Abstract
Malaria is a deadly human disease that is still a major cause of casualties worldwide. In this work, we consider the fractional-order system of malaria pestilence. Further, the essential traits of the model are investigated carefully. To this end, the stability of the model at equilibrium points is investigated by applying the Jacobian matrix technique. The contribution of the basic reproduction number, R0, in the infection dynamics and stability analysis is elucidated. The results indicate that the given system is locally asymptotically stable at the disease-free steady-state solution when R0<1. A similar result is obtained for the endemic equilibrium when R0>1. The underlying system shows global stability at both steady states. The fractional-order system is converted into a stochastic model. For a more realistic study of the disease dynamics, the non-parametric perturbation version of the stochastic epidemic model is developed and studied numerically. The general stochastic fractional Euler method, Runge–Kutta method, and a proposed numerical method are applied to solve the model. The standard techniques fail to preserve the positivity property of the continuous system. Meanwhile, the proposed stochastic fractional nonstandard finite-difference method preserves the positivity. For the boundedness of the nonstandard finite-difference scheme, a result is established. All the analytical results are verified by numerical simulations. A comparison of the numerical techniques is carried out graphically. The conclusions of the study are discussed as a closing note.
Highlights
Malaria is a Latin word which means “foul air”
We present three generalized stochastic fractional techniques to solve the stochastic fractional-order system (15), namely, Euler, Runge–Kutta and a nonstandard finitedifference (NSFD) scheme
We departed from a fractional-order disease model and transformed it into a non-parametric perturbation stochastic model
Summary
Malaria is a Latin word which means “foul air”. Biologically, malaria is an ailment due to the microorganism plasmodium, which is a bug found in the mosquito. P. falciparum is extremely dangerous and fatal, causing a wide range of physical symptoms, such as fever, flu, severe chills, vomiting, muscle aches, headache, nausea, diarrhea, tiredness, low blood pressure, respiratory disorder, cerebral disorder, and hemoglobin in the urine, with some cases showing jaundice and anemia Physicians knew about this disease at least 2000 years ago, and noted that it is very common in marshy areas, where stagnant water is found frequently. In 2020, Olaniyi et al presented an SEIR mathematical model to control malaria among travelers [2]. In 2020, Baihaqi et al proposed an SEIRS p-model to investigate how malaria disease spreads among humans [5]. In 2017, Mojeeb et al presented an SEIR model to investigate the ways to control the mosquito population and eradication of malaria outbreaks [16]. Our scheme will be able to preserve various important properties of the solutions [21,22,23,24,25]
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