SENSITIVITY ANALYSIS OF TREATMENT AND COUNSELING IN A CO INFECTION MODEL OF HIV/AIDS, TUBERCULOSIS AND MALARIA

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1 SENSITIVITY ANALYSIS OF TREATMENT AND COUNSELING IN A CO INFECTION MODEL OF HIV/AIDS, TUBERCULOSIS AND MALARIA Ochieng Ombaka Physical Sciences Department, Chukka University, Kenya ABSTRACT This study presents a co infection deterministic model defined by a system of ordinary differential equations for HIV/AIDS, malaria and tuberculosis. The model is analyzed to investigate the potential impact of counseling and treatment on disease progression by carrying out sensitivity analysis of the reproduction number with respect to counseling and treatment. The sensitivity indices of the reproduction numbers R H, R HM, R HT and R HMT with respect to treatment and counseling for the HIV/AIDS individuals showed that counseling is the most sensitive parameter in controlling the co infections. Keywords: : HIV/AIDS - TB and Malaria, equilibria, stability, bifurcation, sensitivity, counseling, treatment. 1. INTRODUCTION Sensitivity analysis on mathematical disease modeling investigates the potential impact of the model parameters on disease progression by computing the partial derivatives of the reproduction number R0 with respect to the parameters. The basic reproduction number R0 is defined as the average number of secondary infections an infectious individual would cause over his infectious period in an entirely susceptible population. When R o < 1; then an infectious individual is causing, on average, less than one new infection and thus the disease does not invade the population. On the other hand, when R o > 1 then an infectious individual is causing, on average, more than one new infection and thus the disease invades and persist in the population. HIV/AIDS remains one of the leading causes of death in the world with its effects most devastating in sub Saharan Africa. One of the key factors that fuels the high incidence of HIV/AIDS in Sub Saharan Africa is its dual infection with malaria and tuberculosis [16]. Audu et al. [4] investigated the possible impact of co infections of tuberculosis and malaria on the CD4 + cell counts of HIV/AIDS patients and established the following: The healthy control group recorded a median CD4 + cell counts of 789 cells/ μl (789 cells per mm 3 of blood); subjects infected with HIV/AIDS only recorded a median CD4 + cell counts of 386 cell/ μl; subjects co infected with HIV/AIDS and TB recorded a median CD4 + cell counts of 268 cell/ μl; subjects co infected with HIV/AIDS and malaria recorded a median CD4 + cell counts of 211 cell/μl and those co infected with HIV/AIDS, malaria and TB recorded the lowest median CD4 + cell counts of 182 cell/μl. This study explores the joint dynamics of the simultaneous co infections of HIV/AIDS, TB and malaria to nvestigate the potential impact of counseling and treatment on disease progression. 2. MODEL FORMULATION AND DESCRIPTION To study the dynamics of HIV/AIDS, malaria and TB co infection, a deterministic model is formulated described by a system of ordinary differential equations. The model sub-divide the human population into the following epidemiological classes: S H (t) - Susceptible population at time t, I M (t) - Malaria infectives at time t, I H (t) - HIV cases at time t, I A (t) - AIDS cases at time t, I T (t) - TB cases at time t. I HM (t) - Those co infected with malaria and HIV at time t, I AM (t) - Those co infected with malaria and AIDS at time t, I MT (t) - Those co infected with malaria and TB at time t, I HT (t) - Those co infected with HIV and TB at time t, I AT (t) - Those co infected with AIDS and TB at time t, I HMT (t) - Those co infected with HIV, Malaria and TB at time t, I AMT (t) - Those co infected with AIDS, Malaria and TB at time t. The total human population (N H (t)) is therefore denoted by: N H (t) = S H (t) + I M (t) + I H (t) + I A (t) + I T (t) + I HM (t) + I AM (t) + I MT (t) + I HT (t) + I AT (t) + I HMT (t) + I AMT (t). The vector (mosquito) population at time t denoted by N V (t) is sub-divided into the following classes: S V (t) - Vector susceptibles at time t, I V (t) Vector infectives at time t. The total vector population N V (t) is given by N V (t) = S V (t) + I V (t). 78

2 2.1 DEFINITION OF PARAMETERS It is assumed that susceptible humans are recruited into the population at a constant rate either by birth or recovery from malaria and TB. They acquire infection with either HIV/AIDS, malaria or TB and move to the infectious classes. Susceptible mosquitoes are recruited into the mosquito population at a constant rate. They acquire malaria infection following a blood meal feeding on infected malaria humans, becomes infectious and move to the infectious class. The recruitment rate of humans into the susceptible population is denoted by Λ H while that of vectors (mosquitoes) is denoted by Λ V and are both assumed to be constant. The natural death rate of humans is given by d n while that of vectors is given by d v. The death rates due to AIDS, malaria and TB in humans are da, d m and d t respectively. The parameters d am, d mt, d at and d amt account for the combined death rates in the I AM, I MT, I AT and I AMT classes respectively. The parameters r m and r t are the recovery rates from malaria and TB respectively due to effective treatment. It is assumed that the recovered individuals do not acquire temporary immunity to either or both diseases thus become susceptible again. The model assumes that susceptible humans cannot simultaneously get infected with malaria, HIV/AIDS and TB since the transmission mechanics are completely different for the three diseases. The model further assumes that humans acquire HIV/AIDS through sexual contacts between an infective and a susceptible. The average force of infection for HIV/AIDS denoted λ ah is given by where βa is the average transmission probability of HIV/AIDS between an infective and a susceptible per sexual contact and c 1 is the per capita number of sexual contacts of susceptible humans with HIV/AIDS infected individuals per unit time. The parameter δ measures the effectiveness of counseling through condom use and a reduction in the number of sexual partners, where 0 δ 1. Effective counseling reduces the value of the parameter c 1. The model assumes that the classes I HMT, I A, I AM, I AT and I AMT do not transmit the virus due to acute ill health and noticeable AIDS symptoms. Define α 1 as the number of bites per human per mosquito (biting rate of mosquitoes), β m as the transmission probability of malaria in humans per bite thus the force of infection with malaria for humans, denoted λ mh is given by whereas the average force of infection with malaria for vectors, denoted λ mv is given by where β v is the transmission probability of malaria in vectors from any infected human. Finally the average force of infection for TB denoted λ th is given by where β t is the transmission probability of TB in humans and c 2 is the average per capita contact rate of susceptible humans with TB infected individuals. The rate of progression from HIV to AIDS for the untreated HIV cases is p. The parameters θ 1 p, θ 2 p and θ 3 p account for increased rates of progression to AIDS for individuals co infected with HIV - TB, HIV - malaria and HIV malaria - TB respectively where θ 1 < θ 2 < θ 3. Define α as the proportion of the HIV/AIDS infectives receiving effective treatment. This involves the administration of ARV S that keeps the HIV patients from progressing to AIDS while transferring the AIDS patients back to the HIV classes. The modification parameters e h m, e h t and e h mt account for the reduced susceptibility to infection with HIV for individuals in the I M, I T and the I MT classes respectively due to reduced sexual activity as a 79

3 h result of ill health where 1 h e, e 1, e h h m < 1, e 1 m t mt. The parameters e, e, e, m a m h m ht m e at, account for the increased susceptibility to infection with malaria for individuals already infected with AIDS, HIV, HIV - TB and AIDS - TB respectively due to suppressed immune system where e m a > 1, e m h > 1, e m ht > 1, e m at > 1. It is also clear that e m a < e m at and e m h < e m ht. The parameters e t h, e t a, e t mh and e t am account for the increased susceptibility to infection with TB for individuals already infected with HIV, AIDS, HIV - malaria and AIDS - malaria respectively due to suppressed immune system where e t h > 1, e t a > 1, e t hm > 1, e t am > 1. Again e t h < e t hm and e t a < e t am. Malaria and TB does not lead to the depletion of the CD4 + cell counts, however their association with HIV/AIDS leads to a significant reduction in the CD4 + cell counts within an individual leading to faster progression to AIDS. Combining all the aforementioned assumptions and definitions, the model for the transmission dynamics of HIV/AIDS, TB and malaria is given by the following system of differential equations. 2.2 THE MODEL EQUATIONS 80

4 2.3 POSITIVITY AND BOUNDEDNESS OF SOLUTIONS The model system describes living populations therefore the associated state variables are non-negative for all time t > 0. The solutions of this model with positive initial data therefore remain positive for all time t > 0. Proof. Consider the first equation of at time t then From the second equation of at time t then 81

5 We can proceed in a similar manner and show that all the state variables are positive for all time t. be any solution with non-negative initial conditions. The rate of change of the total human population with time is given by: 82

6 2.4. LOCAL STABILITY OF THE DISEASE FREE EQUILIBRIUM In the absence of infection by all the diseases, the model 2.2.1, has a steady-state solution called the disease-free equilibrium (DFE) given by Define F i as the rate of appearance of new infections in the class or compartment i and V i = (V i V i + ), where V i is the rate of transfer of individuals out of compartment i, and V i + is the rate of transfer of individuals into compartment i by all other means. Therefore: The Jacobian of F i and V i at the disease-free equilibrium denoted by F and V respectively is given by: 83

7 84

8 2.5. PARAMETER VALUES FOR THE HIV/AIDS, TB AND MALARIA MODEL 85

9 Figure 1 86

10 2.6 SENSITIVITY ANALYSIS OF TREATMENT AND COUNSELING To investigate the potential impact of counseling and treatment on disease progression, sensitivity analysis of the repro duction numb ers with resp ect to counseling and treatment is carried out. The sensitivity index of R H with respect to δ is given by: The negative sign in equation indicates that there is an expected decline in the rate of new HIV/AIDS infections when counseling is scaled up. Similarly, The sensitivity index of R H with respect to α is given by: Numerical simulations shows that the sensitivity index of R H with respect to treatment is positive indicating that an increase in the proportions of those treated leads to an increase in new HIV cases as shown in figure 2. The negative sign in equations and indicates that there is an exp ected decline in the rate of new malaria and TB cases when treatment is scaled up. Numerical simulations using the parameter values in table shows that the sensitivity index of R H, R M and R T with respect to treatment and counseling yields R = , m = , R r M = and R H = respectively. Assuming that R HMT = max {R M, R T, R H } = R H, rt T R H 87

11 then counseling for the HIV/AIDS individuals is the most sensitive parameter for the control of HIV/AIDS, TB and malaria co infections. 3. CONCLUSION In summery The local stability of the disease free equilibrium was investigated by Theorem 2 by Van, P. and Watmough, J. (2002). The theorem showed that the HIV/AIDS, TB and malaria co infection model have a disease free equilibrium point which is locally asymptotically stable whenever the reproduction number is less than unity. The sensitivity indices of the reproduction numbers R H, R HM, R HT and R HMT with respect to counseling for the HIV/AIDS individuals yields a negative sign indicating that there is an expected decline in the rate of new infections and co infections when counseling is scaled up. Similarly, the sensitivity indices of the malaria reproduction number (R M ) and the TB reproduction number R T with respect to malaria and TB treatment yields a negative sign also indicating that there is an expected decline in the rate of new malaria and TB infections when treatment is scaled up. Numerical simulations of the sensitivity index of R H with respect to ARV treatment yielded a positive gradient indicating that an increase in the proportions of those treated leads to an increase in new HIV cases 3.1. ACKNOWLEDGEMENTS The author is very grateful to Mark Okongo (Department of Physical Sciences) Chuka University - Kenya, for many excellent comments that have enhanced the model as well as the clarity of the paper. 4. REFERENCES [1] L. Abu-Raddad, P.Patnaik, and J. Kublin, Dual infection with HIV and malaria fuels the spread of both diseases in Sub-Saharan Africa, Science, 314(5805), (2006), [2] E, Allman and J. Rhodes, An introduction to Mathematical models in Biology, Cambridge University press: New York, (2004). [3] R. Anderson and R. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press: United Kingdom, (1993). [4] R. Audu, D. Onwujekwe, C. Onubogu, J. Adedoyin, N. Onyejepu, A. Mafe, J. Onyewuche, C. Oparaugo, C. Enwuru, M. Aniedobe, A. Musa, and E. Idigbe, Impact of co infections of tuberculosis and malaria on the C D4+ cell counts of HIV patients in Nigeria, Annals of African Medicine, (2005), 4(1): [5] F. Baryama, and T. Mugisha, Comparison of single - stage and staged progression models for HIV/AIDS models, International Journal of Mathematics and Mathematical sciences.(2007), 12(4): [6] C. Bhunu, W. Garira and Z. Mukandavire, Modeling HIV/AIDS and Tuberculosis Co infection, Bulletin of Mathematical Biology, (2009), 71: [7] Center for Disease Control and Prevention (CDC), Incorporating HIV prevention into the medical care of persons living with malaria : MMWR 2006;55(No. RR-14): Accessed August 22nd [8] O. Diekmann and J. Heesterbeek, Mathematical epidemiology of infectious diseases. Chichester: Wiley, (2000). [9] R. Granich, C. Gilks, C. Dye, K. Decock and B. Wlliam, Universal Voluntary HIV testing with immediate antiretroviral therapy as a strategy for elimination of HIV transmission, Journal of Mathematical Biology, (2008), 28(1): [10] C. Chiyaka, W. Garira and S. Dube, Transmission model of endemic human malaria in a partially immune population, Mathematical and Computer Modelling, (2007). 46: [11] S. Hohman and K. Kim, The impact of HIV and malaria co infection: What is known and suggested avenues for further study. Interdisciplinary perspectives on infectious diseases, (2009), 201(8): [12] B. Kamal,M. David, R. Svetlana, M. Ana, F. Tameru and T. Sharquetta, Mathematical Model of HIV and Malaria Co Infection in Sub-Saharan Africa, Alabama State University: USA, (2007), AL(5), [13] Kenya Demographics profile, Accesed on 3rd August 2015 at w w w.indexmundi.com/k eny a/demog r aphics profile2014, (2014). [14] Kenya National AIDS Control Council Report, Accessed on 03/08/2015 at (2014). [15] D. Kirschner, Dynamics of co infection with Mycobacterium tuberculosis and HIV-1, Theory of Population Biology, (1999), 55: [16] G. Lawi, J. Mugisha and Omolo - Ongati, Mathematical model for malaria and meningitis co-infection among children, Applied Mathematical Sciences, (2011), Vol. 5: 47,

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