Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

The change of susceptibility following infection can induce failure to predict outbreak potential by $\mathcal{R}_{0}$

1 Department of Mathematical Sciences, Shimane University, 1060 Nishikawatsu-cho, Matsue, Japan
2 Division of Bioinformatics, Research Center for Zoonosis Control, Hokkaido University, Sapporo, Hokkaido, Japan, JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332–0012, Japan

Time-varying individual host susceptibility to a disease due to waning and boosting of immunity is known to induce rich long-term behavior of the disease transmission dynamics. Simultaneously, the impact of the time-varying heterogeneity of host susceptibility on the short-term behavior of epidemics is not well-studied despite the availability of a large amount of epidemiological data on short-term epidemics. This paper proposes a parsimonious mathematical model describing the short-term transmission dynamics by taking into account waning and enhancing susceptibility following the infection. In addition to the common classification in the standard SIR model, i.e., "no epidemic" as $\mathcal{R}_{0}\leq1$ or normal epidemic as $\mathcal{R}_{0}>1$, the proposed model also shows the "delayed epidemic" class when an epidemic takes off after the negative slope of the epidemic curve at the initial phase of the epidemic. The condition for each of the three classes is derived based on the obtained explicit solution for the proposed model.
  Figure/Table
  Supplementary
  Article Metrics

References

1. R.M. Anderson and R.M. May, Infectious diseases of humans: dynamics and control, Vol. 28. Oxford: Oxford University Press, 1992.

2. J. Arino, C. C. McCluskey and P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260–276.

3. N. Arinaminpathy, J. S. Lavine and B. T. Grenfell, Self-boosting vaccines and their implications for herd immunity, PNAS, 109 (2012), 20154-20159.

4. M.V. Barbarossa, A. Denes, G. Kiss, Y. Nakata and G. R´'ost, Zs. Vizi, Transmission dynamics and final epidemic size of Ebola Virus Disease outbreaks with varying interventions, PLoS One, 10 (2015), e0131398.

5. W. Dejnirattisai, A. Jumnainsong, N. Onsirisakul, P. Fitton, S. Vasanawathana, W. Limpitikul, C. Puttikhunt, C. Edwards, T, Duangchinda , S. Supasa, K. Chawansuntati, P. Malasit, J. Mongkolsapaya and G. Screaton, Cross-reacting antibodies enhance dengue virus infection in humans, Science, 328 (2010), 745-748.

6. F. Brauer, Backward bifurcations in simple vaccination models, J. Math. Anal. Appl., 29 (2004), 418–431.

7. O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $\mathcal{R}_{0}$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382.

8. O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, The legacy of Kermack and McKendrick, Epidemic Models: Their Structure and Relation to Data, (D. Mollison, ed.) (1995), 95–115.

9. O. Diekmann and R. Montijn, Prelude to Hopf bifurcation in an epidemic model: analysis of a characteristic equation associated with a nonlinear Volterra integral equation, J. Math. Biol., 14 (1982), 117–127.

10. O. Diekmann, H. Heesterbeek and T. Britton, Mathematical tools for understanding infectious disease dynamics, Princeton University Press, 2012.

11. W.J. Edmunds, O.G. Van de Heijden, M. Eerola and N. J. Gay, Modelling rubella in Europe, Epidemiol. Infect., 125 (2000), 617–634.

12. P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48.

13. M.G.M. Gomes, L.J. White and G.F. Medley, Infection, reinfection, and vaccination under suboptimal immune protection: epidemiological perspectives, J. Theor. Biol., 228 (2004), 539–549.

14. D. Greenhalgh, O. Diekmann and M. C. M de Jong, Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity, Math. Biosci., 165 (2000), 1–25.

15. C. Hall, E. Walsh, C. Long and K. Schnabel, Immunity to and frequency of reinfection with respiratory syncytial virus, J. Infect. Dis., 163(1991), 693–698.

16. H.W. Hethcote, H.W. Stech and P. van den Driessche, Nonlinear oscillations in epidemic models, SIAM J. Appl. Math., 40 (1981), 1–9.

17. H. Inaba, Kermack and McKendrick revisited: the variable susceptibility model for infectious diseases, J. Ind. Appl. Math., 18 (2001), 273–292.

18. H. Inaba, Endemic threshold analysis for the Kermack-McKendrick reinfection model, Josai Math. Mono., 9 (2016), 105–133.

19. H. Inaba, Age-structured population dynamics in demography and epidemiology, Springer, 2017.

20. D. Isaacs, D. Flowers, J. R. Clarke, H. B. Valman and M. R. MacNaughton, Epidemiology of coronavirus respiratory infections, Arch. Dis. Child., 58 (1983), 500–503.

21. G. Katriel, Epidemics with partial immunity to reinfection, Math. Biosci., 228 (2010), 153–159.

22. W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics-II. The problem of endemicity. Proceedings of the Royal Society, 138A (1932), 55–83; Reprinted in Bull, Math. Biol., 53 (1991), 57–87.

23. W.O. Kermack and A.G. McKendrick, Contributions to the mathematical theory of epidemics-III. Further studies of the problem of endemicity. Proceedings of the Royal Society, 141A (1933), 94–122; Reprinted in Bull, Math. Biol., 53 (1991), 89–118.

24. R. Kohlmann, A. Salmen and A. Chan, Serological evidence of increased susceptibility to varicella-zoster virus reactivation or reinfection in natalizumab-treated patients with multiple sclerosis, Mult. Scler. J., 21 (2015), 1823–1832.

25. J.R. Kremer, F. Schneider and C.P. Muller, Waning antibodies in measles and rubella vaccinees-a longitudinal study, Vaccine, 24 (2006), 2594–2601.

26. C.M. Kribs-Zaleta and J.X. Velasco-Hernandez, A simple vaccination model with multiple endemic states, Math. Biosci., 164 (2000), 183–201.

27. J.S. Lavine, A. A. King and O. N. Bjørnstad, Natural immune boosting in pertussis dynamics and the potential for long-term vaccine failure, PNAS, 108 (2011), 7259–7264.

28. Y. Nakata, Y. Enatsu, H. Inaba, T. Kuniya, Y. Muroya and Y. Takeuchi, Stability of epidemic models with waning immunity, SUT J. Math., 50 (2014), 205–245.

29. C.A. Siegrist, Vaccine immunology, Vaccines, 5 (2008), 1725.

30. T.J.W. van de Laar, R. Molenkamp, C. V. D. Berg, Frequent HCV reinfection and superinfection in a cohort of injecting drug users in Amsterdam, J. Hepatol., 51 (2009), 667–674.

31. S. Verver, Rate of reinfection tuberculosis after successful treatment is higher than rate of new tuberculosis, Am. J. Resp. Crit. Care, 171 (2005), 1430–1435.

32. G. Wei, H. Bull, X. Zhou and H. Tabel, Intradermal infections of mice by low numbers of African trypanosomes are controlled by innate resistance but enhance susceptibility to reinfection, J.Infect. Dis., 203 (2011), 418–429.

33. S.S. Whitehead, J.E. Blaney, A.P. Durbin and B.R. Murphy, Prospects for a dengue virus vaccine, Nat. Rev. Microbiol., 5 (2007), 518–528.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved