TRANSMISSION OF TYPHOID FEVER THROUGH DELAYED STRATEGIES

Abstract

This study analyzes the transmission of typhoid fever caused by Salmonella typhi using a mathematical model that highlights the significance of delay in its effectiveness. Time delays can affect the nature of patterns and slow down the emergence of patterns in infected population density. The analyzed model is expanded with the equilibrium analysis, reproduction number, and stability analysis. This study aims to establish and explore the nonstandard finite difference (NSFD) scheme for the typhoid fever virus transmission model with a time delay. Ln addition, the forward Euler method and Runge-Kutta method of order 4 (RK-4) are also applied in the present research. Some significant properties, such as convergence, positivity, boundedness, and consistency, are explored, and the proposed scheme preserves all the mentioned properties. The theoretical validation is conducted on how NSFD outperforms other methods in emulating key aspects of the continuous model, such as positive solution, stability, and equilibrium about delay. Hence, the above analysis also shows some of the limitations of the conventional finite difference methods, such as forward Euler and RK-4 in simulating such critical behaviors. This becomes more apparent when using larger steps. This indicated that NSFD is beneficial in identifying the essential characteristics of the continuous model with higher accuracy than the traditional approaches.

Specification

PREAMBLE TO THE DESORPTION
THE FIELD OF INVENTION
This study pertains to the field of mathematical epidemiology, specifically modeling the transmission of infectious diseases like typhoid fever. It incorporates the application of delay differential equations (DDEs) to explore the effects of time delays in the spread of the disease.
BACKGROUND OF THE INVENTION
Typhoid fever is caused by Salmonella typhi, which is transmitted through contaminated food and water. Despite advancements in water sanitation, the disease remains a significant health issue in developing countries. Mathematical models play a crucial role in understanding how the disease spreads and in evaluating the effectiveness of control strategies. Previous models have focused on direct transmission, but few have incorporated the role of time delays, which represent incubation periods or other delays in disease dynamics. This research aims to address that gap by introducing a delay factor into the standard SELR (Susceptible-Exposed-Infectious-Recovered) model.
SUMMARY OF THE INVENTION
This study proposes a modified SEIR model that includes a time delay to better capture the dynamics of typhoid fever transmission. The model demonstrates that time delays can significantly influence the progression of infection in a population. The research compares the performance of different numerical methods (such as Euler and Runge-Kutta) with the proposed Non-Standard Finite Difference (NSFD) scheme, showing that NSFD is superior in maintaining stability, positivity, and accuracy in simulating delayed epidemic models. The model has practical implications for improving the prediction of disease outbreaks and the formulation of public health interventions.
Specifications
o Focus: The research belongs to the field of mathematical epidemiology, focusing on disease transmission modeling. It examines the effects of time delays in the progression of infectious diseases like typhoid fever.
o The study aims to improve the accuracy of typhoid fever transmission models by incorporating time delays, which reflect the incubation period and other delays in disease progression. It also evaluates numerical methods like Non-Standard Finite Difference (NSFD) in comparison to traditional methods like Euler and Runge-Kutta.
o Introduction of delay parameters in the SEIR model to capture the dynamics of disease spread.
o Exploration of stability, positivity, and convergence properties using the NSFD scheme.
o Validation of the model with real-world data and comparison with non-delay models for improved public health strategy formulation.
DESCRIPTION
* Objective: The primary objective of this study is to develop a more accurate mathematical model for understanding the transmission dynamics of typhoid fever by incorporating time delays. These delays represent incubation periods and other factors affecting the disease's progression. The model seeks to improve the accuracy of predictions related to typhoid fever outbreaks and support the design of more effective public health interventions. It also explores various numerical methods, comparing the performance of the Non-Standard Finite Difference (NSFD) scheme against traditional methods like Euler and Runge-Kutta.
• Novel Perspective: The innovative aspect of this research lies in the integration of time-delay differential equations (DDEs) into the classical SE1R epidemic model. Time delays are essential in reflecting real-world dynamics, such as the latency period before an infected individual becomes contagious. The study demonstrates that introducing delays can alter the disease's progression patterns, helping in more precise modeling of epidemic scenarios. Additionally, the research highlights the superiority of the NSFD method over traditional techniques, particularly in maintaining stability, positivity, and convergence in simulations, regardless of step sizes. This novel approach enhances the reliability of predictions in the context of typhoid fever and similar infectious diseases.
We Claim
1. 1. Incorporation of Time Delays: We claim that incorporating time delays into the classical SEIR epidemic model significantly enhances the accuracy of predicting disease transmission, specifically for typhoid fever, by accounting for latency periods and other real-world temporal factors in the infection process.
2. Novel Mathematical Modeling Approach: The study presents a novel use of delay differential equations (DDEs) to mode) infectious diseases, showing that this approach improves the reliability of predictions and more accurately reflects real-world epidemic dynamics compared to traditional non-delay models.
3. Superiority of the NSFD Scheme: We claim that the Non-Standard Finite Difference (NSFD) scheme outperforms traditional methods (like Euler and Runge-Kutta) in terms of maintaining positivity, stability, and convergence, even when larger step sizes are used in the numerical analysis.
4. Enhanced Control Strategy Insights: The study provides valuable insights into the control and management of infectious diseases, claiming that the time-delay model offers better predictions for public health interventions, helping to shape strategies for disease containment and prevention.
5. Validation through Real-World Data: We claim that the proposed model is validated with real-world data from typhoid fever cases, demonstrating that the model accurately simulates the transmission dynamics of the disease and offers reliable projections for future outbreaks.
6. Applicability to Other Diseases: We assert that the modeling approach developed in this study, specifically the integration of time delays and NSFD numerical techniques, can be generalized and applied to other infectious diseases, improving the understanding and control of epidemics across various contexts.

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