Abstract
Hepatitis B (HepB) is one of the most common infectious diseases affecting over two billion people worldwide. About one third of all HepB cases are in China. In recent years, China made significant efforts to implement a nationwide HepB vaccination program and reduced the number of unvaccinated infants from 30 to 10%. However, many individuals still remain unprotected, particularly those born before 2003. Consequently, a catch-up retroactive vaccination is an important and potentially cost-effective way to reduce HepB prevalence. In this paper, we analyze a game theoretical model of HepB dynamics that incorporates government-provided vaccination at birth coupled with voluntary retroactive vaccinations. Given the uncertainty about the long-term efficacy of the HepB vaccinations, we study several scenarios. When the waning rate is relatively high, we show that this retroactive vaccination should be a necessary component of any HepB eradication effort. When the vaccine offers long-lasting protection, the voluntary retroactive vaccination brings the disease incidence to sufficiently low levels. Also, we find that the optimal vaccination rates are almost independent of the vaccination coverage at birth. Moreover, it is in an individual’s self-interest to vaccinate (and potentially re-vaccinate) at a rate just slightly above the vaccine waning rate.
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AC, SM, MN, and VP worked on this manuscript as part of the course MATH/BIOL 380—Introduction to mathematical biology. They wish to acknowledge the help and support of their classmates. The authors would also like to thank the anonymous reviewers for their comments and suggestions that helped to improve the manuscript.
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Appendix A: Equilibria of the Dynamics
Appendix A: Equilibria of the Dynamics
We have to solve the following system of (algebraic) equations.
1.1 Disease Free Equilibrium
In the disease free equilibrium when \(0=L_0=I_0=C_0=R_0\), the system (23)–(28) reduces to
Adding (29) and (30) together yields
By (30),
and thus
Now, we will determine the number of secondary infections caused by a single infected individual. The individual stays infected for a period \(\frac{1}{\mu _0+\gamma _1}\). During that period, at rate \(\beta S_0\), they infect susceptible individuals who become latently infected. Of those, only a fraction of \(\frac{\sigma }{\sigma +\mu _0}\) becomes acutely infected. Moreover, an infected individual can become a carrier with probability \(\frac{q\gamma _1}{\mu _0+\gamma _1}\). The individual stays in the carrier compartment for a period \(\frac{1}{\mu _0+\mu _1+\gamma _2}\). During that period, at rate \(\beta \epsilon S_0\), they infect susceptible individuals who then become latently infected. The carrier also makes other carriers at rate \(\mu \omega v\). Consequently, if l is a number of latently infected individuals caused by a single carrier, we get
and thus
Putting it all together, we get, as in Zou et al. (2010b)
The disease free equilibrium is stable when \(R_0<1\) (Zou et al. 2010b, Theorem 1).
1.2 Endemic Equilibrium
From (28), (26), and (27) we get
From (25),
which, after dividing by \(I^*\ne 0\), yields
From (24),
and finally, by (23)
which yields
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Chouhan, A., Maiwand, S., Ngo, M. et al. Game-Theoretical Model of Retroactive Hepatitis B Vaccination in China. Bull Math Biol 82, 80 (2020). https://doi.org/10.1007/s11538-020-00748-5
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DOI: https://doi.org/10.1007/s11538-020-00748-5