<abstract><p>This paper focuses on a mathematical model for coffee berry disease infestation dynamics. This model considers coffee berry and vector populations with the interaction of fungal pathogens. In order to gain an insight into the global dynamics of coffee berry disease transmission and eradication on any given coffee farm, the assumption of logistic growth with a carrying capacity reflects the fact that the amount of coffee plants depends on the limited size of the coffee farm. First, we show that all solutions of the chosen model are bounded and non-negative with positive initial data in a feasible region. Subsequently, endemic and disease-free equilibrium points are calculated. The basic reproduction number with respect to the coffee berry disease-free equilibrium point is derived using a next generation matrix approach. Furthermore, the local stability of the equilibria is established based on the Jacobian matrix and Routh Hurwitz criteria. The global stability of the equilibria is also proved by using the Lyapunov function. Moreover, bifurcation analysis is proved by the center manifold theory. The sensitivity indices for the basic reproduction number with respect to the main parameters are determined. Finally, the numerical simulations show the agreement with the analytical results of the model analysis.</p></abstract>
In this paper, the mathematical model of the coronavirus pandemic with vaccination is formulated and analyzed to show the impact of severe acute respiratory syndrome coronavirus 2 pathogens in the environmental reservoir. In the model analysis, the vaccination-induced reproduction number which helps us in establishing the local and global stability of COVID-19-free and endemic equilibrium points was derived. The local stability of the COVID-19-free equilibrium is established via the Jacobian matrix and Routh-Hurwitz criteria. In contrast, the global stability of the endemic equilibrium is proved by using an appropriate Lyapunov function. Sensitivity indices are also discussed. The proposed model is extended into the optimal control problem by incorporating three control variables: preventive, medical care, and surface disinfection. Then, the necessary conditions for the optimal control of the disease were analyzed by applying Pontryagin minimum principle. Finally, the numerical simulations indicated that a combination of medical care and surface disinfection strategies is effective in controlling the disease epidemic.
This study concentrates on a nonlinear deterministic mathematical model for the impact of pathogens on human disease transmission with optimal control strategies. Both pathogen-free and coexistence equilibria are computed. The basic reproduction number
R
0
, which plays a vital role in mathematical epidemiology, was derived. The qualitative analysis of the model revealed the scenario for both pathogen-free and coexistence equilibria together with
R
0
. The local stability of the equilibria is established via the Jacobian matrix and Routh-Hurwitz criteria, while the global stability of the equilibria is proven by using an appropriate Lyapunov function. Also, the normalized sensitivity analysis has been performed to observe the impact of different parameters on
R
0
. The proposed model is extended into optimal control problem by incorporating three control variables, namely, preventive measure variable based on separation of susceptible from contacting the pathogens, integrated vector management based on chemical, biological control, ... etc. to kill pathogens and their carriers, and supporting infective medication variable based on the care of the infected individual in quarantine center. Optimal disease control analysis is examined using Pontryagin minimum principle. Numerical simulations are performed depending on analytical results and discussed quantitatively.
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