Understanding the impact of media and latency in information response on the disease propagation: a mathematical model and analysis

Bharat Kaushik; Sumit Kaur Bhatia; Sanyam Tyagi; Shashank Goel

doi:10.1515/em-2024-0011

Article

Understanding the impact of media and latency in information response on the disease propagation: a mathematical model and analysis

Bharat Kaushik , Sumit Kaur Bhatia , Sanyam Tyagi and Shashank Goel

Published/Copyright: January 20, 2025

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From the journal Epidemiologic Methods Volume 14 Issue 1

Abstract

Exposure to the media can have a significant impact on the transmission of infectious diseases. In this study, we examine the role and importance of information intervention in the fight against infectious illnesses. A four-class mathematical model: H _S, H _I, H _R, and H _Z has been developed. The first three classes, denoted as H _S, H _I, and H _R, stand for the population’s subclasses, susceptible, infected, and removed people, respectively. The fourth class, denoted as H _Z, stands for the population’s information density. Another component of this model is time delay (τ), which accounts for the delay in people’s taking preventive action due to their decision-making period between the report and action times. Essential properties, like boundedness and non negativity of the solutions of the model have been studied. We have addressed the existence and local stability of both the equilibrium points: disease-free and endemic. Conditions for global stability of disease-free and endemic equilibrium points have been obtained. Also, the critical value of delay is obtained, which, when crossed leads to a switch in stability and thus Hopf bifurcation. Finally, to support these analytical findings, numerical simulations have been performed. We have explored the relationship between the number of infective people and the rate of illness transmission, information intervention, and reaction intensity. We have also fitted our model to the actual data of COVID-19 of India for the month of April 2021. Based on the findings, policymakers are recommended to create suitable policies that limit the degree of disease transmission and accelerate the dissemination of information.

Keywords: communicable disease model; media effect; equilibrium points; stability analysis; global stability analysis; sensitivity analysis

Corresponding author: Sumit Kaur Bhatia, Amity Institute of Applied Science, Amity University Uttar Pradesh, Noida, U.P., India, E-mail: sumit2212@gmail.com

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The authors state no conflict of interest.
Research funding: No funding was recevied.
Data availability: The data that supports the findings of this study are openly available and the relevant papers have been cited in the article.

Appendix A

T * = M + L e − λ τ ,

where

M = M 11 M 12 M 13 0 M 21 M 22 0 0 M 31 M 32 M 33 0 0 M 42 0 M 44 and L = L 11 0 0 L 14 0 0 0 0 L 31 0 0 L 34 0 0 0 0

with M 11 = r − 2 r H S * κ − β H I * − μ m H Z * , M 12 = − β H S * , M 13 = δ , M 14 = 0 , M 21 = β H I * , M 22 = β H S * − ( ρ + v + γ ) = 0 , M 23 = 0 , M 24 = 0 , M 31 = μ m H Z * , M 32 = γ , M 33 = − ( ρ + δ ) , M 34 = 0 , M 41 = 0 , M 42 = α 0 1 + β 0 H I * 2 , M 43 = 0 , M 44 = − α 1 . L 11 = 0 , L 12 = 0 , L 13 = 0 , L 14 = − μ m H S * , L 21 = 0 , L 22 = 0 , L 23 = 0 , L 24 = 0 , L 31 = 0 , L 32 = 0 , L 33 = 0 , L 34 = μ m H S * , L 41 = 0 , L 42 = 0 , L 43 = 0 , L 44 = 0 .

A 1 = − M 11 − M 33 − M 44 , A 2 = − M 12 M 21 − M 13 M 31 + M 11 M 33 + M 11 M 44 + M 33 M 44 , A 3 = − M 13 M 21 M 32 + M 12 M 21 M 33 + M 12 M 21 M 44 + M 13 M 31 M 44 − M 11 M 33 M 44 , A 4 = M 13 M 21 M 32 M 44 − M 12 M 21 M 33 M 44 , B 1 = 0 , B 2 = 0 , B 3 = − M 21 M 42 L 14 , B 4 = M 21 M 33 M 42 L 14 − M 13 M 21 M 42 L 34 .

Appendix B

Following is an outline of the general MATLAB code to carry out genetic algorithm on a differential equations based model.

PS=100;

Parms=12; % number of parameters to be fitted

optsAns=optimoptions (‘ga’, ‘PopulationSize’, PS,…

‘InitialPopulationMatrix’, (rand(PS, Parms)),…

‘MaxGenerations’, 2000, ‘FunctionTolerance’, 1E-5);

[para, fval, exitflag, output, population, scores]=ga(ftns, Parms, [], [], [], [],…

zeros(Parms, 1), Upperbounds, [], [], optsAns);

Firstly, a fitness function of parameters should be constructed appropriately for any given dynamical model. This fitness function utilizes the solution output of a differential equation solver like ode45 or dde23, evaluates the model outputs at the given data points. This function is fed into the ga() function and initializing the options of ga, one can obtain the fitted parameters.

References

1. Tulchinsky, TH, EA Varavikova. Communicable diseases. The new public health. Amsterdam: Academic Press; 2014:149–236 pp.10.1016/B978-0-12-415766-8.00004-5Search in Google Scholar

2. The Top 10 Causes of Death. Available from: https://www.who.int/news-room/fact-sheets/detail/the-top-10-causes-of-death.Search in Google Scholar

3. Negri, S. Communicable disease control. In: Research handbook on global health law. Edward Elgar Publishing; 2018:265–302 pp.10.4337/9781785366543.00018Search in Google Scholar

4. Zhang, Z, Kundu, S, Tripathi, JP, Bugalia, S. Stability and Hopf bifurcation analysis of an SVEIR epidemic model with vaccination and multiple time delays. Chaos, Solitons Fractals 2020;131:109483. https://doi.org/10.1016/j.chaos.2019.109483.Search in Google Scholar

5. Influenza (Seasonal). Available from: https://www.who.int/news-room/fact-sheets/detail/influenza-(seasonal).Search in Google Scholar

6. WHO COVID-19 Dashboard. Available from: https://data.who.int/dashboards/covid19/deaths?n=c.Search in Google Scholar

7. Al Basir, F. Dynamics of infectious diseases with media coverage and two time delay. Math Models Comput Simul 2018;10:770–83. https://doi.org/10.1134/s2070048219010071.Search in Google Scholar

8. Hethcote, HW. Three basic epidemiological models. In: Applied mathematical ecology. Berlin, Heidelberg: Springer; 1989:119–44 pp.10.1007/978-3-642-61317-3_5Search in Google Scholar

9. Hethcote, HW. The mathematics of infectious diseases. SIAM Rev 2000;42:599–653. https://doi.org/10.1137/s0036144500371907.Search in Google Scholar

10. Kaddar, A. On the dynamics of a delayed SIR epidemic model with a modified saturated incidence rate. Electron J Differ Equ 2009;2009:1–7.Search in Google Scholar

11. Tang, S, Xiao, Y, Yuan, L, Cheke, RA, Wu, J. Campus quarantine (Fengxiao) for curbing emergent infectious diseases: lessons from mitigating A/H1N1 in Xi’an, China. J Theor Bio 2012;295:47–58. https://doi.org/10.1016/j.jtbi.2011.10.035.Search in Google Scholar PubMed

12. Zhang, J, Lou, J, Ma, Z, Wu, J. A compartmental model for the analysis of SARS transmission patterns and outbreak control measures in China. Appl Math Comput 2005;162:909–24. https://doi.org/10.1016/j.amc.2003.12.131.Search in Google Scholar PubMed PubMed Central

13. Bajiya, VP, S Bugalia, JP Tripathi. Mathematical modeling of COVID-19: impact of non-pharmaceutical interventions in India. Chaos 2020;30:113143. https://doi.org/10.1063/5.0021353.Search in Google Scholar PubMed

14. Cay, I. On the local and global stability of an SIRS epidemic model with logistic growth and information intervention. Turk J Math 2021;45:1668–77. https://doi.org/10.3906/mat-2103-116.Search in Google Scholar

15. Kumar, A, Takeuchi, Y, Srivastava, PK. Stability switches, periodic oscillations and global stability in an infectious disease model with multiple time delays. Math Biosci Eng 2023;20:11000–32. https://doi.org/10.3934/mbe.2023487.Search in Google Scholar PubMed

16. Kassa, SM, Ouhinou, A. The impact of self-protective measures in the optimal interventions for controlling infectious diseases of human population. J Math Biol 2015;70:213–36. https://doi.org/10.1007/s00285-014-0761-3.Search in Google Scholar PubMed

17. Liu, Y, Cui, JA. The impact of media coverage on the dynamics of infectious disease. Int J Biomath 2008;1:65–74. https://doi.org/10.1142/s1793524508000023.Search in Google Scholar

18. Misra, AK, Sharma, A, Singh, V. Effect of awareness programs in controlling the prevalence of an epidemic with time delay. J Biol Syst 2011;19:389–402. https://doi.org/10.1142/s0218339011004020.Search in Google Scholar

19. Cui, J, Sun, Y, Zhu, H. The impact of media on the control of infectious diseases. J Dyn Differ Equ 2008;20:31–53. https://doi.org/10.1007/s10884-007-9075-0.Search in Google Scholar PubMed PubMed Central

20. Manfredi, P, A D’Onofrio, editors. Modeling the interplay between human behavior and the spread of infectious diseases. New york: Springer Science & Business Media; 2013.10.1007/978-1-4614-5474-8Search in Google Scholar

21. Gani, SR, Halawar, SV. Optimal control for the spread of infectious disease: the role of awareness programs by media and antiviral treatment. Optim Control Appl Methods 2018;39:1407–30. https://doi.org/10.1002/oca.2418.Search in Google Scholar

22. Avila-Vales, E, Pérez, ÁG. Dynamics of a time-delayed SIR epidemic model with logistic growth and saturated treatment. Chaos, Solitons Fractals 2019;127:55–69. https://doi.org/10.1016/j.chaos.2019.06.024.Search in Google Scholar

23. Li, T, Xiao, Y. Complex dynamics of an epidemic model with saturated media coverage and recovery. Nonlinear Dyn 2022;107:2995–3023. https://doi.org/10.1007/s11071-021-07096-6.Search in Google Scholar PubMed PubMed Central

24. Tipsri, S, Chinviriyasit, W. Stability analysis of SEIR model with saturated incidence and time delay. Int J Appl Phys Math 2014;4:42–5. https://doi.org/10.7763/ijapm.2014.v4.252.Search in Google Scholar

25. Arya, N, Bhatia, SK, Kumar, A. Stability and bifurcation analysis of a contaminated sir model with saturated type incidence rate and Holling type-III treatment function. Commun Math Biol Neurosci 2022;2022:3.Search in Google Scholar

26. Van den Driessche, P, Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math Biosci 2002;180:29–48. https://doi.org/10.1016/s0025-5564(02)00108-6.Search in Google Scholar PubMed

27. Singh, HP, Bhatia, SK, Bahri, Y, Jain, R. Optimal control strategies to combat COVID-19 transmission: a mathematical model with incubation time delay. Results Control Optim 2022;9:100176. https://doi.org/10.1016/j.rico.2022.100176.Search in Google Scholar

28. Wang, X. A simple proof of Descartes’s rule of signs. Amer Math Mon 2004;111:525–6. https://doi.org/10.2307/4145072.Search in Google Scholar

29. Yang, X. Generalized form of Hurwitz-Routh criterion and Hopf bifurcation of higher order. Appl Math Lett 2002;15:615–21. https://doi.org/10.1016/s0893-9659(02)80014-3.Search in Google Scholar

30. La Salle, JP. The stability of dynamical systems. Philadelphia: Society for Industrial and Applied Mathematics; 1976.10.1137/1.9781611970432Search in Google Scholar

31. Coronavirus Data for India. Available from: https://www.worldometers.info/coronavirus/country/india/.Search in Google Scholar

32. Population of India. Available from: https://www.worldometers.info/world-population/india-population/.Search in Google Scholar

33. Ministry of Health and Family Welfare. Available from: https://mohfw.gov.in/.Search in Google Scholar

34. Greenhalgh, D, Rana, S, Samanta, S, Sardar, T, Bhattacharya, S, Chattopadhyay, J. Awareness programs control infectious disease–multiple delay induced mathematical model. Appl Math Comput 2015;251:539–63. https://doi.org/10.1016/j.amc.2014.11.091.Search in Google Scholar

Received: 2024-04-20

Accepted: 2024-12-20

Published Online: 2025-01-20

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Articles in the same Issue

https://doi.org/10.1515/em-2024-0011

Keywords for this article

communicable disease model; media effect; equilibrium points; stability analysis; global stability analysis; sensitivity analysis