Mathematical Modeling of Typhoid Fever Disease Incorporating Unprotected Humans in the Spread Dynamics

Abstract

A deterministic mathematical model of typhoid fever incorporating unprotected humans is formulated in this study and employed to study local and global stability of equilibrium points. The model incorporating Susceptible, unprotected, Infectious and Recovered humans which are analyzed mathematically and also result into a system of ordinary differential equations which are used for interpretations and comparison to the qualitative solutions in studying the spread dynamics of typhoid fever. Jacobian matrix was considered in the study of local stability of disease free equilibrium point and Castillo-Chavez approach used to determine global stability of disease free equilibrium point. Lyapunov function was used to study global stability of endemic equilibrium point. Both equilibrium points (DFE and EE) were found to be local and globally asymptotically stable. This means that the disease will be dependent on numbers of unprotected humans and other factors who contributes positively to the transmission dynamics.

On the Effect of Vaccination, Screening and Treatment in Controlling Typhoid Fever Spread Dynamics: Deterministic and Stochastic Applications

“…The basic reproduction number ( ) of this model system is the rate of transmissibility of typhoid infection in a largely susceptible population in the course of infection. The next generation matrix method is employed to obtain the threshold, see [12,15]. The two typhoid infected class is linearized around the typhoid -free fixed point solution to give the basic reproduction number of model system (1) as time units in their class.…”

“…From (12) - (15), the transformed stochastic model equations are given by: (16) Under the initial conditions S p (0) ≥ 0, V c (0) ≥ 0, I f (0) ≥ 0, C r (0) ≥ 0, R r (0) ≥ 0.…”

“…The model was further extended into an optimal control problem and the cost-effective strategy was analyzed. Stephen and Nkuba [15] investigated the impact of education, vaccination and treatment as a form of effective control in minimizing typhoid fever infection through the formulation of a deterministic model, also, Aji, Aldila and Handari, Moathlhodi and Gosalamang [5,9]. Peter et al [14], employed the approximate method of variational iterative and Runge -Kutta fourth -order method to solve a system of ordinary differential equations describing typhoid dynamics.…”

Let the DOCTOR Decide Whom to Test: Adaptive Testing Strategies to Tackle the COVID-19 Pandemic

“…The SEIR model is a popular epidemiological model for simulating the progression of an epidemic through a population of individuals, and it has been used to successfully simulate the outbreak of many infectious diseases, e.g., Ebola [25], COVID-19 [23, 31], etc. In the standard SEIR model, a population of N individuals is split into four compartments: (i) Susceptible ( S ), i.e., individuals who have never been infected or exposed to the COVID-19 virus; (ii) Exposed ( E ), i.e., individuals who have been exposed to the virus, but are not infectious yet; (iii) Infectious ( I ), i.e., individuals who have been infected and can spread infection to other individuals; and (iv) Recovered ( R ), i.e., individuals who have recovered or died from the virus.…”

Modeling Anthrax with Optimal Control and Cost Effectiveness Analysis

“…Complex models for transmission dynamics of diseases such as periodic orbits, Hoff bifurcations and multiple equilibrium have been proposed and worked on for some time now. They give a concise qualitative illustration of the disease dynamics and better analysis and implications for disease prediction of [5] [6]. [1] investigated the effects of constant vaccination on anthrax model but never considered effects of optimal control.…”