Global Dynamics of a Delayed Fractional-Order Viral Infection Model With Latently Infected Cells

In this paper, we propose a fractional-order viral infection model, which includes latent infection, a Holling type II response function, and a time-delay representing viral production. Based on the characteristic equations for the model, certain sufficient conditions guarantee local asymptotic stability of infection-free and interior steady states. Whenever the time-delay crosses its critical value (threshold parameter), a Hopf bifurcation occurs. Furthermore, we use LaSalle’s invariance principle and Lyapunov functions to examine global stability for infection-free and interior steady states. Our results are illustrated by numerical simulations.

Dynamic Analysis of a Stochastic SEQIR Model and Application in the COVID-19 Pandemic

In this study, a deterministic SEQIR model with standard incidence and the corresponding stochastic epidemic model are explored. In the deterministic model, the reproduction number is given, and the local asymptotic stability of the equilibria is proved. When the reproduction number is less than unity, the disease-free equilibrium is locally asymptotically stable, whereas the endemic equilibrium is locally asymptotically stable in the case of a reproduction number greater than unity. A stochastic expansion based on a deterministic model is studied to explore the uncertainty of the spread of infectious diseases. Using the Lyapunov function method, the existence and uniqueness of a global positive solution are considered. Then, the extinction conditions of the epidemic and its asymptotic property around the endemic equilibrium are obtained. To demonstrate the application of this model, a case study based on COVID-19 epidemic data from France, Italy, and the UK is presented, together with numerical simulations using given parameters.

Time-Delayed Bioreactor Model of Phenol and Cresol Mixture Degradation with Interaction Kinetics

This paper is devoted to a mathematical model for phenol and p-cresol mixture degradation in a continuously stirred bioreactor. The biomass specific growth rate is presented as sum kinetics with interaction parameters (SKIP). A discrete time delay is introduced and incorporated into the biomass growth response. These two aspects—the mutual influence of the two substrates and the natural biological time delay in the biomass growth rate—are new in the scientific literature concerning bioreactor (chemostat) models. The equilibrium points of the model are determined and their local asymptotic stability as well as the occurrence of local Hopf bifurcations are studied in dependence on the delay parameter. The existence and uniqueness of positive solutions are established, and the global stabilizability of the model dynamics is proved for certain values of the delay. Numerical simulations illustrate the global behavior of the model solutions as well as the transient oscillations as a result of the Hopf bifurcation. The performed theoretical analysis and computer simulations can be successfully used to better understand the biodegradation dynamics of the chemical compounds in the bioreactor and to predict and control the system behavior in real life conditions.

Eradication Conditions of Infected Cell Populations in the 7-Order HIV Model with Viral Mutations and Related Results

In this paper, we study possibilities of eradication of populations at an early stage of a patient’s infection in the framework of the seven-order Stengel model with 11 model parameters and four treatment parameters describing the interactions of wild-type and mutant HIV particles with various immune cells. We compute ultimate upper bounds for all model variables that define a polytope containing the attracting set. The theoretical possibility of eradicating HIV-infected populations has been investigated in the case of a therapy aimed only at eliminating wild-type HIV particles. Eradication conditions are expressed via algebraic inequalities imposed on parameters. Under these conditions, the concentrations of wild-type HIV particles, mutant HIV particles, and infected cells asymptotically tend to zero with increasing time. Our study covers the scope of acceptable therapies with constant concentrations and values of model parameters where eradication of infected particles/cells populations is observed. Sets of parameter values for which Stengel performed his research do not satisfy our local asymptotic stability conditions. Therefore, our exploration develops the Stengel results where he investigated using the optimal control theory and numerical dynamics of his model and came to a negative health prognosis for a patient. The biological interpretation of these results is that after a sufficiently long time, the concentrations of wild-type and mutant HIV particles, as well as infected cells will be maintained at a sufficiently low level, which means that the viral load and the concentration of infected cells will be minimized. Thus, our study theoretically confirms the possibility of efficient treatment beginning at the earliest stage of infection. Our approach is based on a combination of the localization method of compact invariant sets and the LaSalle theorem.

Asymptotic and stability analysis of solutions for a Keller Segel chemotaxis model

A kind of Keller Segel chemotaxis model has a wide range of applications, but its coupling relationship is very complex. The commonly used method of constructing the upper and lower solutions is no longer suitable for the model solution, which results in a long time for its analysis. In this paper, we propose a method to analyze the asymptotic behavior and stability of a Keller Segel chemotaxis model. The previous methods of first formally and then rigorously, the asymptotic expansion of these monotone steady states, and then we use this fine information on the spike to prove its local asymptotic stability. Moreover, we obtain the uniqueness of such steady states. The asymptotic behavior of the solution of a Keller Segel chemotaxis model is analyzed, and the asymptotic rate is calculated; According to the limitation of Neumann boundary condition, the complete blow up of chemotaxis model solution and the stability of the initial value of the complete blow up time are studied, and the asymptotic and stability analysis of a kind of Keller Segel chemotaxis model solution is completed. The experimental results show that the proposed method takes less time to solve a kind of Keller Segel chemotaxis model, improves the efficiency of the solution, and the accuracy of the solution is higher.

Stability Analysis of Delayed Age-Structured Resource-Consumer Model of Population Dynamics With Saturated Intake Rate

This article studies nonlinear n-resource-consumer autonomous system with age-structured consumer population. The model of consumer population dynamics is described by a delayed transport equation, and the dynamics of resource patches are described by ODE with saturated intake rate. The delay models the digestion period of generalist consumer and is included in the calorie intake rate, which impacts the consumer’s fertility and mortality. Saturated intake rate models the inhibition effect from the behavioral change of the resource patches when they react to the consumer population growing or from the crowding effect of the consumer. The conditions for the existence of trivial, semi-trivial, and non-trivial equilibria and their local asymptotic stability were obtained. The local asymptotic stability/instability of non-trivial equilibrium of a system with depleted patches is defined by new derived criteria, which relate the demographic characteristics of consumers with their search rate, growth rate of resource in patches, and behavioral change of the food resource when consumer population grows. The digestion period of a generalist consumer does not cause local asymptotical instabilities of consumer population at the semi-trivial and nontrivial equilibria. These theoretical results may be used in the study of metapopulation dynamics, desert locust populations dynamics, prey-predator interactions in fisheries, etc. The paper uses numerical experiments to confirm and illustrate all dynamical regimes of the n-resource-consumer population.

Conformable Fractional SEIR Epidemic Model With Treatment

Background: The present paper is based on models of conformable fractional differential equation to describe the dynamics of certain epidemics. Methods: In this paper we have divided the population in the susceptible, exposed, infectious, recovered and also describe the treatment modalities. Results: The analytical study of the model show two equilibrium points (disease free equilibrium and endemic equilibrium). Conclusion: For both cases local asymptotic stability has been proven. In the conclusion we have presented the numerical simulation.

On an SE(Is)(Ih)AR epidemic model with combined vaccination and antiviral controls for COVID-19 pandemic

In this paper, we study the nonnegativity and stability properties of the solutions of a newly proposed extended SEIR epidemic model, the so-called SE(Is)(Ih)AR epidemic model which might be of potential interest in the characterization and control of the COVID-19 pandemic evolution. The proposed model incorporates both asymptomatic infectious and hospitalized infectious subpopulations to the standard infectious subpopulation of the classical SEIR model. In parallel, it also incorporates feedback vaccination and antiviral treatment controls. The exposed subpopulation has three different transitions to the three kinds of infectious subpopulations under eventually different proportionality parameters. The existence of a unique disease-free equilibrium point and a unique endemic one is proved together with the calculation of their explicit components. Their local asymptotic stability properties and the attainability of the endemic equilibrium point are investigated based on the next generation matrix properties, the value of the basic reproduction number, and nonnegativity properties of the solution and its equilibrium states. The reproduction numbers in the presence of one or both controls is linked to the control-free reproduction number to emphasize that such a number decreases with the control gains. We also prove that, depending on the value of the basic reproduction number, only one of them is a global asymptotic attractor and that the solution has no limit cycles.

Global Stability Analysis of a Bioreactor Model for Phenol and Cresol Mixture Degradation

We propose a mathematical model for phenol and p-cresol mixture degradation in a continuously stirred bioreactor. The model is described by three nonlinear ordinary differential equations. The novel idea in the model design is the biomass specific growth rate, known as sum kinetics with interaction parameters (SKIP) and involving inhibition effects. We determine the equilibrium points of the model and study their local asymptotic stability and bifurcations with respect to a practically important parameter. Existence and uniqueness of positive solutions are proved. Global stabilizability of the model dynamics towards equilibrium points is established. The dynamic behavior of the solutions is demonstrated on some numerical examples.