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We consider a system of differential equations, which describes the free oscillations of a thin-walled conical shell of rotation with a vertex. Based on the analytical theory of systems of differential equations with a small parameter at the highest derivative and equations with a regular singular point, we establish the formal structure of regular integrals of the original equations.

The paper considers a three-dimensional system of differential equations describing the thermoelastic static equilibrium of homogeneous isotropic materials with voids. In the Cartesian coordinate s...

The initial equations for solving the axisymmetric problem are given and consideredboundary conditions on the end surfaces and average section of the design element. As a result, we get a system of partial differential equations that can be solved by the numerical-analytical (modified) method of straight lines. The transformation of the reduced equations of equilibrium in parts, as well as the reduced models of the boundary conditions of the end surfaces and the average section, are shown. As a result, a boundary value problem for the system of reduced differential equations in ordinary derivatives written in the Cauchy form with boundary conditions of the general form is obtained. The thermal conductivity of the cylindrical wall was calculated, the results were compared with analytical calculations and results of other authors, which confirms the reliability of the developed methodology. A computer simulation of the stress-strain state of a cylindrical structural element due to the complex action of temperature, force and kinematic effects was carried out. Important conclusions have been made for the use of the modified method of straight lines, which is free from the complications that arise when using the classical method of straight lines., Приведені вихідні рівняння для вирішення вісесиметричної задачі та розглянуті граничні умови на торцевих поверхнях і середньому перерізі розрахункового елементу. В результаті чого отримуємо систему диференціальних рівнянь в частинних похідних, що розв’язується чисельно-аналітичним (модифікованим) методом прямих. Показано перетворення редукованих рівнянь рівноваги в частинах, а також редуковані моделі граничних умов торцевих поверхонь і середнього перерізу. В результаті отримано граничну задачу для системи редукованих диференціальних рівнянь в звичайних похідних, записані в формі Коші з граничними умовами загального вигляду. Проведено розрахунок теплопровідності циліндричної стінки, результати порівнювалися з аналітичними розрахунками та результатами інших авторів, чим підтверджується достовірність розробленої методики. Проведено комп’ютерне моделювання НДС циліндричного елементу конструкції від комплексної дії температурного, силового та кінематичного впливів. Зроблені важливі висновки для використання модифікованого методу прямих, який позбавлений ускладнень,що виникають при використанні класичного методу прямих.

This paper studies the problem of optimizing the right-hand sides of nonlocal conditions with respect to a system of linear differential equations. Nonseparated nonlocal conditions linearly depend on the point and integral values of the unknown function. In the problem, it is required to determine the optimal values of the right-hand sides of nonlocal conditions. The issues of existence and uniqueness of a solution to a system of differential equations with nonlocal conditions under consideration are investigated, the convexity of the objective functional is proved. The obtained necessary optimality conditions for the values of the right-hand side terms of the nonlocal conditions allow using first-order optimization methods for the numerical solution of the problem. Computational experiments on solving a test problem are given and the results obtained are analyzed.

In this paper we study a linear pursuit differential game described by an infinite system of first-order differential equations in Hilbert space. The control functions of players are subject to geometric constraints. The pursuer attempts to bring the state of system from a given initial state to the origin for a finite time and the evader’s purpose is opposite. We obtain a formula for the guaranteed pursuit time and construct a strategy for pursuer. Also, we obtain a formula for the guaranteed evasion time.

In this paper, a polynomial system of plane differential equations is observed. The stability of the non-hyperbolic equilibrium point was analyzed using normal forms. The nonlinear part of the differential equation system is simplified to the maximum. Two nonlinear transformations were used to simplify first the quadratic and then the cubic part of the system of equations.

In this work we construct an ultracompact star configuration in the framework of Gravitational Decoupling by the Minimal Geometric Deformation approach. We use the complexity factor as a complementary condition to close the system of differential equations. It is shown that for a polynomial complexity the resulting solution can be matched with two different modified-vacuum geometries., 7 pages, 4 figures. Version accepted in EPJC

In this paper, we present the SIR pandemic model to evaluate the susceptible (R), infectious (I), and removed (R) for COVID-19 in Iraq for the period from the 22nd February 2020 to the 26th June 2020. We divided this period to three sub-periods, and for each sub-period, the real data has been cited from scientific trusted references to estimate the parameters that are required to solve the system of differential equations for the SIR pandemic model. Furthermore, the estimations are given by our model for the real active infection, recovery rate against case fatality rate, reproduction number (Figure presented.) and the growth factor of the real accumulation infection have all shown significant similarities between the reported data and the estimations produced by our model. © 2021 Taru Publications.

The kinetics of gas nitriding of Armco iron is studied using an integral diffusion model. A system of differential equations is used to calculate the thickness of the diffusion layers of e- and (e + γ′)-phases on the surface of saturated armco iron. The calculations do not allow for the incubation time, and the profile of the distribution of the concentration of nitrogen over the thickness of the nitrided layer is assumed to be nonlinear. The model is verified by comparing the values calculated within different kinetic approaches to experimental values of the thickness of nitrided layer. The effect of the nitriding potential on the kinetics of gas nitriding of Armco iron at 550°C is studied. The coincidence between the calculated and the experimental data is good.

One of the most important problems of social development is the organization of competition (struggle) of political parties. For the analysis and forecasting of this process, the most convenient method is mathematical modeling. The relevance of the development of a model of the struggle of parties is determined by the importance of the process under consideration for determining the strategy of the life of countries and peoples. In this study, a modified Richardson model (arms race) in the form of a dynamic system of differential equations is used to describe the competition of parties. An analytical solution of a dynamic system of differential equations is obtained using the operator method. To explain the developed method, a specific numerical example is considered. The features of the application of the model of competition of political parties are analyzed. The prospects for further development of the new developed method are determined. The developed method can be used for the analysis of interethnic, religious conflicts, for determining the maturation of an explosive, crisis situation in society. It allows you to analyze the clash of views and interests of individuals.

The purpose of this paper is to clarify the recommendations and to develop a methodology for calculating multi-piston pneumatic motors (pneumatic positioners) for analyzing the movement parameters of a pneumatic drive and taking into account the need for a smooth stop of the output rod. A system of differential equations is obtained, the solution of which gives a description of the dynamics processes occurring during the operation of a multi-piston pneumatic actuator. The software has been developed that allows to obtain a number of graphical dependencies, on the basis of which the designer decides on the expediency of using braking by the method of throttling the inputs and outputs of the pneumatic motor chambers.

Purpose: To develop a model of interaction between banks and insurance companies, allowing for a joint model assessment from the point of view of the possibility of sustainable operation and development. Findings: The article develops the theory of dynamic systems and a numerical method for solving the problem of assessing the results of interaction between banks and insurance companies based on a system of differential equations. The additive and multiplicative types of interaction are considered. The results of interaction are analyzed for various variants of the influence of the activities of the studied subjects. Practical Implications: The proposed approach is applicable to assess the effectiveness of the used model of interaction between banks and insurance companies. Originality/Value: The author's model of interaction between banks and insurance companies is original. The model has no analogues in the scientific literature of the studied subject area. Research Limitations/Future Research: The study proposes a generalized dynamic model of interaction without specifying the content of variables and the procedure for determining the input parameters. Its description and application determine potential areas for further research. Paper type: Theoretical.

"In the paper Operators on infinite dimensional cartesian product, (Analele Univ. Vest Timişoara, Mat. Inform., 48 (2010), 253–263), by I. A. Rus and M. A. Şerban, the authors give a generalization of the Fibre contraction theorem on infinite dimensional cartesian product. In this paper we give an application of this abstract result to an infinite system of differential equations. "

The modern practice of using vibratory machines when working with fine-size light-weight seeds is faced with such an undesirable phenomenon as the impact of aerodynamic forces and moments on the kinematics of vibrational movement of particles of the seed mixture fractions. According to the results of scientific studies devoted to the solution of this problem, only mathematical models of vibrational movement are used, where the aerodynamic factor is taken into account as taking the seeds by airflow. This is typical only for cleaning modes with the rebound of seeds from the vibrating surface. Aerodynamic forces and moments are present in them only as a force of aerodynamic resistance. The action of lateral aerodynamic forces and their moments are not taken into account. Their consideration allows to extend the range of action of the aerodynamic factor on the modes of vibration cleaning (vibroseparation) without rebound (but with sliding and rolling) which are of greater interest in terms of improving the efficiency of processing fine-size seeds. A mathematical model of seed vibration movement taking into account the action of a complete set of aerodynamic forces (dynamic resistance forces and lateral aerodynamic forces) and moments was proposed. This makes it possible to simulate non-lifting modes of vibrational movement of seeds. A system of algebraic equations that are linear with respect to the kinematic parameters of seed movement which was obtained by translating differential equations of movement into a finite-difference form was presented. The possibility of numerical solution of equations of movement by the Euler method was shown. The results of the evaluation of the model adequacy for the processes of vibration separation of tobacco seeds and false flax were presented. As shown by the results of calculations and experiments, the developed model provides an increase in the adequacy of the simulation results by 30% in comparison with the model where the aerodynamic factor is not taken into account

This paper provides a review of developed software capabilities (hereinafter Bioscheme software) as well as a calculation procedure for the internal exposure dose the case of the biokinetic model from an ICRP publication No. 78. Compared to the available software applied by the branch IRSE NNC RK, Bioscheme software allows solution to not only conventional tasks but also simulation and automatic derivation of a function of the intake, accumulation, and decorporation of radionuclides in non-ordinary cases.

The article assesses the impact of wind power plants on the components of the environment, which are compartments of complex landscape systems, taking into account a number of their parameters. A list of impact categories has been made up, which represent the load on the environment; also, for each category, the relative contribution of harmful factors has been identified, taking into account possible scenarios for waste management. For all potential impacts, using the Eco-indicator methodology, ecological profiles have been built, which made it possible to obtain the values of ecological indexes (impacts) and eco-indicators, expressed in eco-points, characterizing the impact of the wind power plant under study. Mathematical modeling of the processes of influence of a separate wind power plant on the subsystems and layers of compartments was carried out, according to the results of which a system of differential equations has been obtained, the input data for which are individual indexes and eco-indicators, as well as statistical information on the functioning of the elements of the hierarchical structure of compartments of a complex landscape system. The structure graphs are formalized using the Kolmogorov system of differential equations. It is proposed to study the dynamics of the structure on the basis of solving a system of differential equations using the fourth-order numerical Runge – Kutta method. By solving the system of equations, it is possible to study (predict) the developmental stages of a complex landscape system in dynamic and stationary modes during the impact of the life cycle of wind turbines on the subsystems and layers of their compartments, in order to optimize human activities to ensure minimal environmental impact. It is proposed to use the presented algorithms as a mathematical support for a cyber-physical system for studying the states of complex landscape systems and assessing the impact of wind turbines on environmental components. Keywords: cyber-physical system; renewable energy sources; complex landscape system; life cycle, ecosystem states; eco-points

A system with two degrees of freedom in the case of a double natural frequency is considered. The unperturbed system consists of two independent oscillators. The system coordinates describe an elliptical trajectory with four orbital elements. An analysis of the action of linear perturbations (forces) on the orbital elements is carried out. Perturbations are subdivided into six types of forces, and for each type of forces a system of differential equations for the orbital elements is obtained. For all six types of forces, a general solution of the system of differential equations in elementary functions is found.

This paper is devoted to the general theory of linear systems of fractional order pseudo-differential equations. Single fractional order differential and pseudo-differential equations are studied by many authors and several monographs and handbooks have been published devoted to its theory and applications. However, the state of systems of fractional order ordinary and partial or pseudo-differential equations is still far from completeness, even in the linear case. In this paper we develop a new method of solution of general systems of fractional order linear pseudo-differential equations and prove existence and uniqueness theorems in the special classes of distributions, as well as in the Sobolev spaces.

We consider a linear inhomogeneous system of differential equations of special form with three random coefficients defined by characteristic functionals. Operator functions generated by the functio...

The emergence of coronavirus disease 2019 (COVID-19) demonstrates the importance of research on understanding and accurately modeling the transmission and spread of pandemic. In this paper, we consider a susceptible-exposed-infected-recovered-deceased (SEIRD) system of differential equations to describe relationship among the number of susceptible individuals, the number of exposed individuals who are transmitting the virus, the number of infected individuals among the exposed people, the number of recovered individuals from those infected, and the number of deaths from those infected in a town, state or country. Based on the empirical results of transmission process of COVID-19 in the United States from April 16th to June 30th, 2020, we consider a few cases of contact rate, incidence rate, recovery rate, and mortality rate to model the transmission and dynamics of the virus. Numerical analysis and analytical method are used to explore the dynamics and prediction of the pandemic.

The development of maritime transportation, in terms of the size and traffic volume of container ships, is increasingly posing technical challenges especially in crowded harbors and narrow canals. To ensure safe and efficient operation, tugboats are used for assisting large ships in such critical areas under the supervision of the responsible pilots. However, the mis-assessment of risk, especially under bad weather conditions, can lead to accidents, economic losses, and even casualties. Accordingly, the need for automated assistant systems is strong. In this paper, we investigate the maneuverability of tugboat-controlled ships under the influence of wind. We provide a characterization of the admissible set of safe wind situations for the execution of a given maneuver in terms of a constrained system of differential equations. Under some restrictions over the type of maneuvers, we prove the star-convexity of the admissible set. For this purpose, the dynamical model of the ship motion is derived, which takes tugboat and wind forces into account. The star-convexity of the sought-after set and the ship model allow for an efficient approximation of this set by optimal control methods. As an application, we propose a visual aid that allows the tugboat pilots to clearly distinguish between safe and hazardous wind situations during maneuver planning.

In part one we prove a theorem about the automorphism of solutions to Ramanujan's differential equations. We also investigate possible applications of the result. In part two we prove a similar theorem about the automorphism of solutions to the first-order system of differential equations associated to the generalised Chazy equation with parameter $k=\frac{3}{2}$., Comment: 16 pages

We study the dependence of the functions of capital (resource) and consumption in the Ramsey–Kass–Koopmans economic model in the case when saving is an identical constant. The system of differential equations describing the evolution of the economic model under consideration is solved in quadratures under the assumptions made. Upper estimates of the consumption function are found based on the obtained solution.

In this paper, we proposed a generalized of Darbo's fixed point theorem via the concept of operators $ S(\bullet; .) $ associated with the measure of noncompactness. Using this generalized Darbo fixed point theorem, we have given the existence of solution of a system of differential equations. At the end, we have given an example which supports our findings.

The article constructs and investigates the system of differential equations that describes the mutual synchronization of coupled self-oscillating chemical systems, defines the synchronization bar in which a synchronous mode exists and determines the time of establishment of the synchronous phase difference.

In this paper, we propose a new formulation of the physical reliability principle and build a reliability model in the form of a system of differential equations taking into account variable loads. A load is applied to the input of the dynamic system, and a probability function of failure-free operation is formed at its output. The conditions for the equivalence of dynamic models are studied. In the presence of self-similarity of damage accumulation processes, the general dynamic model is reduced to an equivalent simplified basic dynamic model. The results obtained can be used in the theory of reliability of systems with variable loads, in the analysis of survival, and in the theory of accelerated and forced tests.

A host-vector disease model with insecticide resistance genes is proposed as a system of differential equations. The resistance-induced reproduction number $${\mathcal {R}}_e$$ is determined and qualitative stabilities analysis are provided. We use the model to study the effects of insecticide resistance of vectors on the spread of the disease. The resistance-induced reproduction number $${\mathcal {R}}_e$$ is compared with the basic reproduction number $$({\mathcal {R}}_0)$$ in the absence of resistant strain to assess the effects of insecticide resistance.

A quasi-stationary Darwin approximation for the system of Maxwell’s equations in inhomogeneous conducting media is studied. An existence and uniqueness theorem for the initial-boundary value problem for the resulting system of differential equations is proved. Estimates of the proximity between the solutions of the quasi-stationary problem under consideration and the corresponding nonstationary problem, depending on the characteristic values of the data, are presented.

The article is devoted to a description of a mathematical model of a tethered high-altitude unmanned platform in which the power of propulsion systems and payload is supplied from a ground-based energy source via cable. The magnitudes and directions of the forces acting on the unmanned vehicle by the cable are determined from the derived system of differential equations which allows calculating the required power transmitted from the ground to the platform, depending on the lift height and the wind load.

In this paper, we consider a system of differential equations with a cylindrical phase space, which is a mathematical model of a phase-locked loop system. Conditions of the existence of limit cycle of the first kind are obtained.

We consider a system of frequency-phase self-tuning whose mathematical model is a system of differential equations. In this paper, existence conditions for asynchronous modes of a phase system are obtained.

In this paper, we consider a system of differential equations describing the free fall of a parachutist and his dropping with the open parachute canopy. The system is studied qualitatively and its possible equilibrium states are examined. Calculations were performed for experimental data obtained from realistic jumps.

We prove a theorem on the existence of a nonzero periodic solution to a system of differential equations using the fixed-point method for a nonlinear operator defined on the product of two compact sets.

Dynamic breakdown of ammonium perchlorate single crystals is numerically simulated. A system of differential equations that describe processes in an equivalent circuit of a pulsed voltage oscillator, a kinetic equation of impact multiplication of electrons, and a heat-balance equation are simultaneously solved. The breakdown strength of ammonium perchlorate is calculated in the dynamic mode versus interelectrode distance and leading edge of a high-voltage pulse. The calculated results are in reasonable agreement with the experimental data on the dynamic breakdown of ammonium perchlorate.

This article investigates the existence and uniqueness of periodic solutions for a new system of differential equations. By employing fixed point theorems for increasing φ-$(h,\tau )$(h,τ)-concave operators, we establish the existence of unique periodic solution for our differential system and then give a monotone iterative scheme to approximate the unique periodic solution. Some examples are presented in the end to illustrate the validity of our main results.

The paper considers the system of second order differential equations of the form: $$\begin{aligned} \frac{\mathrm {d}^2v_{j}}{\mathrm {d}t^2}-v_{j}=f_{j}(t,v(t));~~~v_{j}(0)=v_{j}(T)=0 \end{aligned}$$ where $$t\in [0,T]$$ , $$v(t)=\left( v_{j}(t)\right) _{j=1}^{\infty }$$ . The system is investigated in Banach spaces $$c_{0}$$ and $$\ell _{1}$$ . Using the concept of measures of noncompactness, conditions for the existence of solution are found for the above system. The idea is supported with different examples.

In this paper, a class of rational methods of second to fourth order of accuracy are presented. The methods are developed by considering the concept of the closest points of approximation in its fo...

The ongoing coronavirus epidemic, COVID-19, has spread across the world in matter of weeks, and has led to a global pandemic. The current study examines the propagation of COVID-19 in Wuhan, the epicentre of the outbreak, and Hubei Province in China, across which the spread was initially the fastest. The logistic model were used to analyse the propagation of the epidemic. The results of the study suggest that the logistic model is an adequate model for explaining the propagation of the epidemic. The model gives projections for the limiting cumulative number of cases, cured/discharged, and deaths. A more integrated approach is proposed for further studies, which would consider the number cured and number of deaths as subprocesses of the number of infected cases at any point of time, as a coupled system of differential equations.

In this article, the reliability of a gear–motor system is analyzed on the condition that the failure of the gear leads under certain circumstances to the failure of the motor. The states of the system are modeled by a continuous-time Markov chain. The resulting system of differential equations is solved using the Laplace transform. In this way, the probabilities of the system’s states can be determined for different failure and repair rates of individual components depending on time and comparative analysis of the availability can be performed.

In this research, the main objective is to perform the comparative analysis of numerical methods (Explicit Euler, Runge Kutta 4 and LocallyExact) for the resolution of differential equations. To fulfill the purpose of this study, the system of differential equations of the Lotka-Volterra model was used and the mathematical software Wolfram Mathematica was used. To perform the comparison of the numerical methods the Lotka-Volterra model was solved using the NdSolve command of Mathematica, this result was compared with the Methods Explicit Euler, Runge Kutta 4 and LocallyExact. The results obtained from the phase diagrams and the point table of the interactions indicate that the Runge Kutta 4 method has greater precision, followed by the LocallyExact method. The explicit Euler method draws considerably away from the result of NDSolve. DOI: http://dx.doi.org/10.21017/rimci.2020.v7.n13.a72

Related to earlier work on existence results of best proximity points (pairs) for cyclic (noncyclic) condensing operators of integral type in the setting of reflexive Busemann convex spaces, in this paper, we introduce another class of cyclic (noncyclic) condensing operators and study the existence of best proximity points (pairs) as well as coupled best proximity points (pairs) in such spaces. Then, we present an application of our main existence result to study the existence of an optimal solution for a system of differential equations.

Az alumínium csomagolóeszközök előállítása során az egyik kulcsfontosságú kérdéskör az alakítóerők és az alakítási határállapot -azaz ahol a palack stabilitásvesztése bekövetkezik- vizsgálata. A dolgozat ezen alakadási lépések, továbbá az ezek közben bekövetkező stabilitásvesztést leíró egyenletrendszert és annak végeselemes megoldási kérdéseit tárgyalja. A feladat vizsgálatához kialakítottunk egy térbeli héjmodellt, amely a modálanalízis eredményeiből kiindulva alkalmas a lengésképek alapján megzavart geometria nemlineáris stabilitásvesztési vizsgálatára módosított Riks-módszer segítségével.

The paper presents a general method for constructing a mathematical model of multi-link robot-manipulators. For the 3-DOF manipulator, kinematic equations and differential equations of dynamics are obtained. To study mathematical models of the dynamics of robotic manipulators and applications in software control systems, it is necessary to develop special analytical methods for solving systems of differential equations. An analytical method of transformations applied for the study of mathematical models of the dynamics of 3-DOF robot-manipulators. The method allows to obtain a solution in an analytical form, taking into account all nonlinear components of the system of differential equations. The software package for study of nonlinear mathematical models was developed and implemented. The problem of software control of electric drives of 3-DOF robot-manipulator is considered and control actions are determined that ensure the fulfillment of a given trajectory by the manipulator. The method for constructing and analyzing a mathematical model of manipulators presented in this work can be used to study a wide class of multi-link manipulators with many degrees of freedom.

Este trabajo de grado presenta una investigación de tipo experimental que plantea la solución numérica del modelo de Heston de valoración de opciones. El modelo de Heston para la valoración de opciones financieras, en particular, las opciones europeas comprende sistema de ecuaciones diferenciales estocásticas que busca predecir el valor de la prima que se paga en este tipo de derivados financieros. Esta clase de modelos son muy difíciles de resolver de manera analítica, por lo que requieren simulaciones computacionales en el proceso del cálculo de la prima. Si bien existen algunas fórmulas cerradas para el modelo de Heston, el contar con estrategias numéricas permite considerar modelos mas sofisticados basados en Heston, y para los cuales no existen tales fórmulas cerradas. Para este propósito, primero se requiere de un modelo matemático que simule el comportamiento de las opciones financieras bajo el modelo de Heston de volatilidad estocástica. Índice general Abstract iii Resumen iv Agradecimientos v Contenidos vi Lista de Tablas viii Lista de Figuras ix Introducción 1 1. Preliminares 3 1.1. Conceptos básicos 3 2. Modelos matemáticos en la valoración de opciones 7 2.1. Conceptos básicos de finanzas 7 2.2. Opciones financieras 8 2.3. El modelo de Black-Scholes-Merton 11 2.4. El modelo de Heston de volatilidad estocástica 13 2.5. La EDP asociada al modelo de Heston 15 3. Métodos numéricos para la valoración de opciones 18 3.1. Métodos de árboles 18 3.2. Métodos monte carlo 19 3.3. Método de líneas 19 4. Esquemas tipo Runge-Kutta para el modelo de Heston 21 4.1. EDP con condiciones de frontera 21 4.2. Esquema numérico opciones tipo CALL 22 5. Experimentos Numéricos 27 5.1. Métodos Explícitos 27 5.2. Métodos Implícitos 28 5.3. Efecto del Tamaño de Paso 29 5.4. Efecto del Tiempo de Maduración 30 Conclusiones y Trabajos Futuros 31 Referencias 32 Pregrado Estadístico Estadística

Some complicated events can be modeled by systems of differential equations. On the other hand, inclusion systems can describe complex phenomena having some shocks better than the system of differential equations. Also, one of the interests of researchers in this field is an investigation of hybrid systems. In this paper, we study the existence of solutions for hybrid and non-hybrid k-dimensional sequential inclusion systems by considering some integral boundary conditions. In this way, we use different methods such as α-ψ contractions and the endpoint technique. Finally, we present two examples to illustrate our main results.

In this work, we construct stellar models ba-\break sed on the complexity factor as a supplementary condition which allows to close the system of differential equations arising from the Gravitational Decoupling. The assumed complexity is a generalization of the one obtained from the well known Tolman IV solution. We use Tolman IV, Wyman IIa, Durgapal IV and Heintzmann IIa as seeds solutions. Reported compactness parameters of SMC X-1 and Cen X-3 are used to study the physical acceptability of the models. Some aspects related to the density ratio are also discussed., Comment: References updated. Some new references added. arXiv admin note: text overlap with arXiv:2108.10311

Virus evolution is important because it generates new mutations, which can be harmful to humans. Because the evolution is a process along the time course, and many mathematical tools describe the phenomenon along the time course, thus it is possible to apply a mathematical tool to a virus evolution. Neuraminidase is one of two surface proteins in influenza A virus playing an important role influenza transmission, thus it is important to model its evolution. Yet, a protein sequence should be converted into numerical values in order to be workable in mathematical tools, and a driving force for evolution should be defined. In this study, first we use the amino-acid pair predictability as a measure of driving force for evolution to convert 3828 neuraminidases sampled from 1956 to 2008 into numerical values; second we use a system of differential equations to describe the mutation neuraminidases; and third we use the analytical solution to fit the evolution of neuraminidases. The results show a promising and encouraging trajectory of evolution of neuraminidases along the time course.

A multi-scale mathematical model is proposed, seeking to describe the propagation of Human Immunodeficiency Virus (HIV) in a group of young people between 15 and 24 years of age, through unprotected sexual contact. The uses of antiretroviral therapy (ART) and therapeutic failure are considered to show how the rate of propagation and prevalence are affected. The model consists of a complex network modeling the interactions on the population scale, coupled with the immunological dynamics of each individual, which is modeled by a system of differential equations. The immunological model allows to observe some known facts from the literature, such as to control HIV infection in the immune system, it is necessary to reduce the probability of healthy CD4 T cells becoming infected or increase the probability at which cells of the specific cell response against HIV eliminate infected CD4 T cells. At the population level, it is shown that, to have a high prevalence, it is not necessary for the virus to spread rapidly at the beginning of the simulation time. Additionally, it is observed that a greater number of sexual partners induces higher HIV prevalence. Using ART in the immune system reduces the number of infected CD4 T cells and, consequently, helps to reduce the spread of infection at the population scale. An important result observed in simulations is that the average number of HIV carriers who abandon ART is greater than those who access it. The study adds to the available literature an original simulation model that describes the dynamics of HIV propagation in a population, considering the immune state of people within that population, and serves as a basis for future research involving more detailed aspects aiming for a model closest to reality.