A New Generalized Fractional-Order Derivative and Bifurcation Analysis of Cholera and Human Immunodeficiency Co-Infection Dynamic Transmission

Abstract

In this study, the co-infection of HIV and cholera model has been developed and analyzed. The new fractional-order derivative is applied and the behavior of the solution is interpreted. The order of new generalized fractional-order derivative implication is presented. A new method is incorporated to determine the forward bifurcation at a threshold <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"> <a:mtext> </a:mtext> <a:msub> <a:mrow> <a:mi>R</a:mi> </a:mrow> <a:mrow> <a:mn>0</a:mn> </a:mrow> </a:msub> <a:mo>=</a:mo> <a:mn>1</a:mn> </a:math> . The developed method is used to determine the stability of steady-state points. The full model and the submodels’ disease-free equilibrium are locally asymptotically stable if the corresponding reproduction number is less than one and unstable if the production number is greater than one. The only HIV model exhibits forward bifurcation at the threshold point, <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"> <c:mtext> </c:mtext> <c:msub> <c:mrow> <c:mi>R</c:mi> </c:mrow> <c:mrow> <c:mn>0</c:mn> </c:mrow> </c:msub> <c:mo>=</c:mo> <c:mn>1</c:mn> </c:math> . The numerical simulations solutions obtained using a new generalized fractional-order derivative shows that the total human population size approaches the disease-free equilibrium if the order of the fractional derivative is higher. Also, the simulated results show that the memory effects toward the invading disease are less whenever the order of the fractional derivative is near 0 but higher whenever the order of the fractional derivative is near 1. Furthermore, V. cholerae concentration in the environment increases whenever the intrinsic growth rate increases. The numerical solutions are carried out using MATLAB software.