Optimal Control Problem of a Non-integer Order Waterborne Pathogen Model in Case of Environmental Stressors

Abstract

In this work, we extend a mathematical model, which has been proposed for susceptible and infected compartments together with pathogen population, by including recovered subgroup. It is known that environmental pollution, such as contaminated drinking water and lack of an ordinary toilet, affects individuals and such negative impacts can be defined as "stressors." In order to include the influence of such stressors, susceptible subpopulation has been divided into two groups as the one affected by stress or not. Thus, spread of the disease is expressed in terms of a five-dimensional system. Moreover, we extend this model with the use of a time fractional derivative due to non-local effects of water pollution and we prove that the solution is non-negative and bounded from above. Then, we perform stability analysis for the disease-free equilibrium point. Afterward, the next step is to apply optimal control theory to optimize the decay rate of pathogens and the stress related parameters so that the number of infected individuals and the pathogen population can be minimized. Finally, we present some numerical results to find out the most appropriate control policy and the effect of the fractional order.

Modeling the epidemic control measures in overcoming COVID-19 outbreaks: A fractional-order derivative approach

“…As a measure of fact, it mentioned that fractional derivatives are beneficial for modeling many real-worlds problems due to memory and the universal properties [33] , [34] , [35] , [36] . As a result, the importance and potential application enlarged day by day [37] , [38] , [39] . The fractional-order differential equations supplement new dimensions in the investigation of epidemiological models.…”

Fractional optimal control problem for an age-structured model of COVID-19 transmission

“…Among the models used for epidemic analysis of COVID-19, the majority was formulated using ODE’s, while others were based on fractional calculus. Recall that the fractional calculus is applied in different directions of physics, mathematical biology, fluid mechanics, electrochemistry, signal processing, viscoelasticity, finance and in many others (see for instance [17] , [18] , [19] , [20] , [21] , [22] , [23] , [24] ). Fractional derivatives were used in the literature to monitor the effect of memory on the system dynamics by replacing the normal derivative arrangement with a fractional derivative arrangement.…”

“…The fractional calculus adds an extra dimension in the study of dynamics of epidemiological models, especially for COVID-19 pandemic. Therefore, the fractional version of many epidemic models has been investigated as in Khan and Atangana [21] , Akman Yıldız [24] , Naik et al. [30] , Higazy [31] , Zhang et al.…”

Basic Control Theory for Linear Fractional Differential Equations With Constant Coefficients

“…In particular, there are a growing number of research areas in physics which employ fractional calculus [10] and it has many applications among its different branches, ranging from imaging processing to fractional quantum harmonic oscillator [11]. Recently, in Yıldız [12] the dynamics of a waterborne pathogen fractional model under the influence of environmental pollution has been studied and the solutions of a generalized fractional kinetic equations are obtained [13] using the generalized fractional integrations of the generalized Mittag-Leffler type function. Finally, we highlight that different fractional systems have also been considered in the framework of control theory [14][15][16][17][18].…”