A fractional mathematical model of heartwater transmission dynamics considering nymph and adult amblyomma ticks

Heartwater is a tick-borne illness that affects ruminants and is carried by the amblyomma ticks. The condition may sometimes be deadly. This paper studies the Caputo fractional version of the disease spread in domestic ruminants and amblyomma ticks. We obtain the positivity and boundedness condition through the Laplace transform. The stability state of the proposed model is obtained using the Ulam–Hyers and Ulam–Hyers–Rassias stability conditions. The heartwater-free equilibrium point is obtained. The fractional model is fitted to the heartwater incidence data from 2006 to 2019. The heartwater reproduction number, R₀^τ, from the parameter estimation is R₀^τ=1.9345 with a fitting fractional order, τ, of 0.6990. The residuals from the data fitting were randomly distributed, indicating that the proposed model could be used for further predictions. Furthermore, we observed the dynamic effect of varying the fractional order and noticed that changing the fractional order produces crisscrossed behaviour in infected compartments of domestic ruminants and infected adult amblyomma ticks. Finally, we showed the relative impact of varying the transmission rates and the infectivity potential of peractive, active, and recovered carrier ruminants on the overall dynamics of the disease. Thus, a reduction in the rate of transmission from nymph and adult amblyomma ticks to vaccinated ticks will increase the number of healthy domestic ruminants.