Modelling The Impact of BCG Vaccines on Tuberculosis Epidemics

Transcription

1 Journal of Mathematical Modelling and Application 2014, Vol. 1, No. 9, ISSN: Modelling The Impact of BCG Vaccines on Tuberculosis Epidemics S.A. Egbetade Department of Mathematics and Statistics, The Polytechnic, Ibadan, Nigeria M.O. Ibrahim Department of Mathematics, University of Ilorin, Ilorin, Nigeria Abstract In this paper, we study vaccination by a tuberculosis (TB) dynamics model incorporating treatment, migration and vaccination. We formulate theorem on existence and uniqueness of solution in order to ascertain the existence of our model. Using a stability analysis of the disease-free equilibrium (DFE) and the basic reproduction number, an analytic threshold which dictates when BCG vaccination will increase the stability of the DFE was established. The analysis found that mass vaccination by BCG is advantageous in countries with a high TB burden in order to result in a more stable DFE for disease elimination. Keywords:Tuberculosis; mathematical model; vaccination; existence and uniqueness; stability; reproduction number. basic 1. Introduction Tuberculosis (TB) is a leading cause of infectious mortality (WHO, 2010). Despite many decades of study, the widespread availability of a vaccine and a highly visible WHO effort to promote a unified global control strategy, TB pandemic remains one of the greatest public health disaster of modern times (Cohen and Murray, 2004). In 2007, there were an estimated 13.7 million chronic active TB cases, 9.3 million new cases and 1.8 million deaths mostly in developing countries (WHO, 2004). Africa alone had 8.8 million new TB infections which results in 1.7 million deaths in 2003 (WHO, 2007). Of all African countries, Nigeria has the highest TB burden and is ranked 4 th among 22 high TB burden countries in the world (WHO, 2012). According to the estimates of the World Health Organization (WHO), Nigeria has 311 TB cases for every 100,000 population. Research have shown that more people in the developed world would contract TB because their immune systems are more likely to be compromised due to higher exposure to immune suppressive drugs, substance abuse or AIDS (CDC,2011). Despite some successes associated with using BCG vaccines and some TB treatment therapies, the overall global incidence of TB is rising as a result of the emergence of drug-resistant TB strains and the deadly combination of HIV (human immune deficiency virus) and TB epidemics(porco et.al.,2001). In more than half of all HIV related deaths, tuberculosis is the opportunistic infection which takes advantage of an immune system already compromised by HIV ( Colijn et.al.,2006). Most of the current resurgence of the disease is in Africa, parts of Europe and Asia (WHO,2009). Mathematical models have been used extensively to gain insights into the spread and control of TB epidemics. The dynamics of these models tend to generally be completely determined by a threshold called the basic reproduction number (denoted by ) which is the total number of secondary infections that an average infections individual will induce given that the rest of the total population is susceptible (Dye,2006). In particular, when, a small influx of infected individuals will not generate large TB outbreaks, the disease dies out and the DFE is asymptotically stable. On the other hand, infection will

2 2 S.A. Egbetade, M.O. Ibrahim persist if where a stable endemic equilibrium exists. The presence of forward bifurcation in some TB transmission models have been noted (Kribs-Zaleta and Velasco-Hermandez,2000). This is a situation where the disease-free equilibrium loses its stability and a stable endemic equilibrium appears as. For such models the requirement is necessary and sufficient for disease elimination. The phenomenon where a stable endemic equilibrium co-exists with a stable disease-free equilibrium for is called backward bifurcation and this pattern can be observed in some disease models (Kar and Modal, 2012). In a backward bifurcation, disease control is only feasible if is reduced further to values below another sub-threshold less than unity. In the global fight against tuberculosis, BCG vaccines are the most widely used in the world (CDC, 2011). However, it has always been surrounded by uncertainty. The main uncertainty stems from a lack of consistency in clinical trials of the vaccine s efficacy (i.e. the reduction in the incidence of the disease among people who have received the vaccine compared to the incidence in unvaccinated people (Fine, 1995). Another dubious aspect of BCG vaccination is its possible interference with the detection of latent tuberculosis infection (Menzies, 2000). However, evidence exists that BCG vaccine has proved to be one of the most effective measures of combating the spread of TB epidemics (Trunz et.al., 2006) The World Health Organization (WHO) recommends BCG vaccination of infants in countries with a high TB burden and this policy has had significant public health impact (WHO, 2009). In countries such as Nigeria, Ghana, South Korea and Taiwan, the vaccine is given regularly throughout life (WHO, 2012). In this paper, we study vaccination and define a vaccine-dependent threshold for the benefit of BCG vaccination. We prove local asymptotic stability of the DFE when. From stability analysis of the DFE and the basic reproduction number, we obtain an analytic threshold for the continuation of BCG vaccination and discuss new approaches and strategies for further improvement of BCG vaccines. 2. Mathematical Formulation We consider a society at the brink of eliminating TB. Individuals enter the population at a rate. A proportion,, of individuals entering the population are vaccinated and enter the vaccinated class. We denote classes of suspectible by, latently infected and unvaccinated by, latently infected and vaccinated by, actively infected by and recovered by. Susceptible individuals once infected can either progress quickly to active TB at a rate or develop a latent infection and move slowly to active TB at a rate. Our model is thus given by where is the total size of the population, is the threshold for the benefit of vaccination and all other parameters are as defined below rate at which the vaccine wanes etection rate of active treat ent rate of active

3 Modelling The Impact of BCG Vaccines on Tuberculosis Epidemics 3 natural eath rate eath rate ue to infection = clearance rate for TB induced death = proportion of recruitment due to immigration trans ission rate rate at which suscepti le in ivi uals recover efficacy of vaccine in preventin initial infection efficacy of vaccine in preventin fast pro ression Lemma 2.1: The basic reproduction number for the model system (1) (6) is We consider the disease-free equilibrium state of our model and take parameter values in equations (1) (6) as follows:. We calculate the basic reproduction number as 3. Existence and Uniqueness of Solution We formulate theorem on existence of unique solution of model system (1) (6) and we establish the proof. We consider the system of linear equations below } Writing equation (8) in compact form, we have Theorem 1 ( Derrick and Grossman, 1976 ) Let denote the region and suppose that satisfies the Lipschitz condition whenever the pairs and belong to where is a positive constant. Then, there is a constant such that there exists a unique continuous vector solution of the system (9) in the interval. It is important to note that condition (11) is satisfied by the requirement that continuous and bounded in We now return to our model equations (1) (6). We are interested in the region We look for a bounded solution in this region and whose partial derivatives satisfy and are positive constants. where are

4 4 S.A. Egbetade, M.O. Ibrahim Theorem 2 Let denote the region. Then, equations (1) (6) has a unique solution. We show that are continuous and bounded in. Let Using equation (13), we have the partial derivatives below } These partial derivatives exist, continuous and are bounded in the same way the other derivatives exist and are bounded. Hence, by Theorem 1, the model system (1) (6) has a unique solution. 4. Stability Analysis of the DFE We consider a society at the brink of eradicating TB and assuming there is no immigration. If a population is free of TB, then and noting that is necessary for an equilibrium, we solve the resulting algebraic equations to obtain the DFE ( ) Linearizing our system about the DFE gives the following characteristics equation Solving the resulting characteristics equation from (21) we obtain ou le ei en values which implies that } has negative real parts. Therefore, the DFE is locally asymptotically stable (LAS).

5 Modelling The Impact of BCG Vaccines on Tuberculosis Epidemics 5 5. Basic Reproduction Number We use the formulation of presented in Diekmann and Heesterbeek (2000). To start with, we note that our model has 3 disease states : latently infected and unvaccinated, latently infected and vaccinated and Infectious. We denote by the probability that a new infection will be of type corresponding to, and respectively. The proportion of the population vaccinated at the DFE is given by and the univaccinated and we get the respective probabilities for, and as We define transition rate probability matrix Representing infectivity in the infectious state by, we have ( ) and the basic reproduction ratio of our model is [ ] where ( ) ( ) Setting the coefficient of in equation (28) to zero and further manipulation, we obtain as the threshold for the treatment of active TB through vaccination and vaccination is advantageous if it lowers i.e..

6 6 S.A. Egbetade, M.O. Ibrahim 6. Remark Noting biological considerations, if a country is able to detect and treat TB at a rate which exceeds, the discontinuation of BCG will increase the stability of the DFE. If the detection and treatment rate is below BCG vaccination will result in a more stable DFE. However, available TB data shows that no country is able to eradicate TB completely and policies that promote BCG vaccination should continue to be formulated. Using parameter values,,, we estimate Clearly, (i.e. ). 7. Conclusion In this paper, we discussed a mathematical model of TB with a focus on vaccination for the treatment of active TB. We derive and use it to find a threshold for the benefit of vaccination. Typically, vaccination is advantageous if it lowers and this condition is satisfied when the numerical values obtained for both an are compared. This phenomenon has important public health implications. First, a strategy of continuous BCG vaccination by countries with high prevalence of tuberculosis like Nigeria would result in a more stable DFE for disease eradication. Also, there would be significant reduction of TB incidence if coverage of vaccination is high. However, over the decades clinical experiments have shown that the protective efficacy of BCG often begins to wane in early adolescence. Though there is a relative efficacy in infants, I is still unclear why BCG fails to prevent pulmonary TB in adolescents. Thus, there is a strong need to evelop vaccines that can either oos CG s initial pri in an protective effects or replace CG by other vaccines. The old vaccines like BCG are neither designed to prevent infection nor to achieve eradication but to better prime or infection control.it is necessary that new vaccines should be designed and targeted at preventing initial infection or achieve sterile eradication of mycobacterium tuberculosis. 8. References Blower, S.M., NcLeen, A.R., Porco, T.C., Small, P.M.,Hopewll, P.C., Sanchez; M.A.and Moss, A.R. (1995). The intrinsic transmission dynamics of tuberculosis epidemics. Nat. Med., 1(8), Blower, S.M., Small, P.M. and Hopewell, P.C. (1996). Control strategies for tuberculosis epidemics: new models for old problems. Science, 273 (5274), Blower, S.M. and Gerberding, J.L. (1998). Understanding, predicting and controlling the emergence of drug-resistant tuberculosis: a theoretical framework. J. Mol. Med., 76(9), Blower, S.M. and Chon, T. (2004). Mo ellin the e er ence of the hot zones tu erculosis an the amplification dynamics of drug resistance. Nat. Med., 10(10), Bonah, C. (2005). he experi ental sta le of the CG vaccine: safety efficiency, proof and standards. Brosch, R., Gordon, S.V.,Garnier, T.and Eiglmeier, K. (2007). Genome plasticity of BCG and impact on vaccine efficacy. Proc. NatlAcad Sci., 104(13), Castillo-Chavez, C. and Feng, Z.(1997). To treat or not to treat: the case of tuberculosis. Castillo-Chavez, C. and Feng, Z.( 1998). Global stability of an age-structure model for TB and its applications to optimal vaccination strategies. Math. Biosci.,151 (2),

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