International Journal of Biomathematics

A heroin model with nonlinear incidence rate and age structure is investigated. The basic reproduction number is determined whether or not a heroin epidemic breaks out. By employing the Lyapunov functionals, the drug-free equilibrium is globally asymptotically stable if $R_0\le 1$; while the drug spread equilibrium is also globally asymptotically stable if $R_0>1$. Our results imply that improving detected rates and drawing up the efficient prevention play more important role than increasing the treatment for drug users.

Considering the uncertainty of kelp–abalone–sea cucumber population, an interval model of carbon sink fisheries with multi-trophic levels is proposed. The equilibria of the model are identified and the corresponding stabilities are discussed. And the existence of bionomic equilibrium of the model is investigated. Next the optimal controller is designed to obtain the optimal harvest using Pontryagin’s maximum principle. Numerical simulations are carried to prove the results.

A predator–prey discrete-time model with non-monotone functional response and density dependence is investigated in this paper. By using the comparison theorem of the difference equation, some sufficient conditions are obtained for the permanence of the system with variable coefficients. At the same time, a set of sufficient conditions about permanent of the system with almost periodic coefficients is also set up, which utilizes almost periodic characteristics of the system. Furthermore, the criteria which guarantee the existence of a globally attractive positive almost periodic solution of the system is established. An example is given to illustrate the feasibility of the obtained results.

In this paper, an exponential method is presented for the approximate solutions of the HIV infection model of CD4${}^{+}$T. The method is based on exponential polynomials and collocation points. This model problem corresponds to a system of nonlinear ordinary differential equations. Matrix relations are constructed for the exponential functions. By aid of these matrix relations and the collocation points, the proposed technique transforms the model problem into a system of nonlinear algebraic equations. By solving the system of the algebraic equations, the unknown coefficients are computed and thus the approximate solutions are obtained. The applications of the method for the considered problem are given and the comparisons are made with the other methods.

Intuitionistic fuzzy sets have many applications in different sciences. In this paper we verify one of the applications of intuitionistic fuzzy sets in medical diagnosis according to the ideas of Shannon et al., Wang and Xin, Grzregorzewski, Hung and Yang, and Yang and Chiclana. Actually by using the relationships between intuitionistic fuzzy sets and symptoms of patient we determine the kind of illness and finally we compare the methods.

In this paper, we deal with the problem of optimal control of a deterministic model of hepatitis C virus (HCV). In the first part of our analysis, a mathematical modeling of HCV dynamics which can be controlled by antiretroviral therapy as fixed controls has been presented and analyzed which incorporates two mechanisms: infection by free virions and the direct cell-to-cell transmission. Basic reproduction number is calculated and the existence and stability of equilibria are investigated. In the second part, the optimal control problem representing drug treatment strategies of the model is explored considering control parameters as time-dependent in order to minimize not only the population of infected cells but also the associated costs. At the end of the paper, the impact of combination of the strategies in the control of HCV and their effectiveness are compared by numerical simulation.

In this paper, we consider a stochastic predator–prey model with modified Leslie–Gower and Holling-type II schemes. We analyze long-time behavior of densities of the distributions of the solution. We prove that the densities can converge in $L^1$ to an invariant density or can converge weakly to a singular measure under appropriate conditions.

Inferring gene regulatory networks (GRNs) is a challenging task in Bioinformatics. In this paper, an algorithm, PCHMS, is introduced to infer GRNs. This method applies the path consistency (PC) algorithm based on conditional mutual information test (PCA-CMI). In the PC-based algorithms the separator set is determined to detect the dependency between variables. The PCHMS algorithm attempts to select the set in the smart way. For this purpose, the edges of resulted skeleton are directed based on PC algorithm direction rule and mutual information test (MIT) score. Then the separator set is selected according to the directed network by considering a suitable sequential order of genes. The effectiveness of this method is benchmarked through several networks from the DREAM challenge and the widely used SOS DNA repair network of Escherichia coli. Results show that applying the PCHMS algorithm improves the precision of learning the structure of the GRNs in comparison with current popular approaches.

This paper is concerned with the problem of synchronization analysis for discrete-time coupled neural networks. The networks under consideration are subject to: (1) the jumping parameters that are modeled as a continuous-time, discrete-state Markov process; (2) impulsive disturbances; and (3) time delays that include both the mode-dependent discrete and distributed delay. By constructing suitable Lyapunov–Krasovskii functional and combining with linear matrix inequality approach, several novel criteria are derived for verifying the global exponential synchronization in the mean square of such stochastic dynamical networks. The derived conditions are established in terms of linear matrix inequalities, which can be easily solved by some available software packages. A simulation example is presented to show the effectiveness and applicability of the obtained results.

In this paper, we introduce stochasticity into an SIR epidemic model with vaccination. The stochasticity in the model is a standard technique in stochastic population modeling. When the perturbations are small, by the method of stochastic Lyapunov functions, we carry out a detailed analysis on the dynamical behavior of the stochastic model regarding of the basic reproduction number $R_0$. If $R_0\le 1$, the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model. If $R_0>1$, there is a stationary distribution and the solution has the ergodic property, which means that the disease will prevail.

In this paper, we have investigated a model with three interacting species: non-toxic phytoplankton, toxic phytoplankton and zooplankton with Holling type II and III functional responses over the space and time. The role of toxin producing phytoplankton (TPP) has been studied. We have presented the theoretical analysis of pattern formation in spatially distributed population with local diffusion. The paper highlights the heterogeneity of HABs over space and time. The choice of parameter values and the functional response is important to study the effect of TPP, also it would depend more on the nonlinearity of the system. With the help of numerical simulations, we have observed the spatial and spatiotemporal patterns for plankton system. This study demonstrates that TPP plays an important role in controlling the dynamics. We have observed that prey’s anti-predator efforts promote predator switching. It has been found that high predation of TPP helps for the coexistence of toxic, non-toxic phytoplankton and zooplankton population.

This paper is intended to determine physical parameters describing volumetric heat capacity and thermal conductivity of sea ice in a quasi-linear thermodynamic system using field observations. The quasi-linear thermodynamic system of sea ice with unknown physical parameters is described, and the existence and uniqueness of its solution is proved. Then the physical parameters are taken as control variable, temperature deviations as objective function, and a parameter identification model is established. The existence of its optimal solution is discussed. To solve the identification model, a new algorithm containing genetic algorithm, Hooke–Jeeves algorithm and semi-implicit finite difference scheme is constructed. The physical parameters are calculated using the observations measured at Nella Fjord around Zhongshan Station, Antarctic in CHINARE 2006. For comparability and consistency with other works, a new international standard named TEOS-10 is used. To examine the validity of the identified results, another simulation for temperature profiles in different measurement period is operated. Numerical results show that better simulations of temperature distribution are possible with the identified parameters than EC1993. Therefore not only the identified parameters can be applied in sea ice modeling, but also this study can enrich and supplement observations of sea ice.

Lotka–Volterra population competition model plays an important role in mathematical models. In this paper, Julia set of the competition model is introduced by use of the ideas and methods of Julia set in fractal geometry. Then feedback control is taken on the Julia set of the model. And synchronization of two different Julia sets of the model with different parameters is discussed, which makes one Julia set change to be another. The simulation results show the efficacy of these methods.

To investigate the effect of heterogeneity on the global dynamics of two SIR epidemic models with general nonlinear incidence rate and infection delays, we formulate a multi-group model corresponding to the heterogeneity in the host population and a multi-stage model corresponding to heterogeneous stages of infection. Under biologically motivated considerations, we establish that the global dynamics for each of the two models is determined completely by the corresponding basic reproduction number: if the basic reproduction number is less than or equal to one, then the disease-free equilibrium is globally asymptotically stable and the disease dies out in all groups or stages; if the basic reproduction number is larger than one, then the disease will persist in all groups or stages, and there is a unique endemic equilibrium which is globally asymptotically stable. Then we conclude that the heterogeneity does not change the global dynamics of the SIR model when the incidence rate is a general nonlinear function. Our results extend a class of previous results and can be applied to the other epidemiological models. The proofs of the main results use Lyapunov functional and graph-theoretic approach.

This work is concerned about multiscale models of compact bone. We focus on the lacuna–canalicular system. The interstitial fluid and the ions in it are regarded as solvent and others are treated as solute. The system has the characteristic of solvation process as well as non-equilibrium dynamics. The differential geometry theory of surfaces is adopted. We use this theory to separate the macroscopic domain of solvent from the microscopic domain of solute. We also use it to couple continuum and discrete descriptions. The energy functionals are constructed and then the variational principle is applied to the energy functionals so as to derive desirable governing equations. We consider both long-range polar interactions and short-range nonpolar interactions. The solution of governing equations leads to the minimization of the total energy.

In order to study the fitting of some characteristics like low diameter at breast height (DBH), the height and height to the beginning of crown of the Oak species in two man-made and natural forests of this species in north of Guilan’s forests, Tooshi and Radarposhte forests were selected. A 100% inventory was done on a one-hectare sample plot in Tooshi man-made forests and on a two-hectare sample plot in Radarposhte natural forests. The results from this study showed that, in order to fit the diameter and the height of the Oak trees in man-made stands, Johnson’s S${}_{\mathrm{\text{B}}}$ and normal statistical distributions have more capabilities and the three-parameter gamma and the three-parameter Weibull distributions are appropriate for diameters and heights of the natural Oak trees in Radarposhte forest, respectively.