Fractional-Order Modelling and Optimal Control of Cholera Transmission

Abstract

A Caputo-type fractional-order mathematical model for “metapopulation cholera transmission” was recently proposed in [Chaos Solitons Fractals 117 (2018), 37–49]. A sensitivity analysis of that model is done here to show the accuracy relevance of parameter estimation. Then, a fractional optimal control (FOC) problem is formulated and numerically solved. A cost-effectiveness analysis is performed to assess the relevance of studied control measures. Moreover, such analysis allows us to assess the cost and effectiveness of the control measures during intervention. We conclude that the FOC system is more effective only in part of the time interval. For this reason, we propose a system where the derivative order varies along the time interval, being fractional or classical when more advantageous. Such variable-order fractional model, that we call a FractInt system, shows to be the most effective in the control of the disease.

Fractional Derivative and Optimal Control Analysis of Cholera Epidemic Model

“…Fractional derivatives are one of the most widely applied differential calculus in science and engineering to describe natural and biological phenomena [10,18]. Particularly, in this study, Caputo fractional derivative of order α is applied to describe the dynamics of cholera infection and described by the following equations:…”

“…Proof. To show positivity of solutions we follow the procedures done in [18]. Consider the trajectory of solution along S-axis so that I(0) � 0, R(0) � 0, B(0) � 0, and S(0) > 0. en, the first equation of model ( 2) reduces to the form as follows:…”

“…Due to sudden outbreak of cholera infection and its huge impact on economic resources, optimal control strategies toward infection is highly recommended [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. On the other hand, fractional derivative is one of the highly applicable di erential calculus to describe both biological and physical phenomena [1,[15][16][17][18][19]. Di erent mathematical models of cholera infection are developed starting from the discovery of infection.…”

Mathematical assessment of a fractional‐order vector–host disease model with the Caputo–Fabrizio derivative

“…The application of such mathematical tool can be found in physics, chemistry, biology, and so forth. [11][12][13][14] The two fractional operators, that is, Podlubny 15 and Miller and Ross, 9 have the singularity problem in their kernels. Caputo-Fabrizio (CF) 16 and then Atangana-Baleanu 17 developed new concepts of fractional calculus to resolve the singularity problems of both of the above mentioned fractional operators.…”

“…Such properties enhance the importance of fractional derivatives for understanding various natural phenomena. The application of such mathematical tool can be found in physics, chemistry, biology, and so forth 11–14 . The two fractional operators, that is, Podlubny 15 and Miller and Ross, 9 have the singularity problem in their kernels.…”