Abstract
In this paper, the prey-predator model of five compartments is constructed with treatment given to infected prey and infected predator. We took predation incidence rates as functional response type II, and disease transmission incidence rates follow simple kinetic mass action function. The positivity, boundedness, and existence of the solution of the model are established and checked. Equilibrium points of the models are identified, and local stability analyses of trivial equilibrium, axial equilibrium, and disease-free equilibrium points are performed with the method of variation matrix and the Routh-Hurwitz criterion. It is found that the trivial equilibrium point E o is always unstable, and axial equilibrium point E A is locally asymptotically stable if β k − t 1 + d 2 < 0 , q p 1 k − d 3 s + k < 0 and q p 3 k − t 2 + d 4 s + k < 0 conditions hold true. Global stability analysis of an endemic equilibrium point of the model has been proven by considering the appropriate Lyapunov function. The basic reproduction number of infected prey and infected predators are obtained as R 01 = q p 1 − d 3 2 k β d 3 s 2 / q p 1 − d 3 q p 1 − d 3 2 k s t 1 + d 2 + r s q p 2 k q p 1 − k d 3 − d 3 s and R 02 = q p 1 − d 3 q p 3 d 3 k + α r s q k q p 1 − k d 3 − d 3 s / q p 1 − d 3 2 t 2 + d 4 k , respectively. If the basic reproduction number is greater than one, then the disease will persist in the prey-predator system. If the basic reproduction number is one, then the disease is stable, and if the basic reproduction number is less than one, then the disease dies out from the prey-predator system. Finally, simulations are done with the help of DEDiscover software to clarify results.
Highlights
Mathematical modeling of prey-predator systems of interaction of species have a long history since the original remarkable work done by the Lotka-Volterra Model in the 1920s [1,2,3,4], and the SIR model compartment of systems of population is another vital area of research after the pioneering work of Kermack and Mckendrick [1,2,3,4,5,6,7,8,9]
Many studies focused on the study of disease in prey only [1,2,3, 5, 6, 10, 12, 14,15,16,17,18,19,20], other researchers were interested in the study of disease within the predator population only [18, 21], and there are some studies on diseases in both prey and predators [4, 8, 11]
We proposed and studied infectious disease in both prey and predator interaction of species with treatment given to infected prey and infected predator
Summary
Mathematical modeling of prey-predator systems of interaction of species have a long history since the original remarkable work done by the Lotka-Volterra Model in the 1920s [1,2,3,4], and the SIR model compartment of systems of population is another vital area of research after the pioneering work of Kermack and Mckendrick [1,2,3,4,5,6,7,8,9]. Several mathematical models have been proposed and studied on prey-predator systems [1,2,3,4,5,6, 8,9,10,11,12]. We proposed and studied infectious disease in both prey and predator interaction of species with treatment given to infected prey and infected predator
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