On the fractional order model for HPV and Syphilis using non–singular kernel

Abstract

Most dynamical systems employ the classical integer-order derivatives which yield good results. However, recent trends have shown that the application of arbitrary order derivatives, popularly known as fractional order derivatives produces better and more realistic results to a given real life scenario because they are non-local operators. In this work, a fractional order co-infection model for human papilloma virus (HPV) and Syphilis is considered and studied using the non-singular kernel derivative. With the aid of Banach and Schauder’s fixed point theorems, the existence and uniqueness of the solution is proven. The conditions for the existence and uniqueness of the solution of the model are also established. The model’s disease free equilibrium (DFE) is shown to be locally asymptotically stable when the reproduction number is below one. The fractional order model is shown to be generalized Ulam Hyers–Rassias stable, under some conditions. The predictor–corrector method with convergence of min2,1+φwas used for the numerical simulations. Simulations of the model showed that the fractional derivative greatly influenced the dynamics of both diseases and their co-infection. Furthermore, it was observed that control strategy for single infection (syphilis only prevention strategy or HPV only prevention strategy) did not result in reduction of single infection cases, instead reduces the co-infection of new cases.

Keywords

• HPV
• Syphilis
• Co-infection
• Atangana-Baleanu derivative
• Ulam–Hyers stability