International Journal of Biomathematics

This study describes the consequences of the ciliary motion on the transport of seminal liquid through the ductus efferentes of the male reproductive tract. By assuming the seminal liquid as a couple stress fluid, we have formulated a mathematical model for a two-dimensional flow through an axially symmetric tube whose inner surface is ciliated in the form of a metachronal wave. The governing system consists of nonlinear coupled partial differential equations which is reduced to a system of ordinary differential equations by utilizing the long wavelength approximation in an environment of the inertia free flow. Exact solutions for the velocity distribution, the pressure gradient and the stream function are obtained in terms of the couple stress parameters and the ciliary metachronism. Special attention is given to the pumping and the trapping characteristics due to the cilia motility. The study reveals that the ciliary pumping has to be more efficient to transport a couple stress fluid as compared to a Newtonian fluid. Moreover, the theoretical results for the couple stress fluid are found to be in good agreement with those reported by [T. J. Lardner and W. J. Shack, Cilia transport, Bull. Math. Biophys. 34 (1972) 325–335]. The analytical results are also displayed in graphical form.

In this paper, a differential-algebraic predator–prey system with time delay is investigated, where the time delay is regarded as a parameter. By analyzing the corresponding characteristic equations, the local stability of the positive equilibrium and the existence of Hopf bifurcation are demonstrated. Furthermore, the explicit formulae which determine the stability, direction and other properties of bifurcating periodic solutions are obtained by applying the normal form theory and the center manifold argument. At last, some numerical simulations are carried out to illustrate the feasibility of our main results.

This work studies and numerically simulates a logistic-type model for the dynamics of Chagas disease, which is caused by the parasite T. cruzi and affects millions of humans and domestic mammals throughout rural areas in Central and South America. A basic model for the disease dynamics that includes insecticide spraying was developed in Spagnuolo et al. (2010) [27] and consists of a delay-differential equation for the vectors and three nonlinear ordinary differential equations for the populations of the infected vectors, infected humans and infected domestic mammals. In this work, the vector equation is modified by using a logistic term with zero, one or two delays or time lags. The aim of this study is three-fold: to numerically study the effects of using different numbers of delays on the model behavior; to find if twice yearly insecticide spraying schedules improve vector control; and to study the sensitivity of the system to the delays in the case of two delays, by introducing randomness in the delays. It is found that the vector equation with different number of delays has very different solutions. The “best” day of spraying is the middle of Spring and twice annual sprayings cause only minor improvements in disease control. Finally, the model is found to be insensitive to the values of the delays, when the delays are randomly distributed within rather narrow intervals or ranges centered on the parameter values used in Coffield et al. (2014) [8].

Maximum probability of existence of cancer in human bodies is normally diagnosed very late, so that it is highly cumbersome for physicians to cure. Reliability in predicting cancer at initial stage is always needed, so that curing and medical recovery is possible. In this paper, an investigation was made to diagnose the presence of blood cancer using Markov Chain Monte Carlo (MCMC) trace model, which is most efficient on a wide range of complex Bayesian statistical models. The analysis was carried out using version 18 of SPSS AMOS software. Totally, 19 components were considered from the blood samples of 750 patients. Various factors such as class, age, lymphatics, block of affarc, block of lymph c, block of lymph s, bypass, extravasate, regeneration of, early uptake in, lym nodes dimin, lym nodes enlar, change in lym, defect in node, changes in node, changes in strue special forms, dislocation, exclusion of node, number of nodes in blood cancer are analyzed. The maximum likelihood estimators of the parameters were derived and assessed their performance through a Monte Carlo simulation study. The convergence in prior distribution and posterior distribution takes irregular position in the diagrams and thus blood cancer is diagnosed through this model.

A general nonlinear mathematical model for the viral infection with humoral immunity and two distributed delays is proposed and analyzed. Two bifurcation parameters, the basic reproduction number, $R_0$ and the humoral immunity number, $R_1$ are derived. We established a set of conditions on the general functions which are sufficient to determine the global dynamics of the model. Utilizing Lyapunov functions and LaSalle’s invariance principle, the global asymptotic stability of all equilibria of the model is obtained. An example is presented and some numerical simulations are conducted in order to illustrate the dynamical behavior.

Induction of antitumor immunity by vaccination is one of the major current immunotherapy strategies. We present a mathematical model of the competition between immune cells and mammary carcinogenesis under the effect of Triplex vaccine. The model describes both humoral and cell-mediated immune responses against cancer cells. The control of the cancer cells growth occurs through the application of the pulse vaccination. Here we determine the relationship between the strength of the vaccine and the time required to eradicate cancer cells, and we present some simulations to illustrate our theoretical results, namely, the total cancer cells depletion, which is influenced by competition occurs among the immune and cancer cells.

We analyze the global stability of the coexisting equilibria for several models of commensalism, first by devising a procedure to modify several Lyapunov functionals which were introduced earlier for corresponding models of mutualism, further confirming their usefulness. It is seen that commensalism promotes global stability, in connection with higher-order self-limiting terms which prevent unboundedness. We then use the theory of asymptotically autonomous systems to prove global stability results for models of commensalism which are subject to Allee effects, finding that commensalisms of appropriate strength can overcome the influence of strong Allee effects.

The global dynamics of an SVEIR epidemic model with age-dependent waning immunity, latency and relapse are studied. Sharp threshold properties for global asymptotic stability of both disease-free equilibrium and endemic equilibrium are given. The asymptotic smoothness, uniform persistence and the existence of interior global attractor of the semi-flow generated by a family of solutions of the system are also addressed. Furthermore, some related strategies for controlling the spread of diseases are discussed.

We present in this paper a new mathematical model for the scheduling of angiogenic inhibitors in combination with a chemotherapeutic agent for a tumor. Our model takes into account the process of angiogenesis and the quality of the vasculature discriminating between stable blood vessels and unstable blood vessels. We characterize theoretically the optimal controls on drug distribution to minimize the number of cancer cells at the end of the treatment in a free horizon time problem with restrictions on the total amount of drug doses. Finally, we solve the optimal control problem by using numerical simulations, obtaining as a result that, despite the number of the tumor cells decrease with anti-angiogenic treatment, the best results are reached at the end of the chemotherapy treatment.

In this paper, we concern a class of the generalized delayed stochastic predator–prey models with feedback controls based on discrete observations. The existence of global positive solution is given first. Then we discuss the deterministic model briefly, and establish the necessary conditions and the sufficient conditions for almost-sure extinction and persistence in mean for the stochastic system, where we show that the feedback controls can change the properties of the population systems significantly. Finally, numerical simulations are introduced to support the main results.

A terminal sliding mode (TSM) control with self-tuning gain algorithm is proposed for the synchronization of coronary artery system under the existence of the unmodeled dynamics and the external disturbance. Considering the sliding mode dynamics of system, a criterion of selecting the parameters is derived to reach the point of equilibrium in the finite time. The theoretic analysis based on Lyapunov theory proved that the systems with the proposed TSM control with self-tuning scheme could be stabilized in finite time. The proposed method shows that the drive and response systems are synchronized and states of the response system track the states of the drive system in finite time. This information about the bound of unmodeled dynamics and the external disturbance is not needed in advance through self-tuning the gains of controller. The results for coronary artery system synchronization simulation show that the proposed TSM controller with self-tuning achieves better robustness and adaptation against unmodeled dynamics and the external disturbance, which offer the theory basis on curing myocardial infarction.

In this paper, three-dimensional system of the tuberculosis (TB) model is reduced into a two-dimensional first-order and one-dimensional second-order differential equations. We use the method of Jacobi last multiplier to construct linear Lagrangians of systems of two first-order ordinary differential equations and nonlinear Lagrangian of the corresponding single second-order equation. The Noether's theorem is used for determining conservation laws. We apply the techniques of symmetry analysis to a model to identify the combinations of parameters which lead to the possibility of the linearization of the system and provide the corresponding solutions.

The integrable hungry Lotka–Volterra (hLV) system stands for a prey–predator model in mathematical biology. The discrete-time hLV (dhLV) system is derived from a time discretization of the hLV system. The solution to the dhLV system is known to be represented by using the Casorati determinant. In this paper, we show that if the entries of the Casorati determinant become an extended Fibonacci sequence at the initial discrete time, then those are also an extended Fibonacci sequence at any discrete time. In other words, the extended Fibonacci sequence always appears in the entries of the Casorati determinant under the time evolution of the dhLV system with suitable initial setting. We also show that one of the dhLV variables converges to the ratio of two successive extended Fibonacci numbers as the discrete time goes to infinity.

In the present work, research efforts have focused on investigating codimension two and three Bogdanov–Takens bifurcations of a predator–prey system with additive Allee effect. According to the existence conditions of Bogdanov–Takens bifurcation, we give the associated generic unfolding, and derive the dynamical classification in the perturbation parameter plane using some smooth parameter-dependent transformations of coordinate. Moreover, some numerical examples and simulations are performed to complete and illustrate our results.

In this paper, we study the asymptotic behavior of the positive solutions of a system of the following difference equations: