Abstract
We provide a class of fractional-order differential models of biological systems with memory, such as dynamics of tumor-immune system and dynamics of HIV infection of CD4<sup >+</sup> T cells. Stability and nonstability conditions for disease-free equilibrium and positive equilibria are obtained in terms of a threshold parameter <svg style="vertical-align:-3.25793pt;width:22.5px;" id="M1" height="16.025" version="1.1" viewBox="0 0 22.5 16.025" width="22.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.737)"><path id="x211B" d="M852 669l7 -19q-33 -13 -66 -38q83 -53 83 -139q0 -65 -49 -111q-49 -47 -131 -50v-2q54 -40 54 -84q0 -36 -38 -97t-38 -77q0 -34 32 -34q52 0 145 101l17 -16q-100 -115 -176 -115q-38 0 -65.5 23.5t-27.5 58.5q-1 28 20.5 61t43.5 63t20 51q0 48 -33 48q-30 0 -47 11
q-81 -166 -177 -242.5t-219 -76.5q-77 0 -125 36t-48 87q0 31 17 51t42 20q22 0 36.5 -13.5t14.5 -34.5t-11.5 -34.5t-31.5 -13.5q-23 0 -30 12h-2q0 -28 35 -54q34 -25 85 -25q94 0 152 52q60 53 145 203q75 133 131 214q54 75 121 124q-76 48 -192 48q-150 0 -250 -75
q-101 -75 -101 -174q0 -41 30 -68.5t73 -27.5q158 0 205 245h20q-8 -107 -69 -191q-62 -84 -158 -84q-63 0 -102.5 35.5t-39.5 94.5q0 107 111 191t282 84q124 0 221 -59q38 24 84 41zM814 486q0 64 -48 102q-7 -6 -14 -14t-14.5 -18l-10.5 -14q-4 -5 -35 -55l-6 -11
q-6 -11 -10 -20t-10 -20l-6 -11l-17 -36q-4 -7 -7.5 -14.5l-6 -13t-3.5 -7.5q34 0 60 -17q57 6 92 50q36 43 36 99z" /></g> <g transform="matrix(.012,-0,0,-.012,16.112,15.825)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g> </svg> (minimum infection parameter) for each model. We provide unconditionally stable method, using the Caputo fractional derivative of order <svg style="vertical-align:-0.1254pt;width:9.7749996px;" id="M2" height="7.9499998" version="1.1" viewBox="0 0 9.7749996 7.9499998" width="9.7749996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D6FC" d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204
q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" /></g> </svg> and implicit Euler’s approximation, to find a numerical solution of the resulting systems. The numerical simulations confirm the advantages of the numerical technique and using fractional-order differential models in biological systems over the differential equations with integer order. The results may give insight to infectious disease specialists.
Highlights
Mathematical models, using ordinary differential equations with integer order, have been proven valuable in understanding the dynamics of biological systems
Using fractionalorder differential equations can help us to reduce the errors arising from the neglected parameters in modeling biological systems with memory and systems distributed parameters
We presented a class of fractional-order differential models of biological systems with memory to model the interaction of immune system with tumor cells and with human immunodeficiency virus (HIV) infection of CD4+ T cells
Summary
Mathematical models, using ordinary differential equations with integer order, have been proven valuable in understanding the dynamics of biological systems. The behavior of most biological systems has memory or aftereffects The modelling of these systems by fractional-order differential equations has more advantages than classical integer-order mathematical modeling, in which such effects are neglected. The fractional-order differential equations (FODEs) models seem more consistent with the real phenomena than the integerorder models This is due to the fact that fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Fractional-order differential equations are naturally related to systems with memory which exists in most biological systems. (i) Riemann-Liouville fractional derivative: take fractional integral of order (n − α), and take nth derivative as follows: D∗αf (t) = D∗n Ian−αf (t) , D∗n dn dtn. We provide a class of fractional-order differential models to describe the dynamics of tumour-immune system interactions
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