Modelling the effects of pre-exposure and post-exposure vaccines in tuberculosis control

https://doi.org/10.1016/j.jtbi.2008.06.023Get rights and content

Abstract

Epidemic control strategies alter the spread of the disease in the host population. In this paper, we describe and discuss mathematical models that can be used to explore the potential of pre-exposure and post-exposure vaccines currently under development in the control of tuberculosis. A model with bacille Calmette-Guerin (BCG) vaccination for the susceptibles and treatment for the infectives is first presented. The epidemic thresholds known as the basic reproduction numbers and equilibria for the models are determined and stabilities are investigated. The reproduction numbers for the models are compared to assess the impact of the vaccines currently under development. The centre manifold theory is used to show the existence of backward bifurcation when the associated reproduction number is less than unity and that the unique endemic equilibrium is locally asymptotically stable when the associated reproduction number is greater than unity. From the study we conclude that the pre-exposure vaccine currently under development coupled with chemoprophylaxis for the latently infected and treatment of infectives is more effective when compared to the post-exposure vaccine currently under development for the latently infected coupled with treatment of the infectives.

Introduction

Infectious diseases remain a major cause of death and human misery. More than 95% of these deaths occur in developing countries (Folch et al., 2003, Franco-Paredes and Agudelo, 2002). Mycobacterium tuberculosis (Mtb) is one of the world's most devastating human pathogens. In 2004, more than nine million people developed active tuberculosis (TB) and two million people died from the disease making it the second leading cause of death worldwide (WHO, 2006). Currently, TB is the leading cause of death in HIV-infected individuals owing to the high incidence of dual TB–HIV infections, and TB affects not only the developing world but is also on the rise in parts of Europe (Andersen, 2006). TB is one of the diseases associated with resource-poor countries. It has historically attracted limited attention and investment into the development of new and improved drugs, vaccines and diagnostics all of which are desparately needed (Hussey et al., 2007). As a result, the prevention and treatment of TB has lagged behind most other diseases perceived to be a threat to the West.

The World Health Organisation (WHO, 1999) declared TB as a global emergency due to its medical, social, and economic consequences. Currently global control efforts targeting TB include treatment of latent TB infection to prevent endogenous reactivation, case detection and treatment with directly observed therapy short-course (DOTS) to cure active TB and BCG (bacille Calmette-Guerin) vaccination to prevent infection (WHO, 1999). Currently BCG is the only vaccine in use against TB and its limitations are well recognised (Bloom and Fine, 1994). BCG vaccine prevents the invasive complications of childhood TB such as meningitis, and miliary disease but provides variable protection against adult pulmonary disease. All these strategies have had a limited impact in reducing the burden of TB despite adherence to WHO recommendations on the use of DOTS. The original BCG vaccine was not cloned but was distributed to a number of laboratories worldwide. This resulted in a number of BCG related vaccines with varying phenotypic and genomic characteristics (Behr et al., 1999, Behr, 2002). This might explain why data from clinical and observational studies have shown widely disparate results with BCG vaccination (Iseman, 2000). Safety of the BCG vaccine in infants with compromised immunity is a controversial issue as some infants are suggested to acquire BCG related TB (Husseling et al., 2006).

The failure of the BCG vaccine and the increasing global TB-related mortality has emphasised the urgent need to develop more effective TB vaccines to control this scourge (Hussey et al., 2007). The ideal TB vaccine should be affordable, even in the poorest countries of the world, and it should be more cost-effective than BCG vaccine. It should be easily administered at or soon after birth and be safe, immunogenic and effective in all populations (Von Reyn and Vouya, 2002). It is believed that a novel vaccine would have maximal impact in TB-endemic regions if administered as a booster vaccine in adolescence before the highest risk period for the development of pulmonary TB (Young and Dye, 2006). The first generation of novel prophylactic vaccines have been designed either as live mycobacterial vaccines for replacing BCG or as subunit vaccines for boosting BCG to prevent immunity falling to a point at which it is ineffective (Doherty and Andersen, 2005; Skeiky and Sadoff, 2006).

Here we analyse different scenarios for pre-exposure and post-exposure vaccines currently under development using mathematical models. Some studies have been done in modelling the potential impact of pre-exposure and post-exposure vaccines (Ziv et al., 2004, Lietman and Blower, 2000, Salomon et al., 2000, Gomes et al., 2007). Ziv et al. (2004) did not take into account fast and slow dynamics of TB which this study considers. Most of these studies excluded exogenous reinfection and others dealt with only one intervention strategy making it difficult to compare the effects of pre-exposure and post-exposure vaccines. Our work differs from these studies in that our models in addition to treatment, also consider BCG vaccination at birth and aspects of exogenous reinfection. Further we incorporate chemoprophylaxis into the model with the pre-exposure vaccine currently under development. Thus the work enables the comparison of pre-exposure and post-exposure currently under development vaccines.

This paper is organised as follows. Section 2 presents a description and analysis of the model with BCG vaccination and exogenous re-infection. Section 3 presents the model with the pre-exposure vaccine currently under development coupled with treatment of infectives and chemoprophylaxis for the latently infected. Section 4 presents the model with the post-exposure vaccine currently under development coupled with treatment of the infective population. In Section 5 we present numerical simulations for the models. Finally we present a summary and concluding remarks in Section 6.

Section snippets

Model description

This section describes an epidemiological model for tuberculosis including vaccination with BCG. Said model divides the host population into the following subgroups that are unvaccinated susceptibles Su(t), vaccinated susceptibles Sv(t), exposed individuals with no history of pre-exposure vaccination Eu(t) (latently infected), exposed individuals with a history of pre-exposure vaccination Ev(t) (latently infected), infectives displaying TB symptoms I(t) and those recovered from active TB

Effects of pre-exposure vaccines currently under development

In this section we consider the possible benefits of pre-exposure vaccines currently under development. Pre-exposure vaccines are intended to prevent TB in people who have not been infected with Mtb. This is being driven by the fact that, (i) BCG vaccination is effective only for a short period of time from birth to about 20 years of age, (ii) BCG vaccination makes it difficult for chemoprophylaxis against latent forms of TB to be effected as BCG vaccinated individuals will have the same

Effects of post-exposure vaccines currently under development

In this section we explore the effects of possible post-exposure vaccines currently under development in the control of TB. Currently there are a number of different post-exposure vaccines at different stages of development including therapeutic vaccines, anti-latency vaccines and anti-reactivation vaccines. These include MVA85A which is a recombinant MVA expressing antigen 85A (Doherty and Rook, 2006, Stop TB Partnership, 2008) and Mycobacterium vaccae which is believed to act as an

Numerical simulations

In this section we carry out numerical simulations for model systems (3) and (43) using the fourth order Runge–Kutta numerical scheme coded in C++ programming language. The parameter values that we use for numerical simulations are in Table 1.

In Table 1, CSOZ means Central Statistics Office of Zimbabwe, a* denotes parameter values and ranges from Dye et al. (1999), Dye and William (2000), b* denotes parameter values from Bhunu et al. (2008) and c* denotes parameter values from Bao et al. (2007)

Summary and concluding remarks

Mathematical models have been presented and studied to assess the impact of BCG vaccine, pre-exposure and post-exposure vaccines currently under development. A model with BCG vaccination coupled with treatment of infectives is first presented and threshold quantities determined to assess the role of BCG vaccination in the control of TB epidemic. The centre manifold theory was used to establish the local asymptotic stability of the endemic equilibrium. The basic reproduction numbers obtained

Acknowledgements

C.P. Bhunu would like to acknowledge the financial support given to him by the International Clinical Operational Health Services Research Training Award (ICOHRTA) through the Biomedical Research Training Institute, Zimbabwe. The authors would like to thank the anonymous reviewers for the comments and suggestions which resulted in the improvement of our work.

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