Asymptotic behavior of the solutions for a stochastic SIRS model with information intervention.

In this paper, a stochastic SIRS epidemic model with information intervention is considered. By constructing an appropriate Lyapunov function, the asymptotic behavior of the solutions for the proposed model around the equilibria of the deterministic model is investigated. We show the average in time of the second moment of the solutions of the stochastic system is bounded for a relatively small noise. Furthermore, we find that information interaction response rate plays an active role in disease control, and as the intensity of the response increases, the number of infected population decreases, which is beneficial for disease control.

Droplet Growth Model for Dropwise Condensation on Concave Hydrophobic Surfaces.

In this paper, mathematical models for predicting the droplet growth and droplet distribution of dropwise condensation on hydrophobic concave surface are developed and a theoretical analysis of the results of the model simulation is made. Under the assumptions of the Cassie-Baxter wetting mode and the consideration of noncondensable gases, the droplet growth model is not only established by heat transfer through a single droplet but also considered the thermal conductive resistance of the surface promoting layer. In addition, a droplet distribution model has been built based on the population balance theory. According to the calculation, the main thermal resistance in the droplet growth process is the conductive resistance inside the droplet. With the increase of contact angle, the above-mentioned thermal resistance increases; thus, the higher the hydrophobicity is, the slower the droplet growth and the less the droplet density are. Besides, the lower the temperature of the condensing surface is, the faster the droplets grow and the less the droplet density is. The models provide a mathematical tool for predicting the droplet radius at the initial stage of dewing on the concave surface and contribute to the design of functional surfaces in the field of water harvesting.

Dynamics analysis of a delayed virus model with two different transmission methods and treatments.

In this paper, a delayed virus model with two different transmission methods and treatments is investigated. This model is a time-delayed version of the model in (Zhang et al. in Comput. Math. Methods Med. 2015:758362, 2015). We show that the virus-free equilibrium is locally asymptotically stable if the basic reproduction number is smaller than one, and by regarding the time delay as a bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of the endemic equilibrium. Finally, we give some numerical simulations to illustrate the theoretical findings.

Global dynamics of a model for treating microorganisms in sewage by periodically adding microbial flocculants.

In this paper, a mathematical model for microbial treatment in livestock and poultry sewage is proposed and analyzed. We consider periodic addition of microbial flocculants to treat microorganisms such as Escherichia coli in sewage. Different from the traditional models, a class of composite dynamics models composed of impulsive differential equations is established. Our aim is to study the relationship between substrate, microorganisms and flocculants in sewage systems as well as the treatment strategies of microorganisms. Precisely, we first show the process of mathematical modeling by using impulsive differential equations. Then by using the theory of impulsive differential equations, the dynamics of the model is investigated. Our results show that the system has a microorganismsextinction periodic solution which is globally asymptotically stable when a certain threshold value is less than one, and the system is permanent when a certain threshold value is greater than one. Furthermore, the control strategy for microorganisms treatment is discussed. Finally, some numerical simulations are carried out to illustrate the theoretical results.

State-Dependent Pulse Vaccination and Therapeutic Strategy in an SI Epidemic Model with Nonlinear Incidence Rate.

In this paper, the state-dependent pulse vaccination and therapeutic strategy are considered in the control of the disease. A pulse system is built to model this process based on an SI ordinary differential equation model. At first, for the system neglecting the impulse effect, we give the classification of singular points. Then for the pulse system, by using the theory of the semicontinuous dynamic system, the dynamics is analyzed. Our analysis shows that the pulse system exhibits rich dynamics and the system has a unique order-1 homoclinic cycle, and by choosing p as the control parameter, the order-1 homoclinic cycle disappears and bifurcates an orbitally asymptotical stable order-1 periodic solution when p changes. Numerical simulations by maple 18 are carried out to illustrate the theoretical results.

Caspase-1-Mediated Pyroptosis of the Predominance for Driving CD4[Formula: see text] T Cells Death: A Nonlocal Spatial Mathematical Model.

Caspase-1-mediated pyroptosis is the predominance for driving CD4[Formula: see text] T cells death. Dying infected CD4[Formula: see text] T cells can release inflammatory signals which attract more uninfected CD4[Formula: see text] T cells to die. This paper is devoted to developing a diffusive mathematical model which can make useful contributions to understanding caspase-1-mediated pyroptosis by inflammatory cytokines IL-1[Formula: see text] released from infected cells in the within-host environment. The well-posedness of solutions, basic reproduction number, threshold dynamics are investigated for spatially heterogeneous infection. Travelling wave solutions for spatially homogeneous infection are studied. Numerical computations reveal that the spatially heterogeneous infection can make [Formula: see text], that is, it can induce the persistence of virus compared to the spatially homogeneous infection. We also find that the random movements of virus have no effect on basic reproduction number for the spatially homogeneous model, while it may result in less infection risk for the spatially heterogeneous model, under some suitable parameters. Further, the death of infected CD4[Formula: see text] cells which are caused by pyroptosis can make [Formula: see text], that is, it can induce the extinction of virus, regardless of whether or not the parameters are spatially dependent.

Threshold Dynamics of a Stochastic SIR Model with Vertical Transmission and Vaccination.

A stochastic SIR model with vertical transmission and vaccination is proposed and investigated in this paper. The threshold dynamics are explored when the noise is small. The conditions for the extinction or persistence of infectious diseases are deduced. Our results show that large noise can lead to the extinction of infectious diseases which is conducive to epidemic diseases control.

Global Dynamics of a Virus Dynamical Model with Cell-to-Cell Transmission and Cure Rate.

The cure effect of a virus model with both cell-to-cell transmission and cell-to-virus transmission is studied. By the method of next generation matrix, the basic reproduction number is obtained. The locally asymptotic stability of the virus-free equilibrium and the endemic equilibrium is considered by investigating the characteristic equation of the model. The globally asymptotic stability of the virus-free equilibrium is proved by constructing suitable Lyapunov function, and the sufficient condition for the globally asymptotic stability of the endemic equilibrium is obtained by constructing suitable Lyapunov function and using LaSalle invariance principal.