International Journal of Biomathematics

Expression of cellular genes is regulated by binding of transcription factors to their promoter, either activating or inhibiting transcription of a gene. Particularly interesting is the case when the expressed protein regulates its own transcription. In this paper, the features of this self-regulating process are investigated. In the presented model here, the gene can be in two states. Either a protein is bound to its promoter or not. The steady state distributions of protein during and just before switching from one state to the next state are analyzed. Moreover, a powerful numerical method based on the corresponding master equation to compute the protein distribution in the steady state is presented and compared to an already-existing method. Additionally the special case of self-regulation, in which protein can only be produced, if one of these proteins is bound to the promoter region, is analyzed. Furthermore, a self-regulating gene is compared to a similar gene, which also has two states and produces the same amount of proteins but is not regulated by its protein-product.

Understanding the dynamical mechanism of the host–parasite interactions is one of important issues on host–parasite association. In this paper, we formulate a three-dimensional host–macroparasite system to describe the host–parasite interactions, which includes the logistic growth rate of host population, the important free-living stage and the host fecundity reduction due to parasite infection. The purpose of the paper is to investigate the asymptotical behavior of the system. By using the properties of the solution to non-autonomous linear system, the basic production number ${{ℛ}}_0$ is proved to be a threshold which determines the outcome of the parasites. If ${{ℛ}}_0<1$, the parasite will eventually die out, and if ${{ℛ}}_0>1$ the parasite will be uniformly persistent. Hopf bifurcation of the system is further studied, and sufficient conditions for the Hopf bifurcation are obtained. By using the singular perturbation techniques, the system is separated into two time scales with a faster time scale for the free-living infective particles and a slower time scale for the population dynamics of host and parasite, and then a complete analysis of the dynamics on the slow manifold is conducted. The theoretical results show that the level of aggregation of parasites within host may influence the persistence and stability of the system.

In this paper, a delayed prey–predator model with discontinuous harvesting policies is investigated, in which the predator population consumes the prey according to Hassell–Varley-type functional response. Under some reasonable assumptions on the discontinuous harvesting functions, the local existence and global existence of positive solution in sense of Filippov to the system are obtained. Based on the functional differential inclusions theory and the topological degree theory of set-valued analysis, a series of useful criteria on existence and non-existence of the positive periodic solution is established for the system. Finally, some numerical examples are given to show the applicability and effectiveness of the obtained results. It is worthy to point out that the discontinuous harvesting policies are superior to continuous harvesting policies, which are usually adopted in previous papers.

In the present paper, we investigate a prey–predator system with disease in both prey and predator populations and the predator population is cannibalistic in nature. The model is extended by introducing incubation delays in disease transmission terms. Local stability analysis of the system around the biologically feasible equilibria is studied. The bifurcation analysis of the system around the interior equilibrium is also studied. The sufficient conditions for the permanence of the system are derived in the presence of delays. We observe that incubation delays have the ability to destabilize the cannibalistic prey–predator system. Finally, we perform numerical experiments to substantiate our analytical findings.

Improving thermostability of an enzyme can accelerate the relevant chemical reaction. Thus, the analysis and prediction of thermophilic proteins are conducive to protein engineering and enzyme design. In this study, a novel method based on two-step discrimination was proposed to distinguish between thermophilic and non-thermophilic proteins. The model was rigorously benchmarked on an objective dataset including 915 thermophilic proteins and 793 non-thermophilic proteins. Results showed that the overall accuracy of our method is 94.44% in 5-fold cross-validation, which is higher than those of other published methods. We believe that the two-step discriminated strategy will become a promising method in the relevant field of protein bioinformatics.

This paper presents a mathematical model for bacterial growth, mutations, horizontal transfer and development of antibiotic resistance. The model is based on the so-called kinetic theory for active particles that is able to capture the main complexity features of the system. Bacterial and immune cells are viewed as active particles whose microscopic state is described by a scalar variable. Particles interact among them and the temporal evolution of the system is described by a generalized distribution function over the microscopic state. The model is derived and tested in a couple of case studies in order to confirm its ability to describe one of the most fundamental problems of modern medicine, namely bacterial resistance to antibiotics.

In this paper, we present a modified minimal model for glucose and insulin kinetics model. The model proposed here is a smooth approximation of the original non-smooth minimal model. The dynamical properties like dissipativity, existence of equilibrium and stability of the system at the equilibrium points are investigated. A linear feedback-based control strategy is studied to control the blood glucose level in the situation where the physical system fails to regulate the blood glucose level automatically. A critical control parameter value $k_c$ is determined in terms of the system parameters. Extensive numerical simulation is performed with different parameter sets. Assuming different values for the feedback gain parameter, ranges of physiological parameter $\alpha$ are determined where the feedback gain is sufficient to stabilize the control system.

The physical exercise imposes challenges on the human thermoregulatory system, as heat exchange between the body and environment is substantially impaired, which can lead to decrease in performance and increased risk of heat illness. In view of above a three-dimensional finite element model is proposed to study the effect of different intensities of physical exercise on temperature distribution in peripheral regions of human limbs under moderate climatic conditions. Human limb is assumed to have a cylindrical cross-section. The peripheral region of the human limb is divided into three natural components, namely epidermis, dermis and subdermal tissues. The model incorporates the effect of important physiological parameters like blood mass flow rate, metabolic heat generation, and thermal conductivity of the tissues. Appropriate boundary conditions have been framed based on the physical conditions of the problem. The model is transformed into the discretized variational form and finite element method (FEM) has been employed to obtain the solution. The numerical results have been used to obtain the temperature profiles in the region immediately after exercise for an unsteady state case. The thermal information generated from the model can be useful for developing protocols for improving performance of sportsmen, military persons and labor-intensive workers.

The aim of this paper is to investigate the optimal control problem for finite-dimensional bilinear systems and its application to a chemotherapy model. We characterize an optimal control that minimizes a quadratic cost functional in two cases of constrained admissible controls, then we give sufficient conditions for the uniqueness of such a control, and we derive useful algorithms for the computation of the optimal control. The established results are applied to a cancer chemotherapy bilinear model in order to simulate the optimal treatment protocol using two different approaches: one based on a limited instant toxicity, and the other on a limited cumulative toxicity along the therapy session.

Malaria parasites are strongly dependent on Anopheles mosquitoes for transmission; for this reason, mosquito population dynamics are a crucial determinant of malaria risk. However, temperature and rainfall play a significant role in both aquatic and adult stages of the Anopheles. Consequently, it is important to understand the biology of malaria vector mosquitoes in the study of malaria transmission. In this study, we develop a climate-based, ordinary-differential-equation model to analyze how rainfall and temperature determine mosquito population size. In the model, we consider in detail the influence of ambient temperature on gonotrophic and sporogonic cycles over Amajuba District, Kwazulu-Natal Province, South Africa. In particular, we further use the model to simulate the spatial distribution of the mosquito biting rate over the study region. Our results reflect high seasonality of the population of An. gambiae over the region and also demonstrate the influence of climatic factors on the mosquito population dynamics.

In this paper, we present a mathematical model that describes tumor-normal cells interaction dynamics focusing on role of drugs in treatment of brain tumors. The goal is to predict distribution and necessary quantity of drugs delivered in drug-therapy by using optimal control framework. The model describes interactions of tumor and normal cells using a system of reactions–diffusion equations involving the drug concentration, tumor cells and normal tissues. The control estimates simultaneously blood perfusion rate, reabsorption rate of drug and drug dosage administered, which affect the effects of brain tumor chemotherapy. First, we develop mathematical framework which models the competition between tumor and normal cells under chemotherapy constraints. Then, existence, uniqueness and regularity of solution of state equations are proved as well as stability results. Afterwards, optimal control problems are formulated in order to minimize the drug delivery and tumor cell burden in different situations. We show existence and uniqueness of optimal solution, and we derive necessary conditions for optimality. Finally, to solve numerically optimal control and optimization problems, we propose and investigate an adjoint multiple-relaxation-time lattice Boltzmann method for a general nonlinear coupled anisotropic convection–diffusion system (which includes the developed model for brain tumor targeted drug delivery system).

Ebola virus disease (EVD) has emerged as a rapidly spreading potentially fatal disease. Several studies have been performed recently to investigate the dynamics of EVD. In this paper, we study the transmission dynamics of EVD by formulating an SEIR-type transmission model that includes isolated individuals as well as dead individuals that are not yet buried. Dynamical systems analysis of the model is performed, and it is consequently shown that the disease-free steady state is globally asymptotically stable when the basic reproduction number, ${{ℛ}}_0$ is less than unity. It is also shown that there exists a unique endemic equilibrium when ${{ℛ}}_0>1$. Using optimal control theory, we propose control strategies, which will help to eliminate the Ebola disease. We use data fitting on models, with and without isolation, to estimate the basic reproductive numbers for the 2014 outbreak of EVD in Liberia and Sierra Leone.

The growth of plants and the quality and quantity of crop production depend on the uptake dynamics of water and mineral salts by the roots from the rhizosphere. This water and nutrient dynamics in the rhizosphere is still not well understood. In view of above a mathematical model is proposed to study the effect of water flux on the nitrate dynamics in the rhizosphere of a single maize root. The model incorporates the important mechanisms like diffusion and mass flow of solutes in the rhizosphere. The parameters like soil porosity, soil buffer power, radius of the root have incorporated in the model. Appropriate boundary and initial conditions have been framed based on the physical conditions of the problem. Coaxial circular sector elements have been employed to discretize the circular cross-section of the root. Galerkin finite element method has been employed to obtain the solution. The numerical results have been computed and used to study the effect of water flux on nitrate dynamics at the surface of the root and in the rhizosphere. The change percentage in nitrate concentration at the surface of the root in the rhizosphere of maize root due to change in water flux $10^{-7}$–$10^{-6}$cms${}^{-1}$, $10^{-6}$–$10^{-5}$cm${\mathrm{\text{s}}}^{-1}$ and $10^{-7}$–$10^{-5}$cm${\mathrm{\text{s}}}^{-1}$ is observed to be 0.28–1.21%, 2.91–13.73% and 3.20–15.11% in $\mu$mol${\mathrm{\text{cm}}}^{-3}$ respectively which is quite significant.

We study the spatiotemporal dynamics in a diffusive predator–prey system with time delay. By investigating the dynamical behavior of the system in the presence of Turing–Hopf bifurcations, we present a classification of the pattern dynamics based on the dispersion relation for the two unstable modes. More specifically, we researched the existence of the Turing pattern when control parameters lie in the Turing space. Particularly, when parameter values are taken in Turing–Hopf domain, we numerically investigate the formation of all the possible patterns, including time-dependent wave pattern, persistent short-term competing dynamics and stationary Turing pattern. Furthermore, the effect of time delay on the formation of spatial pattern has also been analyzed from the aspects of theory and numerical simulation. We speculate that the interaction of spatial and temporal instabilities in the reaction–diffusion system might bring some insight to the finding of patterns in spatial predator–prey models.

The paper characterizes the global threshold dynamics of an epidemic model of SIQS type in environments with fluctuations, where the quarantine class is explicitly involved. Criteria are established for the permanence and extinction of the infective in environments with time oscillations. In particular, we further consider an environment which varies periodically in time. The global threshold dynamic scenarios i.e. the existence and global asymptotic stability of the disease-free periodic solution, the existence of the endemic periodic solution and the permanence of the infective are completely characterized by the basic reproduction number defined by the spectral radius of an associated linear integral operator.