A Mathematical Analysis of HIV/TB Co-Infection Model

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1 pplied Mathematics 6 6(4: 65-7 DOI:.593/j.am.664. Mathematical nalysis of IV/B o-infection Model Bolarin G. * Omatola I. U. Department of Mathematics Federal University of echnology Minna igeria bstract In this work we developed and analyzed a mathematical model of IV/B co infection. he model is a first order Ordinary Differential Equations in which the human population is divided into six mutually- exclusive compartments namely; B- Susceptible individuals (S B-Infected individuals (I B-Recovered individuals (R IV-Infected individuals (P o- Infected individuals (P and individuals with IDS (. he equilibrium states were obtained and their stabilities were analyzed by using Bellman and ooke s theorem. he result shows that the endemic equilibrium state is stable and the disease free equilibrium state will be stable if ( c < ( a d. Keywords uberculosis uman Immunodeficiency Virus o infection reatment Equilibrium Stability. Introduction uberculosis (B is a bacterial disease caused by Mycobacterium tuberculosis (tubercle bacilli. ransmission of B occurs by air borne infectious droplets [3]. It typically affects the lungs (pulmonary B but can affect other parts as well (extra pulmonary B []. he World ealth Organization (WO declared B in 993 has a global emergency. igeria ranks th among the high burden in the world []. B is preventable and curable. mathematical model for B has been a useful tool in accessing the spread of the disease as well as its control. See [3] for example. IV stands for uman Immunodeficiency Virus. IV is a virus that gradually attacks the immune system and the immune system is our body s natural defence against illness. If a person becomes infected with IV he or she will find it hard to fight off infections and diseases. he virus destroys a type of white blood cell called -helper cells and makes copies of it inside them. -helper cells are also referred to as D4 cells [6]. here is no cure or vaccine for IDS [5]. owever antiretroviral (R treatment improves health pro-longs life and substantially reduces the risk of IV transmission. In both high-income and low-income countries the life expectancy of patients infected with IV who have access to R is now measured in decades and might approach that of uninfected populations in patients who receive an high IV treatment [5]. o-infection of B and IV is when someone has both IV and B infections. When someone has IV and B IV infection accelerates the activation of tuberculosis and * orresponding author: g.bolarin@futminna.edu.ng (Bolarin G. Published online at opyright 6 Scientific & cademic Publishing. ll Rights Reserved tuberculosis increases the rate at which IV infection develops into IDS [9]. It is worthy of note that IV infection and infection with B bacteria are two different infections as established in [6]. o-treatment of IV related B improves survival especially in patients with D4 counts <5 ells/mm 3 [8]. Globally one-third of the 34 million people living with IV are infected with B [4]. In 4 about 4 people who had both B and IV were estimated to have died in addition to the. million people who died from B alone and 8 deaths from IV alone. So in 4 more people died from B than from IV related infections [5]. [] in his work formulated a sex-structured model to capture the effect of complacency on the dynamics of IV/IDS but did not include how B will affect the mix. But recently a lot of great work has been done in the mathematical modelling of co infection of different pathogens though very little was done in the modelling of IV-B co infection. [3] [] [6] [7] developed a mathematical model of IV/B co infection under B treatment. heir work did not include anti-iv treatment. [4] Developed a B-IV/IDS co infection model and optimal control treatment. In this paper we incorporate all aspects of B transmission dynamics as well IV transmission dynamics to come up with a mathematical model of IV/B co-infection. his paper incorporate antiretroviral therapy for individuals infected with IV. We investigate the effect of treatment on the B in IV/B co infection and also investigate the effect of co treatment of IV/B co infection. he paper is organized as follows. Section is introduction. he model formulation the positive invariant region the positivity of solutions equilibrium states and the basic reproduction number are obtained in Section. he stability analysis of the model and concluding remark are carried out in sections 3 and 4 respectively.

2 66 Bolarin G. et al.: Mathematical nalysis of IV/B o-infection Model. Model Formulation he model subdivides the human population into six mutually exclusive compartments amely B-susceptible individuals (S B-infected individuals who have active B diseases and infectious (I B-recovered individuals (R IV-infected individuals (P co-infected individuals (P and full blown IDS (. d ep dp ( d (.6 Where SIRP P. he force of infection λ associated with B infection is I P λ. he force of infection λ associated with IV infection is P P δ λ. able. Definition of State Variables and Parameters Used in the B/IV Model Symbol Definition Figure. he diagram for the flow of the model S I R P total population number of susceptible (that is no infection number of persons with active B number of persons recovered from B number of persons with IV infection.. Basic ssumptions Individuals can only get B through contact with B infectious individuals (I and P. Individuals may become IV infected only through contact with IV infectious individuals (P and P. he infectious I both B and IV infectious P and IDS individuals are considered too ill to remain sexually active and they are unable to transmit IV through sexual activity. W I P S R P is the active population. o represent this mathematically we have the following system of differential equations: ds dr dp dp rr ( λ λ S (. λ S ( λ a d I (. ai ( λ r R (.3 ( S R λ bp ( λ e d P (.4 ( b d d P (.5 λ I λ P P W ʌ δ number of persons with both IV and B infection number of persons with IDS number of active population constant recruitment rate Probability of transmission of B infection from an active to a susceptible per contact unit time Probability of transmission of IV infection from an infected person to an uninfected person per contact unit time. per capita contact rate for B per capita contact rate for IV atural death rate a b treatment rate of active B individuals treatment rate of infectious B in IV individuals d IDS progression rate for individuals in P e IDS progression rate for individuals in P r d d d d rate at which B recovered individuals become susceptible to B active B induced death rate death rate due to both IV and B infection IV induced death rate IDS induced death rate

3 pplied Mathematics 6 6(4: he Positive Invariant Region he total population size is d d S I R P P ds dr (.7 dp dp d S I R P P d d d d (.8 he positive invariant region can be obtained by using the following theorem. heorem. he solutions of the system of equations (. to (.6 are feasible for t < if they enter the invariant region D. Proof Let D ( S I R P P R 6 be any solution of the system of equations (. to (.6 with non-zero initial conditions. ssuming there is no disease equation (.8 now becomes d (.9 d (. he integrating factor for (. is ( IF e t Multiply both sides of (. by t e gives t d t t e e e (. t t Integrating both sides we have d( e e (. te ( t t e c (.3 ( t ce t (.4 pplying the initial condition ; t h ( (.5 ( t e (.6 herefore as t in (.6 the human population approaches K (hat is K the parameter K is called the carrying capacity. ence all feasible solution set of the human population of the model equations (. to (.6 enter the region ( 6 SIRP P R : S> I R D P P herefore the region D is positively-invariant (that is the solution is positive for all times (t and equations (. to (.6 are epidemiologically meaningful and mathematically well-posed in the domain D. ence in this model it is sufficient to consider the dynamics of flow generated by the model (. to (.6. In addition the existence uniqueness and continuation of results hold for the system..3. Positivity of Solutions Lemma. Let the initial data be S > I( R( P ( P ( { ( ( } D hen the solution set { I R P P } ( S t of the system of equations (. to (.6 is positive for all t > Proof From the first equation (.6 we have ds ( λ λ S ( λ λ S rr (.7 ds ( λ λ S (.8 Separating the variables and integrating both sides we have ds ( λ λ (.9 S ( λ λ ΙnS t c (. ( t ( (. St e λ λ ( λ λ t S( t Ke Where K e Using the initial condition t S( K ( λ λ t herefore S( t S( e From equation (3. λ S ( λ a d I ( λ a d I (. I ( λ a d I (.3 ( λ a d (.4 ( a d t c InI λ (.5 I t Ke λ ( a d t ( (.6 pplying the initial condition t I( K

4 68 Bolarin G. et al.: Mathematical nalysis of IV/B o-infection Model herefore I( t ( λ a d t I( e (.7 Similarly it can be verified that the rest of the equations ω are positive for all > e > for all ω R..4. Equilibrium States t since.4.. he Disease Free Equilibrium State of the Model he equilibrium state with the absence of infection is known as Disease Free Equilibrium or Zero Equilibrium. herefore the disease-free equilibrium state of the model is ( x x x x x x he Endemic Equilibrium State of the Model he equilibrium state of the model is; ( x x x x x x ( λ a d ( λ r ( r bλλ λ ( r ( λ λ r aλ ( ( ( a λ λ a d λ r λ bλλ λ ( λ ( ( ( e d a d r a B λ λ λ λ b ( aλ ( λ a d ( λ r λ b ( λ e d( ( bλλ ( λ r b ( b λλ λ r e d ( aλ ( λ a d ( λ r λ d ( aλ ( λ a d ( λ r λ d ( ( λ a d ( λ r aλ λ b Where (( λ a d ( λ r ( λ λ arλ (( b d d ( λ e d d bλ B (( λ a d ( λ r ( λ λ arλ (( b d d ( λ e d d arλ (( λ a d ( λ r ( λ λ arλ.5. Basic Reproduction umber R the next generation matrix FV is given by λ S ( S R λ bp F λ P λ I ep dp V ( λ a d I ( λ e d P ( b d d P ( d (.8 (.9

5 pplied Mathematics 6 6(4: c c ( a d ( b d d δ δ b ( e d ( b d d (.3 e d e d b d d herefore the basic reproduction number R ρ( FV spectra radius of FV c R (.3 ( a d 3. Stability nalysis of the Model 3.. Stability nalysis of Disease Free Equilibrium (DFE Jacobean matrix of the system of equations at disease-free equilibrium is c δ c δ r c c ( a d J a ( r δ δ ( e d b ( b d d e d ( d he characteristic equation is ( J Iλ herefore From (3. c δ c δ λ r c ( c a d λ a λ ( r δ δ ( e d λ b ( b d d λ or λ ( a d or λ 3 ( r λ c λ λ λ λ λ and < c > ( a d c ( e d d λ or λ 4 ( e d or λ ( b d d or λ ( d < 5 c λ if and only if < ( a d 6 (3. and λ if and only if >

6 7 Bolarin G. et al.: Mathematical nalysis of IV/B o-infection Model c ence the DFE is stable if < ( a d and otherwise unstable. 3.. Stability nalysis of Endemic Equilibrium State (EE he characteristic equation is obtained from the Jacobi a determinant with the eigenvalues λ λ r λ 7 8 a 9 λ λ λ e d d λ ( J Iλ Where ( ( d λ ( λ ( λ (( ( λ ((( ( (( ( ( 33 ar 48 ( ( ( r a 7 5 λ ( (((( 5 7 ( 7 ( 5 ( 36 7 ( (( ( 6 33 ar 5 ( ( r 7 a 7 5 (( ( a 7 5 (( ( r 8 ( a3 5 ra ( (( ( 8 r a58 9 6r (( a7r 85( 8 7 λ ((( ( ( 63 7 (( ( (( ( (( ( ( 5( 6 a7 (( a3 5 ra ( r( 6 a7 (( 6 a8 9 8( ((( a4 5 ( 3 a7 9 8(( r( (( a ( r c( x x5 δ ( x x5 cx δx 3 cx δx c( x x5 4 5 cx δ ( x4 x5 6 ( a d δx cx δx δ ( x 4 x5 δx r δx3 δ ( x4 x5 cx4 δ ( x4 x5 3 5 δ c( x x5 δ cx4 6 ( x x ( e d x7 ( x x b cx4 δ ( x4 x5 c( x x5 δx 8 9 (3.

7 pplied Mathematics 6 6(4: cx 4 δx b d. d -( ext we shall apply the result of Bellman and ooke s (963 to (3. to establish the stability or otherwise of the model we have; F '( (3.3 ( d (((( a4 5 r( 3 a7 9 8(( ( (( ( (( (( F( (( ( ( 5( 6 a7 (( a3 5 ra ( r( 6 a7 (( 6 a8 9 8( (( 6 a8 9 8( 8 7 r (3.4 G ( (3.5 G'( ( d ((( a 7 3 ( ( ( ( (( ( ar 5 ( ( r 7 a 7 5 (( ( a 7 5 (( ( a75 (( ( r 8 ( a3 5 ra ( (( ( 8 r a r (( a7r 85( 8 7 ((( a4 5 r( 3 a7 9 8(( r( ((( ( ( (( ( (( ( (( ( ( 5( 6 a7 (( 6 a8 9 8( (( 6 a8 9 8( 8 7 r (3.6 Since F '( G( F( and G '( ence F( G'( F '( G( (3.7 herefore the non zero equilibrium state is stable. 4. onclusions In this study we developed and analyzed a mathematical model of IV/B co infection. he model subdivides the human population into six compartments namely; Susceptible B individuals S B-infectious individuals I B-recovered individuals R IV-infected individuals P co-infected individuals P and individuals with full blown IDS. he model solution was found to be positive (establishing the fact that human population cannot be negative for all time t > provided that the initial data set is positive. he stability analysis of the Disease Free Equilibrium State (DFE of the model shows that it will be stable if c < ( a d that is the population is sustainable. he analysis of the Endemic Equilibrium State (EES shows that it is stable. his is in conformity with the real life scenario. hat is whenever IV is present the patient may likely be infected with B if proper and timely care is not given. lso early detection of IV and B cases and provision of early treatment can help to control the disease.

8 7 Bolarin G. et al.: Mathematical nalysis of IV/B o-infection Model REFEREES [] Bellman R. and ooke K (963. Differential Equations. cademic Press London. PP -45. [] Bolarin G. ( Mathematical model of omplacency in IV/IDS Scenario: Sex-Structure pproach Leaonardo Journal of Sciences Issue P.-. [3] Bowong S. and ewa J.J ( Global analysis of a dynamical model for transmission of B with a general content. [4] hristiana J.S. and Delfim F.M (5. B-IV/IDS co-infection model and optimal control treatment. Series ISS (print ISS (online. [5] Deeks S.G. Lewin S.R. avlir D.V (3. he end of IDS: IV infection as a chronic disease. he Lancet Vol. 38 Issue {533}. [6] Funke J.D. (3 Mathematical Epidemiology of IV/IDS and uberculosis o-infection. [7] Joyce K.. (5 Mathematical Modeling of uberculosis as an Opportunistic Respiratory o-infection in IV/IDS in the Presence of Protection. pplied Mathematcal Sciences Vol. 9 5 no [9] Mayer K ( Synergistic Pandemics: onfronting the Global IV and uberculosis Epidemics linical infectious Diseases Volume 5 Supplement 3. [] Mandell G. L. Bennett J. E. Raphael D. ( Mandell Douglas and Bennett's principles and practice of infectious diseases (7th Ed.. Philadelphia P: hurchill Livingstone/ Elsevier. [] igeria B Fact Sheet. [] Shah.. and Jyota G (3 Modeling of IV-B co infection ransmission Dynamics. merican Journal of Epidemiological and Infection Disease. pp -7. [3] Roeger W.L. Zhilan F. and arlos. (9. Modeling B and IV o-infections. Mathematical Biosciences and Engineering Volume 6 umber 4. Doi:.3934/mbe pp [4] World ealth Organization (.he Global Plan to Stop B WO Geneva. [5] World ealth Organization (5. Policy on collaborative B/IV activities WO. [6] ID gov. Retrieved /3/5 [8] Kendal M.. and avlir D ( iming of antiretroviral therapy and B. nalysis of a dynamical model for transmission of tuberculosis with a general content.