In this study, the mathematical model of the cholera epidemic is formulated and analyzed to show the impact of Vibrio cholerae in reserved freshwater. Moreover, the results obtained from applying the new fractional derivative method show that, as the order of the fractional derivative increases, cholera-preventing behaviors also increase. Also, the finding of our study shows that the dynamics of Vibrio cholerae can be controlled if continuous treatment is applied in reserved freshwater used for drinking purposes so that the intrinsic growth rate of Vibrio cholerae in water is less than the natural death of Vibrio cholerae. We have applied the stability theory of differential equations and proved that the disease-free equilibrium is asymptotically stable if R 0 < 1 , and the intrinsic growth rate of the Vibrio cholerae bacterium population is less than its natural death rate. The center manifold theory is applied to show the existence of forward bifurcation at the point R 0 = 1 and the local stability of endemic equilibrium if R 0 > 1 . Furthermore, the performed numerical simulation results show that, as the rank of control measures applied increases from no control, weak control, and strong control measures, the recovered individuals are 55.02, 67.47, and 674.7, respectively. Numerical simulations are plotted using MATLAB software package.
In this study, the co-infection of HIV and cholera model has been developed and analyzed. The new fractional-order derivative is applied and the behavior of the solution is interpreted. The order of new generalized fractional-order derivative implication is presented. A new method is incorporated to determine the forward bifurcation at a threshold R 0 = 1 . The developed method is used to determine the stability of steady-state points. The full model and the submodels’ disease-free equilibrium are locally asymptotically stable if the corresponding reproduction number is less than one and unstable if the production number is greater than one. The only HIV model exhibits forward bifurcation at the threshold point, R 0 = 1 . The numerical simulations solutions obtained using a new generalized fractional-order derivative shows that the total human population size approaches the disease-free equilibrium if the order of the fractional derivative is higher. Also, the simulated results show that the memory effects toward the invading disease are less whenever the order of the fractional derivative is near 0 but higher whenever the order of the fractional derivative is near 1. Furthermore, V. cholerae concentration in the environment increases whenever the intrinsic growth rate increases. The numerical solutions are carried out using MATLAB software.
A mathematical model of HIV transmission is built and studied in this paper. The system’s equilibrium is calculated. A next-generation matrix is used to calculate the reproduction number. The novel method is used to examine the developed model’s bifurcation and equilibrium stability. The stability analysis result shows that the disease-free equilibrium is locally asymptotically stable if 0 < R 0 < 1 but unstable if R 0 > 1 . However, the endemic equilibrium is locally and globally asymptotically stable if R 0 > 1 and unstable otherwise. The sensitivity analysis shows that the most sensitive parameter that contributes to increasing of the reproduction number is the transmission rate β 2 of HIV transmission from HIV individuals to susceptible individuals and the parameter that contributes to the decreasing of the reproduction number is identified as progression rate η of HIV-infected individuals to AIDS individuals. Furthermore, it is observed that as we change η from 0.1 to 1 , the reproduction number value decreases from 1.205 to 1.189, where the constant value of β 2 = 0.1 . On the other hand, as we change the value of β 2 from 0.1 to 1 , the value of the reproduction number increases from 0.205 to 1.347, where the constant value of η = 0.1 . Further, the developed model is extended to the optimal control model of HIV/AIDS transmission, and the cost-effectiveness of the control strategy is analyzed. Pontraygin’s Maximum Principle (PMP) is applied in the construction of the Hamiltonian function. Moreover, the optimal system is solved using forward and backward Runge–Kutta fourth-order methods. The numerical simulation depicts the number of newly infected HIV individuals and the number of individuals at the AIDS stage reduced as a result of taking control measures. The cost-effectiveness study demonstrates that when combined and used, the preventative and treatment control measures are effective. MATLAB is used to run numerical simulations.
In this study, a cholera model with fractional derivative and optimal control analysis is presented. Numerical simulation analysis shows that increasing the order of fractional derivatives contributes to updating the memory of the population to control the effects of cholera infection through available controlling techniques. On the other hand, the optimal analysis gives an indication of applying controlling infection with available treatment and prevention techniques. It provides a better mechanism to prevent the happening of cholera infection. Moreover, cost-effectiveness evaluation of cholera contamination intervention with feasible three or four combos of manipulate measures hygiene, vaccination, remedy of infectives, and chlorination indicates that hygiene, vaccination, and chlorination are the desired higher mixture to govern in addition propagation of cholera contamination. Numerical simulations are performed with the MATLAB platform and numerical solutions and results are discussed.
In this study, a mathematical model of the human immunodeficiency virus (HIV) and cholera co infection is constructed and analyzed. The disease-free equilibrium of the co-infection model is both locally and globally asymptotically stable if R 0 < 1 and unstable if R 0 > 1 . The only cholera model and only the HIV model show forward bifurcation if the corresponding reproduction numbers attain a value one. The disease-free equilibria of only the cholera and only the HIV models is locally and globally asymptotically if R 0 < 1 , and the endemic equilibria of only the cholera model and only the HIV model are locally and globally asymptotically stable if the corresponding reproduction number is equal to one. The endemic equilibrium point of the HIV and cholera model is computed, and stability property is shown with numerical simulations. The computed partial derivatives ∂ R 0 h / ∂ R 0 c > 0 show that the increase of one infection contributes to the increase of other infection. Pontryagin’s maximum principle is applied to construct Hamiltonian function, and optimal controls are computed. The optimal system is solved numerically using forward and backward sweep method of Runge Kutta’s fourth-order methods. The numerical simulations are plotted using MATLAB.
In this study, a very crucial stage of HIV extinction and invisibility stages are considered and a modified mathematical model is developed to describe the dynamics of infection. Moreover, the basic reproduction number R 0 is computed using the next-generation matrix method whereas the stability of disease-free equilibrium is investigated using the eigenvalue matrix stability theory. Furthermore, if R 0 ≤ 1 , the disease-free equilibrium is stable both locally and globally whereas if R 0 > 1 , based on the forward bifurcation behavior, the endemic equilibrium is locally and globally asymptotically stable. Particularly, at the critical point R 0 = 1 , the model exhibits forward bifurcation behavior. On the other hand, the optimal control problem is constructed and Pontryagin’s maximum principle is applied to form an optimality system. Further, forward fourth-order Runge–Kutta’s method is applied to obtain the solution of state variables whereas Runge–Kutta’s fourth-order backward sweep method is applied to obtain solution of adjoint variables. Finally, three control strategies are considered and a cost-effective analysis is performed to identify the better strategies for HIV transmission and progression. In advance, prevention control measure is identified to be the better strategy over treatment control if applied earlier and effectively. Additionally, MATLAB simulations were performed to describe the population’s dynamic behavior.
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