Mathematical Assessment of the Impact of the Imperfect Vaccination on Diphtheria Transmission Dynamics
Abstract
:1. Introduction
2. Model Formulation
3. Analysis of the Diphtheria-Vaccine Model
3.1. Basic Properties
3.2. Disease Free Equilibrium and Reproductive Number
3.3. Endemic Equilibrium and Local Stability
3.4. Global Stability Analysis
3.5. Vaccine-Induced Herd Immunity Threshold
4. Numerical Simulations
4.1. Appropriate Model Parameters
4.2. Sensitivity Analysis of the Reproductive Number
4.3. Effect of Vaccination Rate
4.4. Effect of Asymptomatic Individuals on Spreading Diphtheria
5. Discussion and Conclusions
- (i)
- The threshold value called the basic reproductive number under vaccination of the diphtheria-vaccine model, denoted by , is derived by using the next generation method. It is found that the disease-free equilibrium of the diphtheria-vaccine model is globally asymptomatically stable whenever in the sense that routine vaccination against diphtheria can lead to the effective control or elimination of diphtheria if it can bring (and maintain) . Furthermore, the critical rate of vaccination () and the threshold vaccine-induced community herd immunity of the proposed model () are derived. It is found that is identical to the formula of herd immunity (also called community immunity) which is a new result found in this study.
- (ii)
- Based on constructing the suitable Lyapunov functions, it is found that the endemic equilibrium of the proposed model is globally asymptotically stable whenever . The epidemiological implication of these results is that the community transmission of diphtheria can be significantly curtailed if and the disease still persists in the community if , that is, the vaccination program adopted is not effective. The implication of global stability of equilibrium is verified that the solution of the diphtheria-vaccine model converges to the correct equilibria irrespective of the initial sizes of the six state variables. Our simulations show that if the initial sizes of sub-populations have fluctuated, they will affect fast (slow) convergence to a correct equilibrium. This finding is interesting because the initial sizes of sub-populations, especially, susceptible and vaccinated individuals are major factors in either eliminating diphtheria or controlling diphtheria spreads before the next vaccine type is taken.
- (iii)
- The appropriate model parameters given in Table 1 are obtained by comparing the cumulative number of diphtheria cases produced by the diphtheria-vaccine model with the real cases in Thailand in 2018. The sensitivity analysis of has demonstrated that the rate of vaccination is the most sensitive to . Contour plots of suggest the combined control measures should be addressed on the rate of vaccination and the incubation period of asymptomatic individuals. Further, we consider the asymptomatic class as a separate population because this population can spread the infection without being sick themselves. The study results obtained indicate that the incubation period of asymptomatic individuals has an impact on the optimal vaccination coverage level needed for diphtheria eradication, see Table 2. Our simulations also show that when the vaccination coverage level is greater than the threshold value , asymptomatic individuals still persist in the community for some period of time, even though the infected individuals decrease and are eventually eliminated. In epidemiology, this result supports the possibility of asymptomatic infection being related to antibody decay due to waning and not boosting immunity [54]. Although it is well known that a low rate of vaccination has had an impact on the vaccination proportion resulting in the duration of diphtheria protection, this is the first time to investigate the impact of asymptomatic infection on the vaccination coverage for the past and recent vaccination coverage levels have an effect on the duration of diphtheria’s protection, and it is also the cause of discovering the patients in the different age groups [6]. Therefore, our study suggests that the officers involving disease control should be concerned not only with maintaining the coverage level needed for the primary vaccination but they should be concerned to maintain the boosting vaccination coverage level for all adults every 10 years at least the threshold coverage level () in order to significantly halt the spread of diphtheria in the community.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Proof of Theorem 4
References
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Parameter | Definition | Value | Reference |
---|---|---|---|
Transmission rate | 18.5 | data fit | |
a | Proportion of infectious population, | 0.55 | [64] |
Modification parameter, | 0.7 | [64] | |
Rate of vaccination | 0.0406 | [10] | |
r | Birth rate | 0.0101 | [65] |
Rate of progression from the exposed class to either the asymptomatic class or the infected class | 6 | [1] | |
Natural death rate | 0.0011 | [66] | |
Diphtheria mortality rate | [1] | ||
Rate of waning vaccine | [1] | ||
Rate of progression from the recovered class to the vaccinated class | [1] | ||
Recovered rate of asymptomatic individuals | [1] | ||
Recovered rate of infected individuals | [1] | ||
K | Carrying capacity | - |
5.94 | 1.1398 | 0.0476 | 0.8351 | 0.8347 | 1 | 4 | 0 |
4.26 | 1.2657 | 0.0539 | 0.8515 | 0.8509 | 3 | 6 | 2 |
2.70 | 1.3826 | 0.0597 | 0.8640 | 0.8633 | 4 | 5 | 6 |
2.10 | 1.4276 | 0.0619 | 0.8683 | 0.8675 | 4 | 4 | 7 |
1.26 | 1.4905 | 0.0651 | 0.8739 | 0.8730 | 5 | 3 | 10 |
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Kanchanarat, S.; Chinviriyasit, S.; Chinviriyasit, W. Mathematical Assessment of the Impact of the Imperfect Vaccination on Diphtheria Transmission Dynamics. Symmetry 2022, 14, 2000. https://doi.org/10.3390/sym14102000
Kanchanarat S, Chinviriyasit S, Chinviriyasit W. Mathematical Assessment of the Impact of the Imperfect Vaccination on Diphtheria Transmission Dynamics. Symmetry. 2022; 14(10):2000. https://doi.org/10.3390/sym14102000
Chicago/Turabian StyleKanchanarat, Siwaphorn, Settapat Chinviriyasit, and Wirawan Chinviriyasit. 2022. "Mathematical Assessment of the Impact of the Imperfect Vaccination on Diphtheria Transmission Dynamics" Symmetry 14, no. 10: 2000. https://doi.org/10.3390/sym14102000