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Computational and Mathematical Methods in Medicine
Volume 2017 (2017), Article ID 2324518, 16 pages
https://doi.org/10.1155/2017/2324518
Research Article

Modelling and Optimal Control of Typhoid Fever Disease with Cost-Effective Strategies

1Pan African University Institute of Basic Sciences Technology and Innovation, Nairobi, Kenya
2Faculty of Military Science, Stellenbosch University, Stellenbosch, South Africa
3Department of Mathematics, Kenyatta University, Nairobi City, Kenya

Correspondence should be addressed to Getachew Teshome Tilahun

Received 23 February 2017; Revised 19 May 2017; Accepted 22 May 2017; Published 10 September 2017

Academic Editor: Anwar Zeb

Copyright © 2017 Getachew Teshome Tilahun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We propose and analyze a compartmental nonlinear deterministic mathematical model for the typhoid fever outbreak and optimal control strategies in a community with varying population. The model is studied qualitatively using stability theory of differential equations and the basic reproductive number that represents the epidemic indicator is obtained from the largest eigenvalue of the next-generation matrix. Both local and global asymptotic stability conditions for disease-free and endemic equilibria are determined. The model exhibits a forward transcritical bifurcation and the sensitivity analysis is performed. The optimal control problem is designed by applying Pontryagin maximum principle with three control strategies, namely, the prevention strategy through sanitation, proper hygiene, and vaccination; the treatment strategy through application of appropriate medicine; and the screening of the carriers. The cost functional accounts for the cost involved in prevention, screening, and treatment together with the total number of the infected persons averted. Numerical results for the typhoid outbreak dynamics and its optimal control revealed that a combination of prevention and treatment is the best cost-effective strategy to eradicate the disease.