Bifurcation Analysis of a Vaccination Model of Tuberculosis Infection

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1 merican Journal of pplied and Industrial Cemistry 2017; 1(1): 5-9 ttp:// doi: /j.ajaic Bifurcation nalysis of a Vaccination Model of Tuberculosis Infection M. O. Ibraim 1, S.. Egbetade 2, * 1 Department of Matematics, University of Ilorin, Ilorin, Nigeria 2 Department of Matematics and Statistics, Te Polytecnic, Ibadan, Nigeria address: * Corresponding autor To cite tis article: M. O. Ibraim, S.. Egbetade. Bifurcation nalysis of a Vaccination Model of Tuberculosis Infection. merican Journal of pplied and Industrial Cemistry. Vol. 1, No. 1, 2017, pp doi: /j.ajaic Received: ugust 11, 2015; ccepted: ugust 28, 2015; Publised: ugust 31, 2015 bstract: In tis paper, we extend te model of Blower et al. [1] by incorporating certain infection terms suc as vaccinated individuals, treatment rate, waning rate and efficacy rate. bifurcation analysis is performed on te vaccination model by applying a bifurcation metod based on te use of center manifold teory. We determine tresold values and derive sufficient conditions for bot forward and backward bifurcations. Numerical simulations were carried out and bifurcation diagrams are presented as supporting evidences of our analytical results. Te obtained results sow te possibility of occurrence of forward and backward bifurcations even wen te basic reproduction number is less tan one so tat it is now possible for te disease to exist. Tese results suggest te need for more study on te qualitative biological mecanisms responsible for backward bifurcation. Keywords: Matematical Models, Tuberculosis, Bifurcation, Vaccination, Center Manifold Teory, Stability 1. Introduction In analysing disease transmission models, studies ave sown te existence of forward and backward bifurcations in suc models. In a forward bifurcation scenario, as increases troug one, a stable disease-free equilibrium loses its stability and a stable endemic equilibrium appears. Te beaviour of te bifurcation curve is suc tat as we travel along it from te bifurcation point, te level of infection increases as increases. Many epidemic models tat exibit forward bifurcation can be found in te literature [2]. Te penomenon of backward bifurcation is caracterised by multiple endemic equilibria due to te decrease in as te level of infection increases. In oter words, a stable disease-free equilibrium coexists wit one or more stable endemic equilibria for 1. Tis pattern as been noted in numerous models like multi-group models [3], immunity models [4], vaccination models [5], core group models [6] and treatment models [7]. Over te last decade, several papers ave appeared dealing wit a wide range of models tat ave te potential for exibiting forward and backward bifurcations. Te model by Saromi et al [8] studied te presence of backward bifurcation in some HIV vaccination models wit standard incidence function. Te autors noted tat vaccine-induced backward bifurcation in some HIV models wit standard incidence can be removed by using mass action incidence. s a result, te presence or absence of standard incidence may be crucial to te presence or absence of backward bifurcation in HIV vaccination models. In [9], te existence of backward bifurcation in a discrete SIS model wit vaccination was investigated. It was found tat backward bifurcation may occur if te lumped parameter 1. Te disease can persist for 1 and can be eradicated for 1 if a forward bifurcation occurs at 1. However, te disease may persist even wen 1 if a backward bifurcation occurs at 1. Greenalg and Griffits [4] discussed te penomenon of backward bifurcation in a tree-stage model for Bovine Respiratory Syncytial Virus (BRSV) in cattle. It was sown tat te 3-stage model undergoes backward bifurcation for small, is te common per capital birt and deat rate. Several bifurcation diagrams are obtained by fixing some of te parameter values for BRSV wile varying te oters. Te existence of backward bifurcation in te West Nile Virus (WNV) compartmental models as been investigated in [10]. In

2 6 M. O. Ibraim and S.. Egbetade: Bifurcation nalysis of a Vaccination Model of Tuberculosis Infection teir analysis, te autors found tat it is te iger mortality rate of te ost birds due to WNV infection tat determines te occurrence of backward bifurcation. Buonomo and Lacitignola [5] stressed te importance of a nonlinear incidence rate and an imperfect vaccine in te occurrence of backward bifurcation. bifurcation analysis of te model sows te conditions ensuring te presence of eiter forward or backward bifurcation. In [11], a deterministic model of TB witout and wit seasonality was developed. Te objective of te autors was to study te presence of backward bifurcation in te model. Tey observed te existence of backward bifurcation wen te basic reproduction number is less tan unity. Te autors concluded tat te backward bifurcation scenario is caused by te re-infection of latently infected individuals wit te TB disease. Li and Cui [12] investigated te beaviour of a discrete-time SIS model wit nonlinear incidence rate. Te teoretical analysis and numerical simulations of te model demonstrated tat te model exibits a variety of dynamical beaviours suc as backward bifurcation, opf bifurcation, flip bifurcation and caos. 2. Extension and Modification of Blower Model In 1995, Blower et al proposed a model of TB infection dynamics consisting of tree disease states namely susceptibles(), latently infected () and infected (). Te model is given by te following set of 1st-order differential equations (2.1) (1 ) () (2.2) = ( ) (2.3) = natural deat rate = recruitment rate of susceptible individuals = deat rate due to TB = rate of slow progression = rate of fast progression = transmission rate. Te autors perform a qualitative analysis on te model and one of teir main results is tat te disease-free equilibrium (DFE) is globally asymptotically stable if < 1 wile te endemic equilibrium is unstable wen 1. However, disease control measures suc as vaccination and certain infection terms tat play vital role in TB dynamics was not included in te system and consequently bifurcation analysis was not discussed. For tis reason, we extend te model in [1] to include infection parameters suc as vaccination (), waning rate (), treatment rate (), proportion of recruitment due to immigration(), proportion of immigrants tat are vaccinated () and efficacy rate of vaccine (!, # ). Te proposed vaccination model is given by = (1 )(1 ) (2.4) = $1 & (1! ) (2.5) = (1! )(1 # )( ) (2.6) ll te parameters are positive constants wit te following interpretations,, denotes te compartments of susceptible, vaccinated and infected individualsrespectively. denotes treatment rate denotes rate of waning of vaccine denotes proportion of recruitment due to immigration denotes proportion of immigrants tat are vaccinated! denotes efficacy rate of vaccine in protecting against initial infection # denotes efficacy rate of vaccine in slowing down progression to active TB ll oter parameters are as defined in [1]. Te vaccination model (2.4) - (2.6) sall be investigated for existence of forward and backward bifurcations. We derive conditions, in terms of te parameters of te model tat ensure tat eiter forward or backward bifurcation occurs. We apply bifurcation metod introduced in [13] wic is based on te use of center-manifold teory. In addition, we present a detailed numerical verification of te results obtained for bot forward and backward bifurcations. Bifurcation diagrams are presented as supporting evidences of our analytical results. 3. Equilibrium Points and Local Stability Model (2.4) - (2.6) as a disease-free equilibrium ' = (,,) given by ' = ( (!)*)(!)), - (!)*), -./,01 (3.1) Te endemic equilibrium point ' = (,, ) is suc tat =,3(!)4 5 )(!)4 6 )(!)*) /.$! 4 5 &(7)-)(!)87) = (!)*)(!)87), /.(!)4 5 )7)9 =,-(!)4 5 )(!)4 6 ))(!)*)(!)), (!)87)(/.7)8)-) (3.2) (3.3) (3.4) Using te tecnique developed in [14] for calculating, te for te vaccination model (2.4) - (2.6) was calculated as = 7:(!))./;,(!)4 5 )(!)4 6 ) (-.8) (3.5) Now, we focus on te disease-free equilibrium ' and investigate te occurrence of transcritical bifurcation at = 1. Te Jacobian matrix of (2.4) - (2.6) evaluated at te disease-free equilibrium ' is given by? 0 B <(' ) = 0 (1! ) = 0 (3.6)

3 merican Journal of pplied and Industrial Cemistry 2017; 1(1): so tat te eigenvalues C are real and given by C! =,C # = (),C D =.t te bifurcation = 1, we ave = 1 = = 7:-(!))./;,(!)4 5 )(!)4 6 )) (3.7) It follows ten tat te disease-free equilibrium ' is locally stable wen < as it looses its stability wen. 4. Bifurcation nalysis We will make use of Teorem in [13] for te bifurcation analysis of te model system (2.4) - (2.6). Te teorem prescribes te role of te coefficients F and of te normal form representing te system dynamics on te center manifold in decending te direction of te transcritical bifurcation occurring at G = 0. More precisely, if F < 0and 0, ten te bifurcation is forward, if F 0 and 0, ten te bifurcation is backward. Let us consider a general system of ODE's wit a parameter G: H = (H,G), :, M # ( ) (B1) Witout loss of generality, we assume tat H = 0 is an equilibrium for (N1) Teorem 1[13] ssume (I) ' = O P (0,0) is te linearization matrix of system (B1) around te equilibrium H = 0 wit G evaluated at 0. Zero is a simple eigenvalue of ' and all oter eigenvalues of ' ave negative real parts; (II) Matrix ' as a (nonnegative) rigt eigenvector Q and a left eigenvector corresponding to te zero eigenvalues. (III) Let R denotes te S TU component of and F = R,,X^! R,W^! (4.1) Y R,Q W Q 6 4 Z X (0,0), = YP [ YP \ Y R,Q 6 4 Z (0,0), W YP [ Y] Ten te local dynamics of system (N1) around H = 0 are totally determined by Fand. (i) F 0, 0. Wen G < 0 wit G 1,H = 0 is locally asymptotically stable and tere exists a positive unstable equilibrium 0 < G 1,H = 0 is unstable and tere exists a negative and locally asymptotically stable equilibrium; (ii) F < 0, < 0. Wen G < 0, wit G 1,H = 0 is unstable; wen 0 < G 1,H = 0 is locally asymptotically stable and tere exists a positive unstable equilibrium; (iii) F 0, < 0. Wen G < 0 wit G 1,H = 0 is unstable and tere exists a locally asymptotically stable negative equilibrium; wen 0 < G 1,H = 0 is stable and a positive unstable equilibrium appears; (iv) F < 0, 0 Wen G canges from negative to positive, x=0 canges its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable. Now, we investigate te nature of te bifurcation involving te disease-free equilibrium ' at = 1. We apply Teorem 1 to sow tat model system (2.4) - (2.6) may exibit a backward bifurcation wen =. We now consider te Jacobian matrix <(', ) written as? 0 7:-(!))./;(!)4 5 )(!)4 6 ))- B <(', ) = 0 (1! ) = 0 a 7:-(!))./;(!)4 5 )(!)4 6 ))- b Here, te eigenvalues of te above matrix are given by: C! = ; C # = ; C D = 0. (4.2) Tus, C D = 0 is a simple zero eigenvalue and te oter eigenvalues are real and negative. Hence, wen = 1 (or equivalently = ), te DFE ' is a nonyperbolic equilibrium and te assumption (B1) of Teorem is tus verified. We denote by Q = (Q!,Q #,Q D ) a rigt eigenvector associated wit te zero eigenvalue C D = 0.Ten, it follows tat Q! e 7:-(!))./;(!)4 5 )(!)4 6 ) fq D = 0j ( )Q # (1! )Q D = 0 i e 7:-(!))./;(!)4 5 )(!)4 6 ) fq D = 0 g Solving eqn (4.3) for Q!,Q #,Q D, we obtain (!)4 5 )7-6 (-./) (4.3) Q = (,, 7:-(!))./;(!)4 5 )(!)4 6 )) 1 1 (4.4) We now consider te left eigenvector = (!, #, D ) satisfying.q = 0:! = 0 ( ) # = 0 (1! ) # : k;- b. 6 (!)4 5 )(!)4 6 ) i = 0 g j (4.5) By solving (4.5), we ave! = # = 0 and wit D = 1, te left eigenvector is tus given by = (0,0,1) (4.6) We now compute te coefficient F and defined in Teorem.Taking into account system (2.4) - (2.6) and

4 8 M. O. Ibraim and S.. Egbetade: Bifurcation nalysis of a Vaccination Model of Tuberculosis Infection considering only te nonzero components of te left eigenvector, ten from (4.1) we ave and F = D Q! # l# D l # (', 2 )2 D Q! Q # l # D ll (', 2 )2 D Q! Q D l # D ll (', 2 ) D Q # # l# D l # (', 2 ) 2 D Q # Q D YnYo (', 2 ) D Q D # Y6 4 k Yo 6 (', 2 ) (4.7) D Q! YnY8 (', 2 ) D Q # YnY8 (', 2 ) D Q D YoY8 (', 2 ) (4.8)! (1 )(1 ) # (1 ) (1! ) D (1! )(1 # )( )g i j (4.9) By substituting (4.4), (4.6) and (4.9) into (4.7)-(4.8), we get F 2()F, 7:-(!))./;,(!)4 5 )(!)4 6 ) (4.10) F!)-(!)4 5 ) 6 (!)4 6 ) :(!))./;(!)4 5 )(!)4 6 )) (4.11) Since te coefficient is always positive, it is te sign of te coefficient F and consequently te sign of te quantity F wic determines te local dynamics of te disease around te disease-free equilibrium for 1. For our vaccination model to exibit a forward bifurcation, 0 and F (as defined in (4.11) must be positive so tat condition F 0 will be satisfied. In te backward bifurcation situation, te sign of F must be negative for te quantity F 0. For numerical verification of te results in (4.10) and (4.11), we consider te following parameter values for bot forward and backward bifurcations. Forward bifurcation Parameter values are cosen as follows: , ,! # 0.05, 0.3, 0.001, 0.01.Using tese numerical values, F 8.884J 10 D wic is greater tan zero. Hence, F 0. We calculate 6.977J10 )! 0.s a consequence, system (2.4) (2.6) exibits a forward bifurcation. Backward bifurcation To verify te condition F 0, 0 required for backward bifurcation, te following parameter values are considered: 0.01,! # 0.01, 0.004, 0.2, 0.003, 0.03.Ten, using (4.10) and (4.11), J 10 )! 0,F and consequently F 0 in view of (4.10). Next, we investigate te role specifically played by treatment (), transmission (), waning () and efficacy (!, # ) parameters in te occurrence of forward or backward bifurcation. To acieve tis, we present bifurcation diagrams in figures 1-2. Fig. 1. Bifurcation diagram in te plane (, I) for te case Te bifurcation parameter.is te basic reproduction number. Te solid lines denote stability wile te dased lines denoteinstability. Te numerical values for oter parameters are as follows: 0.01, 0.006, 0.5, 0.009,! # 0.03, 1.5. Using tese values, Fw Hence, 2 and te model undergoes a backward bifurcation. Fig. 2. Bifurcation diagram in te plane (, I ). Te solid lines denote stability wile te dased linesdenoteinstability. Parameter values are 0.01, 0.001, 0.039, 0.009,! # 0.6, 0.05, 0.07.system (2.4)- (2.6) exibits a forward bifurcation. Te bifurcation value is

5 merican Journal of pplied and Industrial Cemistry 2017; 1(1): Results and Discussions We ave performed a bifurcation analysis on our vaccination model described in (2.4) - (2.6) by applying te bifurcation metod wic is based on te use of center manifold teory. Conditions ensuring te occurrence of forward or backward bifurcations are derived. For forward bifurcation, te criterion F < 0 and 0 is required F and are bot given by conditions (4.10) - (4.11). In te case of backward bifurcation scenario, te condition F 0, 0 must be satisfied. Te two qualitative conditions are numerically verified using realistic parameter values of te model. Te obtained results validated te bifurcation conditions. In addition, numerical simulations sow tat te existence of a certain kind of bifurcation critically depends on te interplay among te four biological parameters explicitly included in te model. Tese are treatment (), waning (), transmission () and efficacy (!, # ) parameters. If te terapeutic treatment () is above a certain tresold value Qxy a ig transmission rate, a mediocre efficacy rate and a waning rate 0, te model exibits a backward bifurcation. Figure 1 depicts tis situation. Our analysis furter reveals tat if transmission parameter is sufficiently small, treatment rate lies below a certain tresold and an intermediate vaccine efficacy te bifurcation is forward. Te bifurcation diagram describing tis situation is sown in figure Conclusion One of te major undertakings of modeling is to explore te backward bifurcations in te model. Tis is because in a backward bifurcation, disease elimination is no longer feasible for < 1. Since te penomenon of backward bifurcation is possible in our proposed model, we perform bifurcation analysis, we determine tresold values and obtain conditions for bot forward and backward bifurcations. Some numerical simulations were performed to verify our analytical results. From te results, te transmission rate must be sufficiently small, treatment rate must lie below a certain tresold value and an intermediate vaccine efficacy in order to ave a forward bifurcation. backward bifurcation requires a ig transmission rate, a mediocre vaccine efficacy and a treatment rate wic must lie above a certain tresold value. [2] Kribs-Zaleta, C.M. and Velasco-Hernandez, J.X. (2000). simple vaccination model witmultiple endemic states.mat. Biosci. 164(2), [3] Huang, W., Cooke, K.L. and Castillo-Cavez, C. (1992).Stability and bifurcation for amultiple-group model for te dynamics of HIV/IDS transmission.sim J.ppl. Mat. 52, [4] Greenalg, D. and Griffits, M. (2009).Backward bifurcation, equilibrium and stabilitypenomena in a tree-stage extended BRSV epidemic model.j. Mat. Biosci.59, [5] Buonomo, B. and Lacitignola, D. (2011).On te backward bifurcation of a vaccinationmodel wit nonlinear incidence.nonlinear nalysis: Modeling and Control 16(1), [6] Kribs-Zaleta, C.M. (1999). Core recruitment effects in SIS models wit constant totalpopulations.mat. Biosci. 160(2), [7] Wang, W. (2006).Backward bifurcation of an epidemic model wit treatment.mat. Biosci. 201, [8] Saromi, O., Podder, C.N., Gumel,.B., Elbasa, E.H. and Watmoug, J. (2007).Role ofincidence function in vaccine-induced backward bifurcation in some HIV models.matematical Biosciences 210, [9] Sopia, R. and Jang, J. (2008).Backward bifurcation in a discrete SIS modelwit vaccination.journal of Biological Systems 16(4), [10] Wan, H. and Zu, H. (2010). Te backward bifurcation in compartmental models for WestNile Virus.Matematical Biosciences 227, [11] Bowong, S. and Kurts, J. (2012).Modeling and analysis of te transmission dynamics oftuberculosis witout and wit seasonality.nonlinear Dynamics 67, [12] Li, J. and Cui, N. (2013). Bifurcation and caotic beaviour of a discrete-time SIS model.discrete Dynamics in Nature and Society. ttp://dx.doi.prg/ /2013/ [13] Castillo-Cavez, C. and Song, B. (2004).Dynamical models of tuberculosis and teirapplications.mat. Biosci. Engr. 1, [14] van den Driessce, P. and Watmoug, J. (2002).Reproduction numbers and sub-tresoldendemic equilibria for compartmental models of disease transmission.mat. Biosci.180, References [1] Blower, S.M., McLean,.R., Porco, T.C., Small, P.M., Hopewell, P.C., Sancez, M.. andmoss,.r. (1995).Te intrinsic transmission dynamics of tuberculosis epidemics.nat. Med. 1(8),