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Mathematical Analysis of the role of Information on the Dynamics of Typhoid Fever

Nyanga Honda Masasila, Rigobert Charles Ngeleja

TL;DR

This work develops a deterministic compartmental model coupling humans $(S,I,R)$ with an environmental reservoir $B$ to study how information-driven behavior affects typhoid transmission. Using a next-generation framework, it derives the basic reproduction number $R_0$ and analyzes both disease-free and endemic equilibria, establishing local and global stability results via Metzler matrix techniques and Lyapunov functions. The analysis shows that the disease-free state is globally stable when $R_0<1$, while a unique endemic equilibrium exists and is globally stable for $R_0>1$, with numerical simulations illustrating how information ($\rho$) can shift transmission dynamics and peak infection. The findings underscore the importance of accurate information dissemination and precautionary behaviors in controlling typhoid fever in settings where environmental transmission is significant.

Abstract

We consider a deterministic mathematical model to study the role of Information on the dynamics of Typhoid Fever. We analyses the model to study its boundedness and compute the threshold value known as basic reproduction number for determination of number of secondary cases and establishment of the the condition for local and global asymptotic stability of the stationary points. The numerical simulation is used to depict dynamical behavior of Typhoid Fever in the considered population. The results indicate a clear role of information in influencing a behavior change that may in a way lead to an increase in the transmission typhoid fever. The result further show that the increase of the number of individuals with Typhoid fever is greatly influenced by failure of the people to follow health precaution that reduce the spread of the disease. The result necessitate the importance of the government to continue educating and/or giving information to her people on the behavior that in one way or another may lead to the increase of the transmission of typhoid fever which should also be an importance topic to be discussed when planning for any control strategies against the disease.

Mathematical Analysis of the role of Information on the Dynamics of Typhoid Fever

TL;DR

This work develops a deterministic compartmental model coupling humans with an environmental reservoir to study how information-driven behavior affects typhoid transmission. Using a next-generation framework, it derives the basic reproduction number and analyzes both disease-free and endemic equilibria, establishing local and global stability results via Metzler matrix techniques and Lyapunov functions. The analysis shows that the disease-free state is globally stable when , while a unique endemic equilibrium exists and is globally stable for , with numerical simulations illustrating how information () can shift transmission dynamics and peak infection. The findings underscore the importance of accurate information dissemination and precautionary behaviors in controlling typhoid fever in settings where environmental transmission is significant.

Abstract

We consider a deterministic mathematical model to study the role of Information on the dynamics of Typhoid Fever. We analyses the model to study its boundedness and compute the threshold value known as basic reproduction number for determination of number of secondary cases and establishment of the the condition for local and global asymptotic stability of the stationary points. The numerical simulation is used to depict dynamical behavior of Typhoid Fever in the considered population. The results indicate a clear role of information in influencing a behavior change that may in a way lead to an increase in the transmission typhoid fever. The result further show that the increase of the number of individuals with Typhoid fever is greatly influenced by failure of the people to follow health precaution that reduce the spread of the disease. The result necessitate the importance of the government to continue educating and/or giving information to her people on the behavior that in one way or another may lead to the increase of the transmission of typhoid fever which should also be an importance topic to be discussed when planning for any control strategies against the disease.
Paper Structure (21 sections, 5 theorems, 76 equations)

This paper contains 21 sections, 5 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Model development
  3. Model Description
  4. Description of interaction
  5. Variable and parameters and their description
  6. Model Assumption
  7. Model Equation
  8. Properties of the typhoid fever model
  9. Model's invariant region
  10. Positivity of the solution
  11. Analysis of the Model
  12. Disease Free Equilibrium
  13. Computation of the Basic Reproduction Number $R_0$
  14. Steady State and Stability of the Critical Points
  15. Disease Free Equilibrium
  16. ...and 6 more sections

Key Result

Theorem 4.1

All forward model solutions in $R^{4}_{+}$ of the typhoid fever model system are feasible $\forall t \geq 0$ if they go in the invariant region $\Lambda$ for $\Lambda =\omega_1\times\omega_2$

Theorems & Definitions (7)

  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • Theorem 6.1
  • Theorem 6.2
  • Theorem 6.3