A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying

Devendra Kumar,Jagdev Singh,Maysaa Al Qurashi,Dumitru Baleanu

doi:10.1186/s13662-019-2199-9

Abstract

The present paper deals with a new fractional SIRS-SI model describing the transmission of malaria disease. The SIRS-SI malaria model is modified by using the Caputo–Fabrizio fractional operator for the inclusion of memory. We also suggest the utilization of vaccines, antimalarial medicines, and spraying for the treatment and control of the malaria disease. The theory of fixed point is utilized to examine the existence of the solution of a fractional SIRS-SI model describing spreading of malaria. The uniqueness of the solution of SIRS-SI model for malaria is also analyzed. It is shown that the treatments have great impact on the dynamical system of human and mosquito populations. The numerical simulation of fractional SIRS-SI malaria model is performed with the aid of HATM and Maple packages to show the effect of different parameters of the treatment of malaria disease. The numerical results for fractional SIRS-SI malaria model reveal that the recommended approach is very accurate and effective.

Highlights

Malaria is a life-threatening mosquito-borne blood illness in the developing portion of the globe and especially in Asia and Africa
Mathematical modeling of infectious diseases is a very strong tool to understand the dynamical system of disease spreading and control strategies
Theorem 2 The SIRS-SI malaria model involving the Caputo– Fabrizio (CF) fractional operator expressed in Eq (5) has a solution if there exists t0 such that

Summary

Introduction

Senthamarai et al [7] utilized the HAM to examine the spreading of malaria illness in an SIRS-SI model All these approaches and mathematical models have their own limitation due to local nature of integer-order derivatives. Since the integer-order derivative is local in nature, the presented SIRS-SI malaria model (4) does not describe different effects on humans and mosquitoes in efficient manner. To include the memory effects in the description of malaria disease, we extend the model (4) by employing the newly proposed Caputo–Fabrizio fractional derivative as follows: CF Dλ0Sh = αh + βRh – (ωγ1Ih + ξ γ2Im)Sh – (δ + ηh)Sh, CF Dλ0Ih = μIh + (ωγ1Ih + ξ γ2Im)Sh – (ηh + ε + cν)Ih, CF Dλ0Rh = cνIh – (ηh + β)Rh + δSh,.

It is worth observing that n n n

Thus we have

We present the nonlinear operators as

Conclusions