International Journal of Biomathematics

In this paper, we have introduced a six-compartmental epidemic model with hand, foot and mouth disease (HFMD) infection. The total population is divided into six subclasses, namely susceptible, exposed, infective in asymptomatic phase, infective in symptomatic phase, quarantined and recovered class. Some basic properties such as boundedness and non-negativity of solutions are discussed. The basic reproduction number ($R_0$) of the system is obtained using next generation matrix method. Then the deterministic dynamical behaviors of the system are studied. Our study includes the existence and stability analysis of equilibrium points of the system. The sensitivity analysis of our system helps us to find out the parameters of greater interest. Next, we deal with the epidemic model with three controls (two treatment controls with quarantine control). We show that there exists an optimal control, which is effective in controlling the disease outbreak in a cost effective way. Numerical simulation is presented with the help of MATLAB, which shows the reliability of our model from the practical point of view.

The experimentally verified fact that there is a high pressure zone in the lower part of the oesophagus has established that the earlier models fall short of representing the realistic swallowing process in the oesophagus. Since the high pressure is created by gradually increasing amplitudes of peristaltic waves, swallowing of Casson fluid in oesophagus is mathematically remodeled. It is revealed that in the case of exponentially increasing amplitude, pressure is non-uniformly distributed for different cycles. Pressure increases along the entire length of the oesophagus; and finally toward the end of the oesophageal flow, it increases quite significantly, probably to ensure delivery into the stomach. This is a similar observation for Newtonian as well as non-Newtonian fluids but Casson fluids need more pressure; and hence more efforts are required by the oesophagus to transport the fluid forward. When wave amplitude is small, flow rates are small. In such a case, Casson fluid requires higher flow rates for reflux to occur in comparison to Newtonian fluid. This tendency gradually diminishes with increasing amplitude. For a particular value of amplitude, there is no difference; and beyond that the trends are quite opposite. Thus, Casson fluid is found to be less prone to reflux near the wall. It is also concluded that for the Newtonian fluid as well as for the non-Newtonian Casson fluid, reflux is more likely to occur with increasing amplitude and it is further augmented by the addition of amplifying parameter.

Most economic and industrial processes are governed by inherently nonlinear dynamic system in which mathematical analysis (with few exceptions) is unable to provide general solutions; even the conditions to the existence of equilibrium point for the nonlinear dynamic system are simply not established in some special cases. In this paper, based on numerical solution of a nonlinear multi-stage automatic control dynamic (NMACD) in fed-batch culture of glycerol bioconversion to 1,3-propanediol (1,3-PD) induced by Klebsiella pneumoniae (K. pneumoniae), we consider an optimal design of the NMACD system. For convenience, the NMACD system is reconstructed together with the existence, uniqueness and continuity of solutions are discussed. Our goal is to prove the strong stability with respect to the perturbation of initial state for the solution to the NMACD system. To this end, we construct corresponding linear variational system for the solution to the NMACD system, and also prove the boundedness of fundamental matrix solutions to the linear variational system. On this basis, we prove the strong stability appearing above through the application of this boundedness.

We study an SEIRS epidemic model with an isolation and nonlinear incidence rate function. We have obtained a threshold value $R_0$ and shown that there is only a disease-free equilibrium point, when $R_0<1$ and an endemic equilibrium point if $R_0>1$. We have shown that both disease-free and endemic equilibrium point are globally stable.

In this paper, a hepatitis B viral infection model with a density-dependent proliferation rate of cytotoxic T lymphocyte (CTL) cells and immune response delay is investigated. By analyzing the model, we show that the virus-free equilibrium is globally asymptotically stable, if the basic reproductive ratio is less than one and an endemic equilibrium exists if the basic reproductive ratio is greater than one. By using the stability switches criterion in the delay-differential system with delay-dependent parameters, we present that the stability of endemic equilibrium changes and eventually become stable as time delay increases. This means majority of hepatitis B infection would eventually become a chronic infection due to the immune response time delay is fairly long. Numerical simulations are carried out to explain the mathematical conclusions and biological implications.

Water scarcity is not a new issue, neither is water pollution. While 70% of the earth’s surface is covered with water, only 3% of it is available as fresh water. Moreover the pollution of water resources has dramatically increased the problem of water scarcity over the last century. Bioremediation presents a cheap and effective solution of this problem. In particular, halophiles have been found to be effective in hypersaline wastewater treatment. Therefore, in this paper, we propose a nonlinear mathematical model to study the removal of a pollutant using halophiles in the hypersaline environment. The analysis of the model is being carried out using stability theory of differential equations. The results indicate that halophiles not only help in removing the organic pollutant, but also help in conversion of saline water into fresh water. The numerical simulations along with sensitivity analysis are performed to support the analytical results.

In this paper, a class of stochastic Lotka–Volterra system with feedback controls is considered. The purpose is to establish some criteria to ensure the system is globally dissipative in the mean square. By constructing suitable Lyapunov functions as well as combining with Jensen inequality and It$\widehat{\mathrm{\text{o}}}$ formula, the sufficient conditions are established and they are expressed in terms of the feasibility to a couple linear matrix inequalities (LMIs). Finally, the main results are illustrated by examples.

In this paper, we propose a predator–prey system with a competitor for the prey. The model incorporates a constant prey refuge and predation process follows Holling type II response function. Using the Routh–Hurwitz criterion, the sufficient conditions of locally asymptotically stable of all the equilibria are obtained. Furthermore, global stability of the positive equilibrium is investigated by constructing a suitable Lyapunov function. The occurrence of Hopf-bifurcation of the system is shown at a critical value “$m$” and the system can be stabilized by increasing amount of prey refuge. The result includes the sufficient conditions for uniform persistence. Numerical simulations are carried out to illustrate the obtained results and the dependence of the dynamic behavior on the prey refuge.

Recently, Clancy [SIR epidemic models with general infectious period distribution, Statist. Prob. Lett. 85 (2014) 1–5] has shown how SIR epidemics in which individuals’ infection periods are not necessarily exponentially distributed may be modeled in terms of a piecewise-deterministic Markov process (PDMP). In this paper, we present a more detailed description of the underlying PDMP, from which we analyze the population transmission number and the infection probability of a certain susceptible individual.

A nonlinear model representing oxygen diffusion accompanied by the Michaelis–Menten consumption kinetics inside a spherical cell is solved analytically by the differential transform method (DTM) and the modified Adomian decomposition method (MADM). A perfect agreement between the literature data and the results from the proposed solutions is found. The advantages and drawbacks of the two approaches are discussed and their efficiencies are compared through a CPU-time analysis.

A new method of MRI brain segmentation integrates fuzzy $c$-means (FCM) clustering and rough set theory. In this paper, we use rough set algorithm to find the suitable initial clustering number to initial clustering centers for FCM. Then we use FCM to MRI brain segmentation, but the algorithm of FCM has the limitation of converging to local infinitesimal point in medical segmentation. While avoiding being trapped in a local optimum, we use the particle swarm optimization algorithm to restrict convergence of FCM which can reduce calculation. The final experiment results show that improved algorithm not only retains the advantages of rapid convergence but also can control the local convergence and improve the global search ability. The method in this paper is better than that of cluttering performance.

In this paper, the stability and periodicity of memristor-based neural networks with time-varying delays are studied. Based on linear matrix inequalities, differential inclusion theory and by constructing proper Lyapunov functional approach and using linear matrix inequality, some sufficient conditions are obtained for the global exponential stability and periodic solutions of memristor-based neural networks. Finally, two illustrative examples are given to demonstrate the results.

The nonautonomous single-species Kolmogorov system is studied in this paper. Average conditions are obtained for permanence, global attractivity and extinction in the system. Applications of our main results to logistic equation and generalized logistic equation are given. It is shown that our average conditions are improvement of those in Vance and Coddington [J. Math. Biol. 27 (1989) 491–506] and some published literature on the system.

In this paper, the long-time behavior and blowup of solutions for two generalized Ginzburg–Landau type equations with several nonlinear source terms are investigated. First, the conditions in which the solutions of the two equations are positive are analyzed by using Green’s function. According to the characteristics of nonlinear terms, four different functionals are constructed for solving the problems. Finally, we obtain the long-time behavior and finite-time blowup of solutions for the initial and boundary value problem by using these functionals.

Based on a multi-scale view, in this paper, we study an age-structured within-host model with Crowley–Martin functional response for the control of viral infections. By means of semigroup and Lyapunov function, the global asymptotical property of infected steady state of the model is obtained. The results show that when the basic reproductive number falls below unity, the infection dies out. However, when the basic reproductive number exceeds unity, there exists a unique positive equilibrium which is globally asymptotically stable. This model can be deduced to different viral models with or without time delay.