System analysis and mathematical modeling of the disease caused by the 2019-nCoV coronavirus strain

A particular problem at the beginning of the second millennium is the epidemic caused by coronaviruses. In 2002-2003, new strains of viruses appeared – SARS-CoV SARS, in 2012, MERS-CoV. In 2019-2020, the epidemic caused by the coronavirus strain 2019-nCoV, which belong to the Beta-CoV group, is a threat. Different types of coronaviruses infect humans, cats, birds, dogs, cattle, pigs and hares, bats, camels and other animals. Coronaviruses have a single-stranded RNA genome, which encodes 4–5 structural proteins, including proteins of the outer membrane, matrix and small membrane. The complete 2019-nCoV genome is stored online at GenBank: MN908947.3. Coronaviruses have some unique features in RNA transcription. The RNA minus chain serves as a matrix for the synthesis of both new genomic and subgenomic RNAs. To develop a model of human resistance to the disease caused by the coronavirus family, elements, connections and ways of protecting the Human-Virus-Environment system were identified. The destructive functions of sixteen non-structural and structural proteins of the strain 2019-nCoV are considered. Deterministic and statistical models of the development of the danger of infection of the cells have been developed. A parameterized system for protecting a person from coronavirus


Introduction
One of the most important problems of 2019-2020 is the search for optimal methods for creating protective functions of a person from coronavirus damage. The use of systems analysis methods is one of the promising ways to consider and solve the problems of combating viral infection in the face of uncertainty [1]. Currently, about 40 types of viruses are known, combined into 4 subfamilies. They bear this name because of the structure of their outer shell in the form of a crown. Coronaviruses -viruses with a positive RNA chain (+ RNA), in comparison with other RNA viruses, have an exceptionally large genome and have a unique replication mechanism. Viral particles of strain 2019-nCoV have a spherical shape with a diameter of 60-120 nm. The height of the protrusions is 9-12 nm.

Development of a model of the Human-Virus-Environment (HVE) system
Among viruses, coronaviruses are of particular danger. When developing a vaccine for the coronavirus family, it is necessary to have a methodology for studying the characteristics and assessing the state of human life. To formalize the elements of the HVE system, we introduce the following sets ( Fig. 1  The relationship between the elements of the system is defined by the binary relations R (1), by which we can understand the functional relations, preferences, successions, and others, reflecting the essence of the relationship between the elements of the HVE system: Binary relations can be decomposed into more complex ones with the introduction of an additional variable C, called a state. We decompose the binary relations R into two subsets: a healthy (1) and an infected (2) cell. A cell, like any real system, functions in certain physiological conditions determined by the environment. In this case, system (1) is transformed as follows: The security of the HVE system and the state (C) of each element as a whole depend on their properties, which change during the life cycle of the HVE system.
The state of the HVE system is determined by the properties of the elements of the system (CV, CE, CH). The nature of the change in these properties during the life cycle of the HVE system, as a rule, worsens due to cell aging and a decrease in protective abilities. Therefore, to build the dependences of real state changes in the HVE system, it is necessary to know some functional ℑ that would determine the safe state of the HVE system and its properties: The state of the CV destructive elements of coronaviruses affects the state of the CHVE system: α-, β-, γ-, δ-CoVs, which have certain chemical and biological properties; on the own state of CH cells, depending on the protective functions of the cell, climatogeographic features of human habitation and relationships with animals -CE.

Study of the model for assessing the state of the HVE system
The state of the HVE system is described by the functional characterizing the state of the arrays of elements (4). We are investigating human security in more detail. The state of human safety (CH) depends on the safety index of one's own properties (ΘН); changes in the safety index from the influence of coronaviruses (ΔΘН(V)); from a change in the safety indicator, depending on changes in environmental parameters (ΔΘН(E)) and is determined by the formula Let us describe in more detail the elements of equation (5).
The intrinsic properties of a cell (ΘН) are determined by the susceptibility to destruction (modification) of nucleic acid elements that ensure the storage and transmission of genetic information (DNA and RNA) from exposure to coronavirus.
The change in the safety indicator ΔΘН(V) depends on the group and strain of coronaviruses and is described by the equation Of particular danger is the 2019-nCoV strain, which has a high destructive feature of the modification of nucleic acid elements, cell degradation, cleavage of polypeptides and other functions [2]. Consider the destructive functions of sixteen non-structural and structural proteins (nsp) of the coronavirus strain 2019-nCoV: where: nsp13 -RNA degradation [9]; nsp15 -endoribonuclease [10].
The general destructive functions of the coronavirus strain 2019-nCoV can be described by function (7). Human safety from the influence of coronaviruses can be estimated by the formula (6).

Necessary and sufficient conditions for human infection with coronavirus
The necessary condition for the transition from one state to another of the HVE system is the presence of coronavirus (CoVs), and the reduced distance (x) to the source of danger (coronavirus), the spread of the epidemic (w) and exposure time (τ) are sufficient (table 1).  The relationships between the probabilistic states of the HVE system are presented in Fig.  2, from which it can be seen that the HVE system can transition from a safe state (1) to a dangerous (2), infectious (3) state and vice versa, with an increase in the body's immune system, the cell does not allow penetration and reproduction of the coronavirus in it. The transition from the infectious state to the state of cell death (4) is final, since the cell is destroyed at this point in time. The system of differential equations for the probability graph of states pi and transitions λij of the HVE system is shown in  Formalization of the space of hazard parameters i CoVs allows us to assess the degree of safety of human infection: Thus, the general indicator for many sources of coronaviruses is calculated as the arithmetic mean or equal to zero if at least one of the sources of coronavirus is dangerous: Using equations (8) and (9), one can determine the epidemiological state in the studied region (city).

Statistical model of cell infection
Coronaviruses have some unique features in RNA transcription, protein composition, and assembly mechanisms. They penetrate the cell through adsorption endocytosis. After that, the attachment of genomic RNA to ribosomes occurs, which leads to the synthesis of viral RNAdependent RNA polymerase. Upon transcription of genomic RNA, a complementary minus strand of RNA of full length is formed. Its synthesis is completed 5-6 hours after infection. We compile a statistical model of cell infection with respect to probabilistic function ( ).
With the known laws of the distribution of a random variable using the system of equations (12), it is possible to determine the probability of infection of the cell in order to take timely measures to prevent the protection of the population.

Development of a system for protecting a person from coronavirus damage
Active methods are currently underway to develop a vaccine for the 2019-nCoV virus strain. Schematically depict the cell protection system shown in Fig. 3. The safety of the HVE system according to the parameters of the danger source of coronaviruses will be ensured in case of inequalities: The safety assessment of the HVE system according to the i-th source of danger of the coronavirus, taking into account the protection factors, will be determined by the expression (13) taking into account inequalities (