Abstract
The World Health Organization is yet to realise the global aim of achieving future-free and eliminating the transmission of respiratory diseases such as H1N1, SARS and Ebola since the recent reemergence of Ebola in the Democratic Republic of Congo. In this paper, a Caputo fractional-order derivative is applied to a system of non-integer order differential equation to model the transmission dynamics of respiratory diseases. The nonnegative solutions of the system are obtained by using the Generalized Mean Value Theorem. The next generation matrix approach is used to obtain the basic reproduction number mathcal {R}_{0} . We discuss the stability of the disease-free equilibrium when mathcal {R}_{0} < 1, and the necessary conditions for the stability of the endemic equilibrium when mathcal {R}_{0} > 1 . A sensitivity analysis shows that mathcal {R}_{0} is most sensitive to the probability of the disease transmission rate. The results from the numerical simulations of optimal control strategies disclose that the utmost way of controlling or probably eradicating the transmission of respiratory diseases should be quarantining the exposed individuals, monitoring and treating infected people for a substantial period.
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