International Journal of Biomathematics

Infection of human immunodeficiency virus (HIV) is determined through the decay of healthy CD4+ T-cells in a well-mixed compartment, such as a bloodstream. A mathematical model is considered to illustrate the effects of combined drug therapy, i.e. reverse transcription plus protease inhibitor, on viral growth and T-cell population dynamics. This model is used to explain the existence and stability of infected and uninfected steady states in HIV growth. An analytical technique, called variational iteration method (VIM), is used to solve the mathematical model. This method is modified to obtain the rapidly convergent successive approximations of the exact solution. These approximations are obtained without any restrictions or the transformations that may change the physical behavior of the problem. Numerical simulations are computed and exhibited to illustrate the effects of proposed drug therapy on the growth or decay of infection.

This paper presents an application of finite element method to study the thermoregulatory behavior of three layers of human dermal parts with varying properties. The investigation of temperature distributions in epidermis, dermis and subcutaneous tissue together with Crank–Nicholson scheme at various atmospheric conditions was carried out. The finite element method has been applied to obtain the numerical solution of governing differential equation for one-dimensional unsteady state bioheat equation using suitable values of parameters that affect the heat transfer in human body. The outer skin is assumed to be exposed to cold atmospheric temperatures and the loss of heat due to convection, radiation and evaporation has been taken into consideration. The important parameters like blood mass flow rate, metabolic heat generation rate and thermal conductivity are taken heterogeneous in each layer according to their distinct physiological and biochemical activities. The temperature profiles at various nodal points of the skin and in vivo tissues have been calculated with respect to the severe cold ambient temperatures. The conditions under which hypothermia, non-freezing and freezing injuries develop were illustrated in the graphs.

The paper addresses the analysis of nonlinear dynamical models of some microbial growth processes. Equilibrium points, stability analysis, and structural properties are studied for different bioprocesses with various kinetics structures. First, a simple micro-organism growth process on a single limiting substrate is widely analyzed. Second, a microbial growth process combined with an enzyme-catalyzed reaction is investigated. The analysis shows that these kinds of bioprocesses have multiple equilibria, stable or unstable, operational or non-operational. The partition of nonlinear model in linear and nonlinear parts via some structural properties leads to kinetic decoupling and facilitates the equilibria and stability analysis. The performed research is useful for model reduction and for the design of observers and control algorithms. To illustrate the study results, several numerical simulations are provided.

This study considers SEIVR epidemic model with generalized nonlinear saturated incidence rate in the host population horizontally to estimate local and global equilibriums. By using the Routh–Hurwitz criteria, it is shown that if the basic reproduction number ${{ℛ}}_0<1$, the disease-free equilibrium is locally asymptotically stable. When the basic reproduction number exceeds the unity, then the endemic equilibrium exists and is stable locally asymptotically. The system is globally asymptotically stable about the disease-free equilibrium if ${{ℛ}}_0<1$. The geometric approach is used to present the global stability of the endemic equilibrium. For ${{ℛ}}_0>1$, the endemic equilibrium is stable globally asymptotically. Finally, the numerical results are presented to justify the mathematical results.

In this paper, a stochastic model of plague is first studied by subspace identification. First, the discrete model of plague is obtained based on the classical model. The corresponding stochastic model is proposed for the existence of stochastic disturbances. Second, for the model, the parameter matrices and noise intensity are obtained. Finally, the simulations of the model show that the subspace identification is more precise than least square method.

In this paper, we extensively studied a mathematical model of biology. It helps us to understand the dynamical procedure of population changes in biological population model and provides valuable predictions. In this model, we establish a variety of exact solutions. To study the exact solutions, we used a fractional complex transform to convert the particular partial differential equation of fractional order into corresponding partial differential equation and modified exp-function method is implemented to investigate the nonlinear equation. Graphical demonstrations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very effective, unfailing, well-organized and pragmatic for fractional PDEs and could be protracted to further physical happenings.

The present study deals with the introduction of an alteration in Legendre wavelets method by availing of the Picard iteration method for system of differential equations and named it Legendre wavelet-Picard method (LWPM). Convergence of the proposed method is also discussed. In order to check the competence of the proposed method, basic enzyme kinetics is considered. Systems of nonlinear ordinary differential equations are formed from the considered enzyme-substrate reaction. The results obtained by the proposed LWPM are compared with the numerical results obtained from Runge–Kutta method of order four (RK-4). Numerical results and those obtained by LWPM are in excellent conformance, which would be explained by the help of table and figures. The proposed method is easy and simple to implement as compared to the other existing analytical methods used for solving systems of differential equations arising in biology, physics and engineering.

Based on literature [J. Q. Li, Z. E. Ma and F. Q. Zhang, Stability analysis for an epidemic model with stage structure, J. Appl. Math. Comput. 9 (2008) 1672–1679], incorporating the recovery of the infected population with the length of the infectious periods, a modified epidemic disease SIS model with delay and stage was investigated. First, the criteria keeping stability with delay were given. Next, in order to lower the level of the infected individuals and minimize the cost of treatment, mixed, early and late therapeutic strategies were introduced into our model, respectively. Then we investigated the existence and uniqueness of optimal controls. And then, we expressed the unique optimal control in terms of the solution of the optimality systems. Finally, by numerical simulations, several important results were acquired: (1) The terminal time influenced the early optimal control largely. In detail, for a shorter terminal time it was optimal to initiate treatment with maximal effort at the start of the epidemic and continue treatment with maximal effort until the switch time was arrived. But for a longer terminal time, the maximal treatment effort need not be a prerequisite at the start or end of the epidemic but it was obligatory at the metaphase of the epidemic. (2) For our SIS model, minimizing the total infectious burden of the disease can be achieved by only early optimal treatment tactics. (3) For a disease with a shorter infectious period time, more cost would be spent to control the disease in order to achieve the optimal control objective. Otherwise, a relative lower cost would be to control the disease with a longer infectious period.

Complex dynamics of modified Hastings–Powell (HP) model (phytoplankton-zoo-plankton-fish) with Holling type IV functional response and density-dependent mortality (closure terms) for top predator species is investigated in this paper. Closure terms describe the mortality of top predator in plankton food chain models. Modified HP model with Holling type IV functional response gives rise to similar type of chaotic dynamics (inverted “teacup attractor”) as observed in original HP model with Holling type II functional response. It is observed that introduction of nonlinear closure terms eliminate chaos and system dynamics becomes stable. Observation of this paper support the “Steele–Henderson conjecture” that, nonlinear closure terms eliminate or reduces limit cycles and chaos in plankton food chain models. Chaotic or stable dynamics are numerically verified by Lyapunov exponents (LE) method and Sil’nikov eigenvalue analysis and also illustrated graphically by plotting bifurcation diagrams. It is assumed that mortality of fish population, caused by higher-order predators (which are not explicitly included in the model) is not constant, rather it exhibits random variation throughout the year. To incorporate the effect of random mortality of fish population, white noise term is introduced into the original deterministic model. It is observed that the corresponding stochastic model is stable in mean square when the intensity of noise is small.

In this paper, a circular three-layer flow model is proposed to study mucus transport in the airways due to air motion caused by mild forced expiration or mild coughing. Mucus is represented by four-parameter viscoelastic fluid, a combination of Maxwell and Voigt elements, whereas air and serous fluid are taken as Newtonian fluids (incompressible). The pressure gradient generated in the fluid layers is assumed to be given by a time-dependent function representing mild forced expiration or mild cough in the airways causing laminar flow. The effect of slip velocity at the mucus–serous interface caused by the presence of surfactant and at the top surface caused by immotile cilia are also taken into account. The roles of rheological properties of mucus on its transport are studied. The effect of serous fluid and its viscosity on mucus transport is also considered.

We investigate a generalized form of the Boussinesq equation, relevant for nerve pulse propagation in biological membranes. The generalized conditional symmetry (GCS) method is applied in order to obtain the conditions that enable the equation to admit a special class of second-order GCSs. For the case of quadratic nonlinearities, we outline a new class of invariant solutions.

In this paper, we explain why the chaotic mutation (CM) model of J. M. Bahi and C. Michel (2008) simulates the genes mutations over time with good accuracy. It is firstly shown that the CM model is a truly chaotic one, as it is defined by Devaney. Then, it is established that mutations occurring in genes mutations have indeed a same chaotic dynamic, thus making relevant the use of chaotic models for genomes evolution. Transposition and inversion dynamics are finally investigated.

In this paper, a nonautonomous stochastic food-chain system with functional response and impulsive perturbations is studied. By using Itô’s formula, exponential martingale inequality, differential inequality and other mathematical skills, some sufficient conditions for the extinction, nonpersistence in the mean, persistence in the mean, and stochastic permanence of the system are established. Furthermore, some asymptotic properties of the solutions are also investigated. Finally, a series of numerical examples are presented to support the theoretical results, and effects of different intensities of white noises perturbations and impulsive effects are discussed by the simulations.

In this paper, a generalized impulsive model of hematopoiesis with infinite delays and linear harvesting term is investigated. The main purpose of this paper is to study the existence, uniqueness and exponential stability of the positive pseudo-almost periodic solutions, which improve and extend some known relevant results. Moreover, an example is given to illustrate the main findings.

A periodic pulse differential equation model of tumor immunotherapy is established by considering the periodic and transient behavior of infusing immune cells. Using comparison theorem and Floquet multiplier theory of the impulsive differential equation, the boundedness of the model solution, the existence and stability of the free-tumor periodic solution are given. Furthermore, the persistence of the system is analyzed. Numerical simulations are carried to confirm the main theorems.

In this work, we construct a stage-structured single population system with winter hibernation and impulsive effect in polluted environment. All solutions of the investigated system are proved to be uniformly ultimately bounded. The conditions of the population-extinction solution of the investigated system are obtained. The permanent condition of the investigated system is also obtained. Finally, numerical analysis is inserted to illustrate the results. Our results indicate that the environmental pollution will reduce biological diversity of the natural world. Our results also provide reliable tactic basis for the practical biological resource management.