From Population Monitoring to a Mathematical Model: A New Paradigm of Population Research

Logofet, Dmitrii O.; Ulanova, Nina G.

doi:10.1134/s2079086422030057

Cited by 7 publications

(3 citation statements)

References 95 publications

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“…The growing application of population models in assessing the dynamics of real biological objects necessitates greater accuracy and detail in the description. Simple models, in which individuals are considered to be identical, are replaced by models with the internal structure of a population, where individuals may differ in the values of various parameters [1,2]. Distributed models are of particular interest.…”

Section: Introductionmentioning

confidence: 99%

“…The use of models with an age structure allows for a more detailed and accurate description of the behavior of a population in cases where the parameters that determine its dynamics depend significantly on age. Here, the "age" can mean various indicators that change over time: time since birth (demography and population dynamics) [1][2][3][4], time since the infection (epidemiology, including immunity dynamics) [4][5][6], time since cell division (tissue growth) [7], etc. Population models with an age structure allow us to explain phenomena such as demographic (population) waves [8], to consider the dynamics of immunity [5,6], and to formalize the relationship between rates of evolution and rates of environmental change [9].…”

Section: Introductionmentioning

confidence: 99%

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COVID-19: From Limit Cycle to Stable Focus

Соколов

Voloshinov

2023

Mathematics

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The study aims at investigating a new fundamental property of infectious diseases with natural adaptive immunity that weakens over time—qualitative change (bifurcation) in the behavior of the “virus vs. human” system with an increase in contagiousness. Numerical experiments with a model of the COVID-19 epidemic in Moscow have demonstrated that when the reproduction number R0 is about 4, a qualitative change (bifurcation) occurs in the behavior of the virus–human system. Below this value, the long-term forecast tends toward undamped oscillations; above it, the forecast shows damped oscillations: the amplitudes of epidemic waves decrease gradually, with a constant, very high background level of morbidity that keeps natural immunity near 100%. To confirm this result analytically, we use an original modification of the Euler–Lotka renewal equation, which describes the dynamics of infected patients distributed by disease duration (time since infection) and accounts for immunity. To construct a bifurcation diagram, which illustrates the dependence of the equilibrium stability on the parameter R0, we linearize the equation in the vicinity of the equilibrium point and examine its numerical approximation (discrete form). This approximation can be interpreted as a Leslie model, with the matrix elements dependent on the parameter R0. By examining the roots of the corresponding Lotka polynomial, we can assess the stability of the equilibrium point and verify the basic assumption about the change in the properties of the system with increasing R0—about the transition from undamped oscillations to damped ones. For the bifurcation diagram, we use the functions obtained from the simulation of the COVID-19 epidemic in Moscow. However, observations of the epidemic in other cities and countries support the primary finding of our study regarding the attenuation of epidemic waves.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

COVID-19: From Limit Cycle to Stable Focus

Соколов

Voloshinov

2023

Mathematics

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“…In addition to age-specific population models, stage-specific models have also been developed. Since we would like to dedicate another study to the monitoring in stage-structured models (using observer design), in addition to the classical model of Lefkovitch [28], here we only refer to the very recent surveys by Logofet and Salguero-Gómez [29] and Logofet and Ulanova [30] (see also Logofet and Klochkova [31]). Stage-specific models, however, may display structural properties needing different constructions for observer design.…”

Section: Introductionmentioning

confidence: 99%

Monitoring in a Discrete-Time Nonlinear Age-Structured Population Model with Changing Environment

et al. 2022

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This paper is a contribution to the modeling–methodological development of the application of mathematical systems theory in population biology. A discrete-time nonlinear Leslie-type model is considered, where both the reproduction and survival rates decrease as the total population size increases. In this context, the monitoring problem means that, from the observation of the size of certain age classes as a function of time, we want to recover (estimate) the whole state process (i.e., the time-dependent size of the rest of the classes). First, for the linearization approach, conditions for the existence and asymptotic stability of a positive equilibrium are obtained, then the discrete-time observer design method is applied to estimate an unknown state trajectory near the equilibrium, where we could observe a single age class. It is also shown how the observer design can be used to detect an unknown change in the environment that affects the population dynamics. The environmental change is supposed to be generated by additional dynamics (exosystem). Now, the Leslie-type model is extended with this exosystem, and the observer design is applied to this extended system. In this way, an estimation can be obtained for different (constant or periodic) environmental changes as well.

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