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

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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 und Shashank Goel

Veröffentlicht/Copyright: 20. Januar 2025

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Aus der Zeitschrift Epidemiologic Methods Band 14 Heft 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.

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Received: 2024-04-20

Accepted: 2024-12-20

Published Online: 2025-01-20

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

Schlagwörter für diesen Artikel

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