The SIR model of the epidemic spread is used for consideration the problem of the competition of two viruses having different contagiousness. It is shown how the more contagious strain replaces over time the less contagious one. In particular the results can be applied to the current situation when the omicron strain appeared in population affected by the delta strain.

No abstract

Equations for infection spread in a closed population are found in discrete approximation, corresponding to the published statistical data, and in continuous time in the form of delay differential equations. We consider the epidemic as dependent upon four key parameters: the size of population involved, the mean number of dangerous contacts of one infected person per day, the probability to transmit infection due to such contact and the mean duration of disease. In the simplest case of free-running epidemic in an infinite population, the number of infected rises exponentially day by day. Here we show the model for epidemic process in a closed population, constrained by isolation, treatment and so on. The four parameters introduced here have the clear sense and are in association with the well-known concept of reproduction number in the continuous susceptible--infectious--removed, susceptible--exposed--infectious--removed (SIR, SEIR) models. We derive the initial rate of infection spread from the published statistical data for the initial stage of epidemic, when the quarantine measures were absent. On this basis, we can found the corresponding basic reproduction number mentioned above. Our approach allows evaluating the influence of quarantine measures on free pandemic process that leads to the time-dependent rate of infection and suppression of infection. We found a good correspondence of the theory and reliable statistical data. The initially formulated discrete model, describing epidemic course day by day is transferred to differential form. The conditions for saturation of epidemic are found by solving the delay differential equations. They differ essentially from ones in SIR model due to finite delay, typical for COVID-19. The proposed model opens up the possibility to predict the optimal level of social quarantine measures. The model is quite flexible and it can be extended to more complex cases.

No abstract

Self-consistent and universal description of friction and diffusion for Brownian particles in such various systems as dusty plasma and active particles (e.g., cells in biological systems) is presented. Generalized friction function is determined to describe the friction force itself as well as a drag force in the case of non-zero driven ion velocity in plasmas.Brownian dynamics nowadays is in the focus of interest due to the wide fields of applications: physicalchemical systems, dusty plasmas, various objects in biological systems. Existence of the Einstein relation and even the correct specific forms of the Fokker-Planck equation for such systems are not still completely clarified. Recently the friction and diffusion coefficients as functions of the grain velocity V were rigorously calculated for dusty plasmas [1]. Due to ion absorption by grains the friction coefficient can become negative [2]. In this work we develop a general approach based on the probability transition (PT) [3], to the calculation of the velocity dependent friction and diffusion coefficients for such different systems as dusty plasmas (within the well known model with a fix grain mass) and active particles, e.g. cells.The friction and diffusion coefficients β(V ) and D(V ) = D (V ) (tensorial character of D for the cases considered here is negligible) are determined via the probability transition w(P, q) asHere (Pq) is the scalar product in velocity space of the dimension s and P ≡ M V.Therefore it is impossible to define the friction and diffusion coefficients independently not only for the processes, which describe the systems close to thermodynamic equilibrium, when the Einstein relation is a priori valid, but also for the systems in which there is stationary, but non-equilibrium state, or for the systems far from equilibrium. For the Boltzmann-type collisions between light particles (conventionally atoms) and grains with the masses, respectively, m and M (m M ), the PT function w(P, q) and the coefficients β(V ) and D(V ) can be calculated for an arbitrary cross-section [3]. The Einstein relation is violated only for extremely high grain velocity values V > T /m, where T is the atomic temperature. For spherical grains of radius a the interpolation relation between the diffusion and friction coefficients with the exact velocity asymptotes is: