The contagion dynamics of vaccine skepticism

Abstract

In this manuscript, we discuss the spread of vaccine refusal through a non-linear mathematical model involving the interaction of vaccine believers, vaccine deniers, and the media sources. Furthermore, we hypothesize that the media coverage of disease-related deaths has the potential to increase the number of people who believe in vaccines. We analyze the dynamics of the mathematical model, determine the equilibria and investigate their stability. Our theoretical approach is dedicated to emphasizing the importance of convincing people to believe in the vaccine without getting into any medical arguments. For this purpose, we present numerical simulations that support the obtained analytical results for different scenarios.

Keywords

• vaccine refusal
• mathematical model
• stability

References

1. [1] A. Antos, M-L. Kwong, T. Balmorez, A. Villanueva, and S. Murakami, Unusually high risks of COVID-19 mortality with age-related comorbidities: An adjusted metaanalysis method to improve the risk assessment of mortality using the comorbid mortality data, Infect. Dis. Rep. 13 (3), 700–711, 2021.
2. [2] J. Arino, C.C. McCluskey, and P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, J. Appl. Math. 64 (1), 2002.
3. [3] C.T. Bauch and D.J.D. Earn, Vaccination and the theory of games, Proc. Natl. Acad. Sci. 101 (36), 13391–13394, 2004.
4. [4] C.T. Bauch, A.P. Galvani, and D.J.D. Earn, Group interest versus self-interest in smallpox vaccination policy, Proc. Natl. Acad. Sci. 100 (18), 10564–10567, 2003.
5. [5] C. Betsch, F. Renkewitz, T. Betsch, and C. Ulshöfer, The influence of vaccine-critical websites on perceiving vaccination risks, J. Health Psychol. 15 (3), 446–455, 2010.
6. [6] M-G. Cojocaru, C.T. Bauch, and M.D. Johnston, Dynamics of vaccination strategies via projected dynamical systems, Bull. Math. Biol. 69 (5), 1453–1476, 2007.
7. [7] O. Diekmann, J.A.P. Heesterbeek, and J.A.J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (4), 1990.
8. [8] K. Dietz and J.A.P. Heesterbeek, Daniel Bernoullis epidemiological model revisited, Math. Biosci. 180 (1), 1–21, 2002.

9. [9] X. Duan, S. Yuan, and X. Li, Global stability of an SVIR model with age of vaccination, Appl. Math. Comput. 226, 528-540, 2014.
10. [10] J.E. Edwardes, Vaccination and small-pox in England and other countries showing that compulsory re-vaccination is necessary, RCP Library, J. A. Churchill, 1892. Wiley Digital Archives: The Royal College of Physicians. Accessed 2021.
11. [11] D. Esernio-Jenssen and P. Offit, Don’t hesitate – vaccinate!, J. Child Adolesc. Trauma 13 (3), 337–341, 2020.
12. [12] S. Funk, E. Gilad, and V.A.A. Jansen, Endemic disease, awareness, and local behavioural response, J. Theor. Biol. 264 (2), 501–509, 2010.
13. [13] L. Gardner, E. Dong, K. Khan, and S. Sarkar, Persistence of US measles risk due to vaccine hesitancy and outbreaks abroad, Lancet Infect. Dis. 20 (10), 1114–1115, 2020.
14. [14] M. Gölgeli, A mathematical model of hepatitis B transmission in Turkey, Commun. Fac. Sci. 68 (2), 1586–1595, 2019.
15. [15] M. Gölgeli and F.M. Atay, Analysis of an epidemic model for transmitted diseases in a group of adults and an extension to two age classes, Hacet. J. Math. Stat. 49, 921–934, 2020.
16. [16] M.J. Hornsey, J. Lobera, and C. Díaz-Catalán, Vaccine hesitancy is strongly associated with distrust of conventional medicine, and only weakly associated with trust in alternative medicine, Soc. Sci. Med. 255, 113019, 2020.
17. [17] L. Huo, J. Jiang, S. Gong, and B. He, Dynamical behavior of a rumor transmission model with Holling-Type II functional response in emergency event, Phys. A: Stat. Mech. Appl. 450, 228–240, 2016.
18. [18] W.O. Kermack and A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Series A 115 (772), 700–721, 1927.
19. [19] M. Kröger, M. Turkyilmazoglu, and R. Schlickeiser, Explicit formulae for the peak time of an epidemic from the SIR model: Which approximant to use?, Phys. D: Nonlinear Phenom. 425, 132981, 2021.
20. [20] M. LaCour and T. Davis, Vaccine skepticism reflects basic cognitive differences in mortality-related event frequency estimation, Vaccine 38 (21), 3790–3799, 2020.
21. [21] K.M. Lisenby, K.N. Patel, and M.T. Uichanco, The role of pharmacists in addressing vaccine hesitancy and the measles outbreak, J. Pharm. Pract. 34 (1), 127–132, 2021.
22. [22] N.E. MacDonald, Vaccine hesitancy: Definition, scope and determinants, Vaccine 33 (34), 4161–4164, 2015.
23. [23] M. Martcheva, An Introduction to Mathematical Epidemiology, Springer-Verlag GmbH, 2015.
24. [24] MATLAB-Mathworks, MATLAB version 9.4.0.813654 (R2018a), The Mathworks, Inc., Natick, Massachusetts, 2018.
25. [25] C.J.E. Metcalf, V. Andreasen, O.N. Bjørnstad, K. Eames, W.J. Edmunds, S. Funk, T.D. Hollingsworth, J. Lessler, C. Viboud, and B.T. Grenfell, Seven challenges in modeling vaccine preventable diseases, Epidemics 10, 11–15, 2015.
26. [26] A.K. Misra, A. Sharma, and J.B. Shukla, Modeling and analysis of effects of awareness programs by media on the spread of infectious diseases, Math. Comput. Model. 53 (5-6), 1221–1228, 2011.
27. [27] A.K. Misra, A simple mathematical model for the spread of two political parties Nonlinear Anal.: Model Control 17 (3), 343–354, 2012.
28. [28] M. Motta, S. Sylvester, T. Callaghan, and K. Lunz-Trujillo, Encouraging COVID-19 vaccine uptake through effective health communication, Front. Polit. Sci. 3, 2021.
29. [29] A. Mukhopadhyay, A. De Gaetano, O. Arino, Modeling the intra-venous glucose tolerance test: A global study for a single-distributed-delay model, Discrete Contin. Dyn. Syst. Ser B. 4 (2), 407–417, 2004.
30. [30] X. Nan and K. Madden, HPV vaccine information in the blogosphere: How positive and negative blogs influence vaccine-related risk perceptions, attitudes, and behavioral intentions, Health Commun. 27 (8), 829–836, 2012.
31. [31] F. Nyabadza, T.Y. Alassey, and G. Muchatibaya, Modelling the dynamics of two political parties in the presence of switching, SpringerPlus 5 (1), Art. No. 1018, 2016.
32. [32] S.B. Omer, D.A. Salmon, W.A. Orenstein, M.P. deHart, and N. Halsey, Vaccine refusal, mandatory immunization, and the risks of vaccine-preventable diseases, N. Engl. J. Med. 360 (19), 1981–1988, 2009.
33. [33] N.N. Pelen and M. Gölgeli, Vector-borne disinformation during disasters and emergencies, Phys. A: Stat. Mech. Appl. 596, 127157, 2022.
34. [34] S. Samanta, S. Rana, A. Sharma, A.K. Misra, and J. Chattopadhyay, Effect of awareness programs by media on the epidemic outbreaks: A mathematical model, Comput. Appl. Math. 219 (12), 6965-6977, 2013.
35. [35] E. Shim, J.J. Grefenstette, S.M. Albert, B.E. Cakouros, and D.S. Burke, A game dynamic model for vaccine skeptics and vaccine believers: Measles as an example, J. Theor. Biol. 295, 194–203, 2012.
36. [36] E. Shim, B. Kochin, and A. Galvani, Insights from epidemiological game theory into gender-specific vaccination against rubella, Math. Biosci. Eng. 6 (4), 839–854, 2009.
37. [37] T.C Smith, Vaccine rejection and hesitancy: A review and call to action, Open Forum Infect. Dis. 4 (3), 2017.
38. [38] E. Tornatore, P. Vetro, and S.M. Buccellato, SIVR epidemic model with stochastic perturbation, Neural. Comput. Appl.24 (2), 309–315, 2012.
39. [39] TUIK, Adrese dayalı nüfus kayt sistemi sonuçları, 2019, Tech. report, 2020.
40. [40] M. Turkyilmazoglu, Explicit formulae for the peak time of an epidemic from the SIR model, Phys. D: Nonlinear Phenom. 422, 132902, 2021.
41. [41] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (1-2), 29–48, 2002.
42. [42] P. Verger and E. Dubé, Restoring confidence in vaccines in the COVID-19 era, Expert Rev. Vaccines 19 (11), 991–993, 2020.
43. [43] WHO, Ten threats to global health in 2019, World Health Organization, 2019.
44. [44] Y. Xiao, T. Zhao, and S. Tang, Dynamics of an infectious diseases with media/ psychology induced non-smooth incidence, Math. Biosci. Eng. 10 (2), 445–461, 2013.
45. [45] D. Zhao and S. Yuan, Persistence and stability of the disease-free equilibrium in a stochastic epidemic model with imperfect vaccine, Adv. Differ. Equ. 2016 (280), 2016.