Spatiotemporal Dynamics of Tuberculosis Disease and Vaccination Impact in North Senatorial Zone Taraba State Nigeria

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1 IOSR Journal of Matheatics (IOSR-JM ISSN: Volue, Issue (Sep-Oct. 1, PP 1- Spatioteporal Dynaics of Tuberculosis Disease and Vaccination Ipact in North Senatorial Zone Taraba State Nigeria 1 A. A. Mooh, Jaes, J. Yakoko, A. Tahir and Chukkol Y. B. 1,,, Departent of Matheatics, Modibbo Adaa University of Technology, P.M.B. 76, Yola, Adaawa State, Nigeria. Abstract: In this work, The researchers present Passively Iune Infant ( V -Susceptible class( S - Infection class( I -Recovery class ( R odel to study the dynaic of tuberculosis transission and vaccination ipact in North Senatorial Zone, Taraba State, Nigeria. The copartent of the odel is presented in a syste of ordinary differential equations. Quantitative analysis of the odel was done to investigate the equilibriu and stability of the odel. An analytical approach was used to deterine their Disease Free equilibriu and the Epideic equilibriu state. The stability of the epideic equilibriu is tested using Bellan and Cooke s theore. The odel had two equilibriu position: The disease free equilibriu which was asyptotically stable for R ( e and the endeic equilibriu which was locally asyptotically stable for as it satisfies the Bellan and Cooke s condition for stability i.e. J. Key words: Tuberculosis (TB, Vaccination, Infection, Equilibriu analysis, Stability analysis. I. Introduction Tuberculosis is ostly transitted through air by infected persons coughing, sneezing or spiting with pulonary tuberculosis. The probability of transission per contact, per relevant unit tie is general quite low. The bacteria ay lodge in the person s lung and ultiply. If the iune syste in the lung is able to fight the bacteria and render it inactive (wall off as a result of vaccination or treatent fro TB, the person will develop latent TB which is not infectious and cannot har others. Infected individuals ay reain asyptoatic over their entire live (Latent TB. Active TB (the clinical disease can develop into pulonary and extra pulonary for. Extra pulonary TB is coon in children while pulonary TB is frequent in adult. Mycobacteriu tuberculosis, the casual agent of the disease is transitted alost exclusively via pulonary cases. Cases arising within Five (5 years fro first infection are classified as priary tuberculosis while cases arising after five (5 years fro first infection are known as secondary tuberculosis. The asyptoatic individual ay reain at latent for a long period of tie due to vaccine durations and the antibody defense deposit in the body unless cured by treatent with antibiotics such as Ethanbutol, Rifanpacine, B6 e.t.c. if the bacteria later becoe active due to breakdown of the iune syste, the person is syptoatic and infectious and can be cured (recovered if treated otherwise die fro the infection. The person when cured becoes susceptible as he is likely to be re-infected on contact with an infected person. Recent research efforts have been geared towards studying the heterogeneous factors in the population with a view to incorporate the into the odel. The efforts that have been ade by researchers and their findings on various aspects of TB disease dynaics falls under the following subheadings. i. Transission ii. Vaccination against tuberculosis iii. Protective efficacy of the vaccine and duration of iunity iv. Infection v. Treatent vi. Prevention Most early transission odels were deterinistic but recent resurgence of tuberculosis (TB in develop countries and increase in cases of casualty contact and public transport suggested a odel based on thorough understanding of the dynaics of the disease. Castillo-Chavez and Feng (1998 also consider an age-structured odel, in this case with agedependent transission rates. They include one for of latency, as in their earlier work, and this odel does not 1 Page

2 Spatioteporal Dynaics Of Tuberculosis Disease And Vaccination Ipact In North Senatorial Zone contain re-infection. They study vaccination, and define a vaccine-dependent R threshold R (. They prove stability of the disease-free equilibriu when R ( < 1, and the existence of an endeic steady state when R( > 1, and discuss analytically-deterined optial vaccination strategies. These turn out to be either vaccination at a single age, or vaccination in precisely two age classes. Aparacio et al ( develop a generalized household odel, which took close and casual contact into account. The house cluster coprises of social networks (Faily eber, office ates, classates, any person who have prolong contact with an active case. The basic reproductive nuber for the odel is nk Ro k Where the transission rate, n is the size of the cluster, natural ortality rate, the total percapita reoval rate fro infected and k is the progression rate to active TB. BCG-induced iunity develops about six weeks after vaccination. Experiental studies indicate that the echanis of protection by BCG vaccination consists in reduction of the heatogenous spread of bacilli fro the site of priary infection Sith & Harding (1979 ediated by eory T lyphocytes induced by the first exposure to BCG. There is no evidence that BCG reduces the risk of becoing infected with tuberculosis bacilli, but it prevents fors of tuberculosis depending on heatogenous spread of the bacillus Heibeck (199. This inhibition of the heatogenous spread of bacilli thus reduces the risk of iediate disease and of disease due to reactivation. Because there is reduction in risk of iediate disease, but not of infection, there is a difference in the protective effect of BCG, depending on the type of tuberculosis infection. Myint et al. (1987. The best ethod for deterining the protective efficacy of a vaccine is a prospective, randoized, double-blind, placebo-controlled trial. These studies are difficult and expensive, and have rarely been perfored. WHO has recently sponsored studies to evaluate the protective efficacy of BCG iunization in infants and/or children by two low-cost ethods: case-control studies Sith (198, 1987 and contact studies Ten Da (1987. These studies have recently been reviewed by Milstien and Gibson (1989, who concluded that there is good evidence that the efficacy of odern BCG vaccines is in the range of 6% to 9% for disseinated tuberculosis or eningitis in young children, but soewhat lower for other fors of priary tuberculosis disease. They also found no evidence that one BCG preparation tested was ore efficacious than any other under the conditions of the trials, and thus no evidence to support the choice of one preparation or anufacturer of BCG over another on the basis of protective efficacy. A recent atched case-control study in Bangkok Sirinavin et al. (1991 found an adjusted protective efficacy for neonatal BCG vaccination of 8%, and provided evidence to support the thesis that the accuracy of tuberculosis diagnosis, the types of tuberculosis, the length of tie after vaccination, and the household tuberculosis exposure contribute to variation in the reported protective efficacy of neonatal BCG vaccination. There is a need for the developent of a single in vitro test capable of predicting the induction of iune resistance of huans to infection or disseination of M. tuberculosis. Despite the difficulty in interpretation of data, several trials have shown that efficacy of BCG is highest aong the youngest recipients, and that it falls with increasing age at vaccination Fine et al. Tuberculosis Prevention Trial (1979. It is therefore difficult to assess the ipact of BCG vaccination prograes. However, when BCG iunization of newborns was stopped in Sweden, the incidence of the disease in infants raised six fold Roanus (1987. There are several publications which suggest that the protection BCG provides against tuberculosis is a function of the relative iportance of disease due to endogenous reactivation as copared to re infection fro the outside. Thus, BCG protects against heatogenous spread of infection Fine (1988, Ten Da (198. This suggestion predicts the greater protective efficacy seen against iliary tuberculosis and tuberulous eningitis, copared with pulonary tuberculosis Sith (1987. II. Prevention and Control for Tuberculosis Dye et al (1998 also have developed an age-structured odel, with the goal of exploring TB control under the DOTS strategy and under iproved case detection and cure. Their odel uses discrete tie rather than a partial differential equation fraework. It includes two latent classes (slow and fast with a set portion of new infections oving into each, as well as re-infection. They validate their odel by coparing their results to annual risk of infection data fro the Netherland. They then siulate different epideiological situations. They conclude that if tuberculosis is stable, and if HIV is not included in the odel, then WHO targets of 7% case detection and 85% cure would reduce the incidence rate by 11% (range 8-1 per year and the death rate by 1% (9-1 per year. The effect would be saller if TB were already in decline. They conclude that DOTS has greater potential today (in developing countries than it did 5 years ago in developed countries, but that case detection and cure rates ust be iproved. 15 Page

3 Spatioteporal Dynaics Of Tuberculosis Disease And Vaccination Ipact In North Senatorial Zone Saloon et al. (6 extend this odel, calibrating it to natural TB epideics and then studying the effects of hypothetical new treatents with differing durations. In their odel, drug resistance could result fro transission and acquisition. They first calibrate the odel to represent TB in South-East Asia. They conclude that if ore rapid treatents for TB were available, with reduced infectious periods, this could significantly reduce TB ortality and incidence, particularly if these becoe available soon. There are, of course, other alternatively-structured odels. Salpeter and Salpeter (1998 used TB incidence data to derive a function for the tie delay fro initial infection to active disease, and develop an integral equation for R.This is not a dynaic epideiological odel for TB, but it is relevant to TB odels because the variable delays and widely ranging estiates of R are iportant. They report R near 1, which was associated with tie scales of approxiately 1 years. III. Model Forulation The odel is going to divided into four classes of population under study naely; i. The passively iune infant ii. Susceptible Class iii. Infected Class Iv. Recovery Class Based on this, the researchers divided the populations into four copartents represented as: Model Diagra population of the Individuals are recruited into Figure 1: Scheatic representation of VSIR Model V class via natural birth at the rate Q through passive vaccination, the V decreases due to natural death at rate and oveent of the individuals into result of warning out of vaccines used in passive vaccination at the rate. The population of the increases due to coing of individuals fro the the V and R classes at the rates and. S as a S The population of S class decreases due to the oveent of individuals into the infected class at the rate and natural death at a rate. The population of the I decreases due to treatent against TB at the rate and The population of the R natural death at the rate and death as result of TB infection at the rate. increases due to oveent of individuals at the rates fro individuals into the Definition of paraeters Q = Natural Birth rate S at the rate and natural death at rate V Passively iune infant at the tie t S I Susceptible class at the tie t Infected class at tie t R Reoved class at tie t Natural death rate Rate of the duration of vaccine efficacy TB contact rate I and decreases due to oveent of 16 Page

4 Spatioteporal Dynaics Of Tuberculosis Disease And Vaccination Ipact In North Senatorial Zone Death as a result of TB infection Rate of recovery fro TB infection Rate at which individuals becoe susceptible Model Equation The odel above is thus described by the following set of ordinary differential equations dv ds di dr Q ( V V ( I S S S I ( I I ( R IV. Equilibriu State of the Model (1 ( ( ( Let V 1, S, I, R Equations 1- becoes, d d d d 1 Q ( ( 1 ( 1 ( Disease Free Equilibriu (DFE (5 (6 (8 (7 Fro (5 d1 Q ( 1 Q ( 1 Q ( 1 Fro (6 d ( ( ( Substituting (9 into (1 we have, (9 (1 17 Page

5 Spatioteporal Dynaics Of Tuberculosis Disease And Vaccination Ipact In North Senatorial Zone Q ( ( Q ( ( Q ( ( (11 Fro (7 d ( ( ( Therefore, either, (1 OR ( ( (1 Substituting (1 into (1 gives, Q ( ( (1 Substituting (1 into (8 gives, Hence, the disease free equilibriu (DFE states are as follows Q Q ( 1,,,,,, ( ( ( Epideic Equilibriu State (EE Substituting (8 into (11 Q ( ( ( Recall fro (8 ( ( Substituting (1 into (1 Q ( ( ( ( ( (15 (16 17 Hence, the Epideic Equilibriu (EE states are as follows: Q 1 ( 18 Page

6 Spatioteporal Dynaics Of Tuberculosis Disease And Vaccination Ipact In North Senatorial Zone ( Q ( ( ( Q ( ( ( ( ( IV. Stability Analysis J ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Disease Free Stability Analysis The characteristic equation above takes the for: ( ( ( ( ( ( ( (18 Either, 1 ( OR ( OR ( Q ( ( ( Q ( ( ( Q This iplies the for ( to be asyptotically stable, ( ( ( where Q ( ( is the nuber of susceptible individuals produced. OR ( ( Since then R e( 1,R e(,r e(,r e(, then it is asyptoatically stable. (19 ( (1 19 Page

7 Spatioteporal Dynaics Of Tuberculosis Disease And Vaccination Ipact In North Senatorial Zone Stability Analysis of the Epideic Equilibriu When applying Bellan and Cooke s (196, the result of Bellan and Cooke s theore state that, if ( iy, y I is separated into real and iaginary parts ( iy F( y ig( y and all zeros of ( z have negative real parts then the zeros of F( y and Gy ( are real and alternative then: ' ' F( G ( F ( G( ( ( is applied to the characteristic equation (18. We consider (18 in the for H( And we obtain: H ( ( ( ( ( ( ( ( Also, by applying Bellan and Cooke s theore on above, setting iw, we have H( iw F( w ig( w Where Fw ( and ig( w are real and iaginary part of H( iw. Substituting iw ( to have H ( iw ( iw( ( ( ( iw iw ( iw F ( w ( ( ( ( w G( w w( ( ( ( F ' ( w w G ' ( w ( ( ( ( Setting w F ( ( ( ( ( Substituting the values of and into F ( we have, Q ( ( F ( ( ( ( ( ( ( ( ( F ' ( 6 G ' ( ( ( ( ( Substituting the values of and into G '( we have, ( ' Q G ( (1 ( ( ( (7 G( (8 We have fro Bellan and Cooke s theore (196 that: The condition R e( ; is given by the inequality: ' ' F( G ( F ( G( (9 The equation (9 is the stability condition for the Epideic Equilibriu. But fro (6 and (7 we have F ' ( and G( (5 Page

8 POPULATION Spatioteporal Dynaics Of Tuberculosis Disease And Vaccination Ipact In North Senatorial Zone Hence, equation (9 becoes ' F( G( Let ' J F( G ( Then the Epideic equilibriu state is stable when J. V. Results and Discussion Data collected fro Tuberculosis and Leprosy unit, Ministry of Health, Taraba state, Nigeria, was used to obtain the siulation for the odel. Using hypothetical values pertinent to Taraba State, the researchers obtain the following results using Maple software as shown below: ( Q β θ μ Ψ J REMARKS STABLE STABLE STABLE STABLE STABLE STABLE STABLE STABLE STABLE STABLE UNSTABLE Figure : Result of stability analysis of Epideic Equilibriu State. Siulations The first exaple is concern with the estiation properties of the VSIR odel given by equation (5 (8 in the presence of low contraction, while the second exaple is concern with the estiation properties of the VSIR odel given by equation (5 (8 in the presence of high contraction. The VSIR odel is described by the following paraeters taken fro TB outbreak in North Senatorial zone of Taraba state. Q=.9, =.1 &.1, =.1, =.15, =.5, =., =.. The initial conditions are given by V ( =19,59, S ( =88,71, I ( =56, R ( =7. 9 LOW CONTRACTION Beta = (.1 FOR TB IN NORTH SENATORIAL ZONE OF TARABA STATE, NIGERIA 8 7 Infected Reoved 6 5 Vaccinated Susceptible TIME (Years Figure : Graphical profile showing low contraction rate 1 Page

9 POPULATION Spatioteporal Dynaics Of Tuberculosis Disease And Vaccination Ipact In North Senatorial Zone 9 8 GRAPHICAL PROFILE FOR HIGH Beta = (.1FOR TB IN NORTH SENATORIAL ZONE OF TARABA STATE. Reoved 7 Infected 6 5 Vaccinated 1 Susceptible TIME(Years Figure : Graphical profile showing high contraction rate Figure shows the graphical profile of the VSIR odel for =.1 (low contraction rate. It was observed that the population of the infected class increases steadily until when T =1., then it decreases steadily. While figure shows the graphical profile of the VSIR odel for =.1 (high contraction rate. It was observed that the population of the infected class increase exponentially until T =.5 then, decreases steadily. VI. Conclusion Conclusively, we will like to conclude that, Progression to active TB aong the population is epideiologically significant and interventions should focus on both vaccinations and treatent of infected persons, therefore, effort should be ade to iniize the contraction rate for the sustenance of North Senatorial Zone of Taraba state. Anti-tuberculosis, treatent of adults is crucial in controlling the epideic and intervention easures such as frequent vaccination and adhering to TB treatent should be proposed and ipleented, they should target progression to active TB for those infected. Recoendations: Tuberculosis is an airborne disease and therefore prevention and control of tuberculosis could be achieved if the following easures are adopted: 1. Ensure good, adequate, hygiene easures of health are put in place.. Preventive vaccines such as Becillus Calette Guerin (BCG should be adinistered to particularly children and pregnant others.. All health care centres involve in TB treatent should adopt the Direct Observation Treatent short course (DOTS strategy for treatent of patients.. Governent should create awareness and sensitize the public on transission and spread of TB. 5. Prevent or control exposure to infected anials and infected persons. 6. Governent should ebark on contact tracing for newly infected individuals, so that they can be placed on TB treatent therapy iediately. Reference [1] Aparicio, J.P., Capurri, A.F., Castillo-Chavez, C. (, Transission and dynaics of Tuberculosis on generalized household, J. Theoretical Biology. [] Bellan R. and Cooke K.C. (196, Differential difference equation. London. Acadeic press. [] Castillo-Chavez, C. and Feng, Z., (1998, Global stability of an age-structure odel for TB and its applications to optial vaccination strategies. Math Biosci,151(: [] Dye, C., Garnett, G. P., Sleean, K. and Willias, B. G.(1998, Prospects for Worldwide tuberculosis control under the who dots strategy. Directly observed short-course therapy. [5] Fine PEM (1988. BCG vaccination against tuberculosis and leprosy.br Med Bull; : [6] Fine PEM, Ponnighaus J.M., Maine N.P. (1986, The relationship between delayed type hypersensitivity and protective iunity induced by ycobacterial vaccines in an.syposiu on the Iunology of Leprosy, Oslo, Norway. Page

10 Spatioteporal Dynaics Of Tuberculosis Disease And Vaccination Ipact In North Senatorial Zone [7] Heibeck J. Sur la vaccination préventive de la tuberculose par injection sous-cutanée de BCG chez les élèves infirières de l hôpital Ulleval à Oslo (Norvège. [8] Milstien JB, Gibson JJ. (1989., Quality control of BCG vaccines by the World Health Organization: a review of factors that ay influence vaccine effectiveness and safety. Bull WHO; 68:9-18. [9] Myint TT, Win H, Aye HI-I, (1987, Case-control study on evaluation of BCG vaccination of newborn in Rangoon, Bura. Ann Trop Paediatr 1987;7: [1] Ndaan I.(1 A deterinistic atheatical odel of Tuberculosis disease dynaics, M. TECH thesis, F.U.T Minna. [11] Roanus V. (1987, Tuberculosis in bacillus Calette-Guerin iunized and non iunized children in Sweden: a tenyear evaluation following cessation of general bacillus Calette-Guerin iunization of the newborn in [1] Saloon, A. (6, Prospects for advancing tuberculosis control efforts through novel therapies. PLoS Med, (8:e7,1.171/journal.ped.7. [1] Salpeter, E. E. and Salpeter, S. R. (1998, Matheatical odel for the epideiology of tuberculosis, with estiates of the reproductive nuber and infection delay function. A J Epideiol, 17(:98 6. [1] Sirinavin S, Chotpitayasunondh T, Suwanjutha S,(1991. Protective efficacy of neonatal Bacillus Calette-Guerin vaccination against tuberculosis. Pediatric Infect Dis J1991;1: [15] Sith P.G.(198 Case-control studies of the efficacy of BCG against tuberculosis, In: Proc V Work Conf Tub Resp Dis, Singapore. 1 st Edition. [16] Sith P.G.(1987 Case-control studies of the efficacy of BCG against tuberculosis, In: Proc V Work Conf Tub Resp Dis, Singapore. nd Edition. [17] Sith, D.W., Harding, G.E.(1979, Are current strains of BCG protective in anials? Docuent TRI/SCG/ Geneva: World Health Organization, [18] Ten Da H. G.(1987, Contact studies on the effectiveness of BCG vaccination in children. In: Proc V Work Conf Tub Resp Dis, Singapore, Tokyo. Page