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ABSTRACTThe structural stability of the hepatitis C model with the proliferation of infected and uninfected hepatocytes is investigated in this paper. The model is clinically verified to accurately reflect the viral dynamics of hepatitis C. It is demonstrated that the limit transition from the hepatitis C model to the Riccati system coupled with the diffusive terms leads to the elimination of the quadratic terms. Such a novel effect leads to the introduction of the concept of the deformed order-1 solitary solutions. The generalized operator of differentiation is used to construct the deformed order-1 solitary solutions to the Riccati system coupled with the diffusive terms. Finally, it is demonstrated that the Riccati system coupled with the diffusive terms admits non-deformed order-1 solitary solutions, which proves the structural instability of the hepatitis C model with the proliferation of infected and uninfected hepatocytes.

Solitary solutions to a two-tumor metastasis model represented by a system of multiplicatively coupled Riccati equations are considered in this paper. The interaction between tumors is modelled by a one-sided diffusive coupling when one coupled Riccati system influences the other, but not the opposite. Necessary and sufficient conditions for the existence of solitary solutions to the composite system of Riccati equations are derived in the explicit form. Computational experiments are used to demonstrate the transitions from one steady-state to another via non-monotonous trajectories.

A scheme for the integration of $ \, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)} $-type fractional differential equations (FDEs) is presented in this paper. The approach is based on the expansion of solutions to FDEs via fractional power series. It is proven that $ \, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)} $-type FDEs can be transformed into equivalent $ \left(\, {}^{C} \mathit{\boldsymbol{{D}}}^{(1/n)}\right)^n $-type FDEs via operator calculus techniques. The efficacy of the scheme is demonstrated by integrating the fractional Riccati differential equation.

An analytical and computational framework for the derivation of solitary solutions to biological systems describing the cooperation and competition of species and expressed by the system of Riccati equations coupled with multiplicative terms is presented in this paper. It is demonstrated that relationships between these solitary solutions can be either direct or inverse. Thus, an infinitesimal perturbation of one population would lead to an infinitesimal change in the other population – if only both solitary solutions are coupled with the direct relationship. But, in general, that is not true if solitary solutions are coupled with the inverse relationship – an infinitesimal perturbation of one population may result into a non-infinitesimal change in the other population. Necessary and sufficient conditions for the existence of solitary solutions are derived in the space of the system's parameters and initial conditions.

It is well known that solutions of many ordinary differential equations allow presentation in the power series form. In the paper we introduce and analyze rigorous conditions under which the later series (solutions) can be reduced to a finite sum of standard functions. Some areas of practical applicability of the proposed algorithm are discussed. First Published Online: 14 Oct 2010

A meta-model of diffusively coupled Lotka-Volterra systems used to model various biomedical phenomena is considered in this paper. Necessary and sufficient conditions for the existence of n th order solitary solutions are derived via a modified inverse balancing technique. It is shown that as the highest possible solitary solution order n is increased, the number of nonzero solution parameter values remains constant for solitary solutions of order n > 3 . Analytical and computational experiments are used to illustrate the obtained results., Competing Interests: Competing interestsThe authors declare that they have no competing interests., (© The Author(s) 2021.)

This paper aims to develop a novel computational technique for the detection of the transit through the anaerobic threshold. This technique uses only cardiac intervals derived from the electrocardiogram and is based on algebraic relationships between RR and QRS intervals. Electrocardiograms are measured during the load and the recovery processes. Algebraic relationships between cardiac intervals are used not only to identify the anaerobic threshold but also to characterise individual features of the person during the transit through the threshold. The ratio between carbon dioxide and oxygen in the exhaled air is used to validate the results. The algebraic relationship between cardiac intervals serves as a stand-alone indicator for both the determination of the anaerobic threshold and the characterization of the performance of the person during the load and the recovery processes., Competing Interests: The authors have declared that no competing interests exist.

This paper presents the concept of perfect matrices of Lagrange differences which are used to analyze relationships between RR and JT intervals during the bicycle ergometry exercise. The concept of the perfect matrix of Lagrange differences, its parameters, the construction of the load function and the corresponding optimization problem, the introduction of internal and external smoothing, embedding of the scalar parameter time series into the phase plane-all these computational techniques allow visualization of complex dynamical processes taking place in the cardiovascular system during the load and the recovery processes. Detailed analysis is performed with one person's RR and JT records only-but the presented techniques open new possibilities for novel interpretation of the dynamics of the cardiovascular system.

An analytical and computational framework for the derivation of solitary solutions to biological systems describing the cooperation and competition of species and expressed by the system of Riccati equations coupled with multiplicative terms is presented in this paper. It is demonstrated that relationships between these solitary solutions can be either direct or inverse. Thus, an infinitesimal perturbation of one population would lead to an infinitesimal change in the other population - if only both solitary solutions are coupled with the direct relationship. But, in general, that is not true if solitary solutions are coupled with the inverse relationship - an infinitesimal perturbation of one population may result into a non-infinitesimal change in the other population. Necessary and sufficient conditions for the existence of solitary solutions are derived in the space of the system's parameters and initial conditions.