Mathematical Modeling of Typhoid Fever Disease Incorporating Unprotected Humans in the Spread Dynamics
Julia Wanjiku Karunditu *
Department of Mathematics and Actuarial Science, Catholic University of Eastern Africa, Box 62157-00200, Nairobi, Kenya.
George Kimathi
Department of Mathematics and Actuarial Science, Catholic University of Eastern Africa, Box 62157-00200, Nairobi, Kenya.
Shaibu Osman
Department of Mathematics and Actuarial Science, Catholic University of Eastern Africa, Box 62157-00200, Nairobi, Kenya.
*Author to whom correspondence should be addressed.
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.
Keywords: Basic reproduction number, invariant region, positivity of solution, mathematical model, unprotected humans, disease free equilibrium, endemic equilibrium point, global stability