A non-integer order model for Zika and Dengue co-dynamics with cross-enhancement

Abstract

A novel fractional derivative model with nine compartments is formulated to investigate the transmission dynamics of zika and dengue co-infection. The Atangana–Baleanu fractional derivative in the Caputo sense was employed. The conditions for a unique solution are identified, and the solutions’ positivity and boundedness are demonstrated. The disease-free equilibrium point (DFE) and basic reproduction number, R₀, were obtained. The DFE was shown to be locally asymptotically stable when the basic reproduction number is less than one. Zika-associated reproduction number, R_0z, and dengue-associated reproduction number, R_0d, were estimated to be 1.0144 and 1.1724, respectively. The system was shown to be generalized Ulam Hyers–Rassias stable, and the Adam–Bashforth method was used to provide its’ numerical solution. Sensitivity analysis using the Latin Hyper-cube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) (|PRCC|> 0.45) with 200 runs was carried out using various variables as response functions per time. The most significant parameters were found to be zika human-to-human transmission rate, $\beta$ _hz1, vector death rate, $\mu$ _v, zika recovery rate, $\gamma$ _hz1 and dengue vector-to-human transmission rate, $\beta$ _hd. Real data from Espirito Santo in Brazil is used to validate the model and fit needed parameter values. Numerical simulations illustrated the impact of varying the fractional order derivative, recovery rates, transmission rates, and cross-enhancement parameters on the infected human compartments. The zika Human-to-human transmission rate, $\beta$ _hz1, was found to be a very significant parameter in the control of zika disease transmission. Increasing the vector death rate, $\mu$ _v, was more important in curbing dengue prevalence and incidence than the attainment of recovery from the dengue disease, and the absence of the zika Vector-to-human transmission rate, $\beta$ _hz3, was almost insignificant in the presence of the zika Human-to-human transmission rate, $\beta$ _hz1, for disease eradication. This study suggested control measures and strategies to decrease the dengue and zika human-to-human transmission rates.