

• *c*_{1}: Number of contacts made by a person from active TB class

• *c*_{2}: Number of contacts made by a person from latent TB class

• *c*_{3}: Number of contacts made by a person having only HIV and no TB

• *c*_{4}: Number of contacts made by a person having HIV and TB both

• *α*: Rate with which all type of infective develop AIDS.

• *ν*_{1}: Rate of progression of individuals from the latent TB class to the active class who have only LTB and no HIV.

• *ν*_{2}: Rate of progression of individuals from the LTB class to the active TB class who have LTB and HIV/AIDS both.

• *γ*_{1}: treatment rate of latent TB individuals

• *γ*_{2}: treatment rate of infectious (active TB) individuals.

The model equations are given below:

We name this above set of twelve equations as system (1).

With

So, the feasible region for the system is

Let be the equilibrium point of the model given above.

Since, the recruitment term B can never be zero and population cannot vanish, therefore there is no trivial equilibrium point like

So, let

Then system of equations at this point gives

So, we can see that there is a disease free equilibrium at

Let

Therefore,

where and are column matrices given by

and

The derivatives and at disease free equilibrium point , are partitioned as

where *F* and *V* are 10 × 10 matrices given by

Therefore, basic reproduction number = spectral radius of

### 3. Stability of Disease Free Equilibrium

The Jacobian of model equations at the disease free equilibrium point can be written as

The disease free equilibrium is stable if all the eigen values of the Jacobian matrix of the system under consideration have negative real parts.

Here, trace (*J*) is clearly negative.

Now for det (*J*), on solving we reach the relation

For det(*J*) > 0, we have

or

This shows that the disease free equilibrium is locally asymptotically stable if otherwise unstable.

### 4. Sensitivity Analysis

Sensitivity indices of to all the different parameters tell us that how crucial each parameter is to the disease spread. This helps us choose the right parameter(s) responsible for making the scenario endemic.

The values of different parameters used in model formulation are given in Table 1, ^{[8, 9, 10, 11, 12]}.

[For formula refer A.1.]

We calculate the sensitivity indices for those parameters on which the value of basic reproduction depends. The results are given in Table 2.

These results show that the total population size, new recruitments as susceptible, contact rate with tuberculosis patients and probability of transmission of TB have a constant effect on the scenario. The most important parameters reflected here and need to be addressed are rate of progression of LTB to ATB and their treatment rates. So, if we try to control tuberculosis cases and treat the active TB cases more promptly then we can reduce the level of intensity of this lethal co-epidemic.

### 5. Numerical Simulation

This technique help us foresee the future trends of an epidemic and thus help us be ready for future or take safety measures today for making a better future. Here, we take a sample population of 35000. The results of simulation in different compartments are shown in following Figure 2 - Figure 5:

**Fig**

**ure**

**2.**Simulation of Susceptible, LTB, HIV and LTB with HIV

**Fig**

**ure**

**3.**Simulation of ATB, ATB with HIV, TTB and TTB with HIV

**Fig**

**ure**

**4.**Simulation of AIDS, AIDS with LTB, AIDS with ATB and AIDS with TTB

**Fig**

**ure**

**5.**Population dynamics in various compartments in next 40 years

### 6. Results and Discussion

Here, we formulated a mathematical model using ordinary differential equations for HIV-TB co-infection scenario. We divided the entire population into twelve compartments. Then a relation for basic reproduction number in established. Steady state conditions are derived which show that the disease free equilibrium is locally asymptotically stable only if. Sensitivity analysis results tell us that we need to work more rigorously in order to control this co-epidemic. Numerical simulation is done using MATLAB. Figure 5 shows the trends of population in different compartments in next 40 years.

### Acknowledgement

This research is supported by UGC project scheme #41-138612012(SR).

### Appendix

A.1 The normalised forward sensitivity index of a variable, u, that depends continuously on a parameter, p, is defined as

### References

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[2] | Morris Q., “Analysis of a Co-Epidemic Model”, SIAM, pp 121-133. | ||

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[3] | Pawlowski A. et al, “Tuberculosis and HIV Co-Infection”, PLoS Pathogens, Vol. 8, issue 2, 2012. | ||

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[4] | Escombe A. R. et al, “The infectiousness of tuberculosis patients co-infected with HIV”, PLoS Medicine, Vol. 8, issue 9, 2008. | ||

In article | |||

[5] | Sharomi O. et al, “Mathematical analysis of the transmission dynamics of HIV/TB co-infection in the presence of treatment”, Mathematical Biosciences and Engineering, Vol. 5, No. 1, pp 145-174, 2008. | ||

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[6] | Bauer A. L. et al, “The effect of HIV-1 infection on latent tuberculosis”, Math. Model. Nat. Phenom. Vol. 3, No. 7, pp 229-266, 2008. | ||

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[8] | Shah N. H. and Gupta Jyoti, “Mathematical Modelling of Pulmonary and Extra-pulmonary Tuberculosis”, International Journal of Mathematical Trends and Technology”, Vol 4 (9), 158-162, 2013. | ||

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[9] | Shah N. H. and Gupta Jyoti, “Modelling and Analysis of HIV/AIDS Menace using Differential Equations”, Journal of Advances in Mathematics, Vol 3 (2), 190-200 October, 2013. | ||

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[10] | www.who.int/. | ||

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[11] | http://www.cdc.gov/. | ||

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[12] | UNAIDS Report 2010. | ||

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