Bifurcation analysis of a network-based SIR epidemic model with saturated treatment function

Dynamical behaviors of a delayed SIR information propagation model with forced silence function and control measures in complex networks

Due to the advanced network technology, there is almost no barrier to information dissemination, which has led to the breeding of rumors. Intended to clarify the dynamic mechanism of rumor propagation, we formulate a SIR model with time delay, forced silence function and forgetting mechanism in both homogeneous and heterogeneous networks. In the homogeneous network model, we first prove the nonnegativity of the solutions. Based on the next-generation matrix, we calculate the basic reproduction number R 0 . Besides, we discuss the existence of equilibrium points. Next, by linearizing the system and constructing a Lyapunov function, the local and global asymptotically stability of the equilibrium points are found. In the heterogeneous network model, we derive the basic reproduction number R 00 through the analysis of a rumor-prevailing equilibrium point E ∗ . Moreover, we conduct the local and global asymptotic stability analysis for the equilibrium points according to the LaSalle's Invariance Principle and stability theorem. As long as the maximum spread rate β is large enough, the rumor-prevailing point E ∗ is locally asymptotically stable when R 00 > 1 . Additionally, it hits that the system exists bifurcation behavior at R 00 = 1 due to the newly added forced silence function. Later, after adding two controllers to the system, we research the problem of optimal control. Finally, aimed at authenticating the above theoretical results, a serious of numerical simulation experiments are carried out.

Global dynamics of two-strain epidemic model with single-strain vaccination in complex networks

Contagious pathogens, such as influenza and COVID-19, are known to be represented by multiple genetic strains. Different genetic strains may have different characteristics, such as spreading more easily, causing more severe diseases, or even evading the immune response of the host. These facts complicate our ability to combat these diseases. There are many ways to prevent the spread of infectious diseases, and vaccination is the most effective. Thus, studying the impact of vaccines on the dynamics of a multi-strain model is crucial. Moreover, the notion of complex networks is commonly used to describe the social contacts that should be of particular concern in epidemic dynamics. In this paper, we investigate a two-strain epidemic model using a single-strain vaccine in complex networks. We first derive two threshold quantities, R 1 and R 2 , for each strain. Then, by using the basic tools for stability analysis in dynamical systems (i.e., Lyapunov function method and LaSalle's invariance principle), we prove that if R 1 < 1 and R 2 < 1 , then the disease-free equilibrium is globally asymptotically stable in the two-strain model. This means that the disease will die out. Furthermore, the global stability of each strain dominance equilibrium is established by introducing further critical values. Under these stability conditions, we can determine which strain will survive. Particularly, we find that the two strains can coexist under certain condition; thus, a coexistence equilibrium exists. Moreover, as long as the equilibrium exists, it is globally stable. Numerical simulations are conducted to validate the theoretical results.

Stability behavior of a two-susceptibility SHIR epidemic model with time delay in complex networks

Taking two susceptible groups into account, we formulate a modified subhealthy-healthy-infected-recovered (SHIR) model with time delay and nonlinear incidence rate in networks with different topologies. Concretely, two dynamical systems are designed in homogeneous and heterogeneous networks by utilizing mean field equations. Based on the next-generation matrix and the existence of a positive equilibrium point, we derive the basic reproduction numbers R 0 1 and R 0 2 which depend on the model parameters and network structure. In virtue of linearized systems and Lyapunov functions, the local and global stabilities of the disease-free equilibrium points are, respectively, analyzed when R 0 1 < 1 in homogeneous networks and R 0 2 < 1 in heterogeneous networks. Besides, we demonstrate that the endemic equilibrium point is locally asymptotically stable in homogeneous networks in the condition of R 0 1 > 1 . Finally, numerical simulations are performed to conduct sensitivity analysis and confirm theoretical results. Moreover, some conjectures are proposed to complement dynamical behavior of two systems.

A biological mathematical model of vector-host disease with saturated treatment function and optimal control strategies

The aims of this paper to explore the dynamics of the vector-host disease with saturated treatment function. Initially, we formulate the model by considering three different classes for human and two for the vector population. The use of the treatment function in the model and their brief analysis for the case of disease-free and endemic case are briefly shown. We show that the basic reproduction number (< or >) than unity, the disease-free and endemic cases are stable locally and globally. Further, we apply the optimal control technique by choosing four control variables in order to maximize the population of susceptible and recovered human and to minimize the population of infected humans and vector. We discuss the results in details of the optimal controls model and show their existence. Furthermore, we solve the optimality system numerically in connection with the system of no control and the optimal control characterization together with adjoint system, and consider a set of different controls to simulate the models. The considerable best possible strategy that can best minimize the infection in human infected individuals is the use of all controls simultaneously. Finally, we conclude that the work with effective control strategies.

Dynamical analysis of a mean-field vector-borne diseases model on complex networks: An edge based compartmental approach

In this paper, we propose a concise method to investigate the global dynamics of a mean-field vector-borne diseases model on complex networks. We obtain an explicit formula of the basic reproduction number by a renewal equation. We show that the model exhibits a threshold dynamics in terms of the basic reproduction number by constructing subtle Lyapunov functions. Roughly speaking, if the basic reproduction number R₀<1, the vector-borne diseases die out; otherwise, it persists. Moreover, numerical simulations show that vector-control is an effective measure for slowing down the spread of vector-borne diseases.