Liver cirrhosis is the liver scarring in human body which causes the liver organ failure to function its regular activities effectively and normally. In this paper, we proposed and analyzed the combined effect of hepatitis B virus (HBV) infection and heavy alcohol consumption on the progression dynamics of liver cirrhosis. In order to study the progression dynamics of cirrhosis and to describe the effect of alcohol intake variation on a chronic hepatitis B patients a deterministic model and a logistic function are considered, respectively. The detailed proof of the positivity, boundedness, and biological feasibility of the proposed model is presented. The disease endemic equilibrium point and the existence of bifurcation were investigated in detail. We established and proved the existence theorems for forward and backward bifurcations, respectively. Finally, we performed large scale numerical simulations to verify the analytic work and the result of the numerical simulations reveal that heavy alcohol consumption significantly accelerates the progression of liver cirrhosis in chronic hepatitis B infected individuals.

Objective Liver cirrhosis, which is considered as the terminal stage of liver diseases, has become life-threatening among non-communicable diseases in the world. Viral hepatitis (hepatitis B and C) is the major risk factor for the development and progression of chronic liver cirrhosis. The asymptomatic stage of cirrhosis is considered as the compensated cirrhosis whereas the symptomatic stage is considered as decompensated cirrhosis. The latter stage is characterized by complex disorder affecting multiple systems of liver organ with frequent hospitalization. In this paper, we formulate system of fractional differential equations of chronic liver cirrhosis with frequent hospitalization to investigate the dynamics of the disease. The fundamental properties including the existence of positive solutions, positively invariant set, and biological feasibility are discussed. We used generalized mean value theorem to establish the existence of positive solutions. The Adams-type predictor-evaluate-corrector-evaluate approach is used to present the numerical scheme the fractional erder model. Results Using the numerical scheme, we simulate the solutions of the fractional order model. The numerical simulations are carried out using MATLAB software to illustrate the analytic findings. The analysis reveals that the number of decompensated cirrhosis individuals decreases when the progression rate and the disease’s past states are considered.

In this paper, we present a nonlinear deterministic mathematical model for malaria transmission dynamics incorporating climatic variability as a factor. First, we showed the limited region and nonnegativity of the solution, which demonstrate that the model is biologically relevant and mathematically well-posed. Furthermore, the fundamental reproduction number was determined using the next-generation matrix approach, and the sensitivity of model parameters was investigated to determine the most affecting parameter. The Jacobian matrix and the Lyapunov function are used to illustrate the local and global stability of the equilibrium locations. If the fundamental reproduction number is smaller than one, a disease-free equilibrium point is both locally and globally asymptotically stable, but endemic equilibrium occurs otherwise. The model exhibits forward and backward bifurcation. Moreover, we applied the optimal control theory to describe the optimal control model that incorporates three controls, namely, using treated bed net, treatment of infected with antimalaria drugs, and indoor residual spraying strategy. The Pontryagin’s maximum principle is introduced to obtain the necessary condition for the optimal control problem. Finally, the numerical simulation of optimality system and cost-effectiveness analysis reveals that the combination of treated bed net and treatment is the most optimal and least-cost strategy to minimize the malaria.

We proposed in this study a deterministic mathematical model of malaria transmission with climate variation factor. In the first place, fundamental properties of the model, such as positivity of solution and boundedness of the biological feasibility of the model, were proved whenever all initial data of the states were nonnegative. The next-generation matrix method is used to compute a basic reproduction number with respect to the disease-free equilibrium point. The Jacobian matrix and the Lyapunov function are used to check the local and global stability of disease-free equilibriums. If the basic reproduction number is less than one, the model’s disease-free equilibrium points are both locally and globally asymptotically stable; otherwise, an endemic equilibrium occurs. The results of the sensitivity analysis of the basic reproduction numbers were obtained, and its biological interpretation was provided. The existence of bifurcation was discussed, and the model exhibits forward and backward bifurcations with respect to the first and second basic reproduction numbers, respectively. Secondly, using the maximum principle of Pontryagin, the optimal malaria reduction strategies are described with three control measures, namely, treated bed nets, infected human treatment, and indoor residual spraying. Finally, based on numerical simulations of the optimality system, the combination of treatment and indoor spraying is the most efficient and least expensive strategy for malaria eradication.

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