A model for the spread of the COVID-19 pandemic and its induced changes in individual sectors of the economy has been developed. An approach to studying the interconnection of the food production system, food transportation, and the socio-economic sphere during the pandemic conditions using a three-sector Lorenz model has been proposed. Research has been conducted on the impact of the pandemic on changes in the supply-demand balance in these sectors of the economy. A model of the Tcell immune response has been developed, considering the peculiarities of its course in COVID-19, when excessive inflammation leads to cytokine release syndrome (cytokine storm), causing damage to vital organs and disruption of the immune system. The model considers the influence of cellular energetics on the regulation of the immune response and the amplification of inflammation.
In the 2021 Davos Forum report [1], it was noted that infectious diseases rank first among the highest risks influencing the next decade. On April 7, 2024, a total of 774,699,366 people was infected worldwide, and 7,033,430 people died [2]. The global changes of the past decades have so closely intertwined various spheres of social organization that the global COVID-19 pandemic has significantly impacted many processes in the world economy. This has initiated a chain of changes leading to disruptions in socio-economic connections at the local, regional, and global levels [3[.
In an effort to control the epidemic process, governments have implemented lockdowns and other restrictions on economic activity, which have closed many businesses, limited domestic travel, closed borders to the movement of labor and certain food products, and introduced requirements for social distancing and curfews. Anti-epidemic measures are becoming a heavy burden on the economy as they begin to affect a complex set of ecological, economic, political, and social processes, which have enormous consequences for the lives of individuals, societal wellbeing, economic activity, and food security. In formulating effective strategies to normalize the economic situation amidst the pandemic, many governments are faced with the difficult problem of how to provide support to the economy in the short term while avoiding unwanted inflationary consequences and risks that could threaten financial stability in the medium term.
The uniqueness of the COVID-19 virus is associated with the lack of a single viewpoint on the nature of its origin and dissemination pathways. This leads to a situation of uncertainty that requires special mathematical methods used to investigate improbable or unique events, where the stochastic nature of the object under study, or the incompleteness of the sample, reduces the effectiveness of traditional statistical methods. Therefore, the task of developing models that consider the impact of random disturbances becomes particularly relevant in forecasting the development of the pandemic, as these disturbances become additional factors increasing the level of structural disruptions in the epidemic system. 0000-0001-9892-054X (K. Atoyev); 0000-0001-6550-2237 (P. Knopov) © 2024 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
At the same time, it becomes clear that forecasting the development of pandemics is impossible without the development of mathematical models to study the negative impact of global epidemics on economic development. These models should consider both epidemic and economic factors and allow the determination of effective strategies to minimize the number of casualties and economic losses from the epidemic under various scenarios of its development. They should also enable a systematic analysis of responses to challenges for sustainable development in the face of new bio-threats. The aim of this work is to develop precisely such mathematical models for studying the negative impact of global pandemics on economic development and to conduct a system analysis of responses to challenges for sustainable development in the face of new bio-threats.
In [4], an approach to studying the interconnection of water, food, energy resources, food transportation, and medical consequences of a pandemic was proposed using a multisectoral Lorenz model. This model unifies sectors of the economy that are similarly described in a single structure, with each sector considered in terms of productivity levels, the number of jobs, and the level of structural disruptions. Through modeling, conditions for the emergence of deterministic chaos were identified, and trajectories of changes in socio-economic factors were calculated, allowing for the reduction of the number of structural disruptions. This reduction is achieved by altering the balance between supply and demand in creating jobs and production in relevant sectors of the economy. The study of the model allowed tracing how changes in the balance of supply and demand in interconnected systems lead to the emergence of a chaotic stable attractor.
The SIR epidemic model used frequently to study the spread of COVID-19. It describes the interaction of three population groups: the healthy (S), the infected (I), and those who recovered (R) [5]. To investigate the interrelationship between changes in the economy and the epidemic process, these two approaches are combined. This allows connecting the SIR model with parameters whose dynamics are determined using a model of interconnections in the food production, its transportation, and the socio-economic sphere. It is assumed that the level of disruptions is proportional to the level of disruptions in the operation of production systems in different sectors of the economy.
Let's consider three
manufacturing systems (MS) related to food
production, its
transportation, and medical sector. We will use the Lorenz model to study interrelated sectors of
the economy [4]. Let ( ), ( ) and ( ) be normalized levels of productivity, employment
quantity, and level of structural disruptions respectively for the food production system (i = 1),
food transportation (i = 2), and the socio-economic sphere (medical infrastructure) (i = 3). Let's
assume that different sectors of the economy compete with each other for labor, while changes
in climatic conditions, financial, and political instability introduce random disturbances ( ),
becoming additional factors increasing the level of structural disruptions in the socioeconomic
system [4]. Since processes in different sectors of the economy proceed at different rates, we will
scale time in them by introducing parameters ε . The model has the following form:
ε
ε
= σ ( − ) + δi ̇ ,
= − + δ ̇ ,
ε
= [ ( 1, 2, 3) − ] − + δ ̇ ,
(
Research on the canonical Lorenz model shows that increasing the parameters leads to the
emergence of turbulence in the model (
The results obtained in [4] allow us to relate the parameters σ , , with the characteristics
of the sectors of the economy in the following way:
σ = (α1 β2 )/(α2 γ2 ), = (β1 γ1 )/(β2 γ2 ), = ζ /(α2 γ2 ) ,
where α1 and α2 are parameters characterizing adaptive capabilities; β1 are demands for
the activity of the i-th MS, normalized per unit of the material production system, considering the
workplace in the corresponding industry of production ; β2 are supplies, normalized per unit
of function of the i-th MS ; γ1 are demands for an increase in number of jobs, normalized per
unit of ; γ2 are supplies of jobs involved in providing , normalized per unit of ; ζ are the
specific rate of growth in the number of disruptions. For the study of the epidemic process, we
will use the modified SIR model, which has the following form:
(
Let us consider the initial wave of the epidemic, when there are no individuals with
immunity to the virus. Let ( ) = 0, and let the functions ( 3), 1( 3) and 2( 2) have the
following form:
( 3) = (1 + 3/ 1),
1( 3) = 1 + 2 32/( 3+ 32), 2 ( 2) = 1 + 2 / ,
(
In figures 1-2, the results of simulating the initial wave of the epidemic are shown for various initial numbers of infected individuals and varying recovery rates 2.
Figure 3 displays the results of modeling when a portion of the sick who died ( 0) is a function of the level of disruptions in the medical sector ( 3) and is defined as follows: 0( 3) = 0[1 + 4 3/( 5 + 3)].
, in all figures are measured by the number of people, are measured by the number of disruptions, time is measured in months.
In [4], the investigation focused on how changes in the parameter affect the formation of
system (
In figure 4, it is shown how the increase in the parameter 3, which characterizes the level of impact of the pandemic on the functioning of the medical sector, contributes to the emergence of stochastic regimes (time measured in months). In the absence of such influence ( 3 = 0), damped oscillations of the variable 3 are observed.
At 3 = 100, the variable 3 reaches a steady level. Then it only deviates from it, depending on the changes in 2. At 3 = 850 and 3 = 1000, cyclic oscillations begin, which dampen when the level of infected individuals decreases to level that was before the beginning of the epidemic. At 3 = 1250 and 3 = 1500, stable chaotic oscillations arise.
In figure 5 the results of modeling changes in the levels of disruptions in the food production system 1, food transportation 2, and medical infrastructure 3 during the epidemic are shown. During the simulation, the total population was chosen to be 34 million people. With increasing values of the parameters , the number of infected individuals also increases. Periodic trajectories transition to chaotic ones upon reaching the bifurcation value of this parameter.
There exists a mutual negative influence. On one hand, the state of the economy affects the ability to implement effective measures against the pandemic. On the other hand, the infection rate influences the state of the economy. Figure 6 presents the results of modeling with variations in parameters . The increase in these parameters simulates a decrease in job supply levels in sectors of the economy related to food production, its transportation, and medical infrastructure. We considered the case where these parameters are equal to each other. The growth of leads to a significant change in the phase portrait of the system, resulting in a reduction in the levels of structural disruptions in various sectors of the economy and a decrease in the peak number of infected individuals 2 in the chosen initial modeling interval. With growing the overall number of infected individuals 2 initially decreases and then begins to rise again.
When modeling, we used a standard package for solving differential equations - the GNU
Octave environment version 4.4.1, which is licensed under the GNU GPL and can run on Linux,
macOS, BSD and Windows operating systems. Identification of model parameters was carried out
using data [2]. According to these data, the peak of the epidemic in Ukraine was reached after 3
months, with the number of deaths reaching about 100,000 after 5 months when the first wave
of the epidemic subsided. The total number of infected individuals during this time was over 3.5
million. These data align well with the results of modeling the first wave of the pandemic's
development in Ukraine. To estimate the parameters of model (
One of the most serious complications of COVID-19 is an excessive immune response, in which the level of pro-inflammatory cytokines responsible for regulating intercellular and intersystemic interactions sharply increases (a "cytokine storm").This leads to the disruption of the immune system, an increase in the level of free radicals, causing multiple damages to internal organs (lungs, heart, kidneys, blood vessels, liver, gastrointestinal tract, brain), severe multiorgan failure, and alters the balance of synthesis and expenditure of energy in cells, posing a lethal risk. Unfortunately, the specific mechanism of the cytokine storm remains undefined, making research on the impact of the immune system on the course of COVID-19 highly relevant. Typically, when modeling the epidemic process, the key parameters determining the dynamics of epidemics include: 1) the average incubation period of the disease; 2) the average active period when the patient is contagious; 3) the average recovery period; 4) the average period until death. It is important to note that these constants in epidemiological models are functions of the state of the patient's immune system, which plays a crucial role in fighting infection. However, there are currently no epidemiological models that allow for the consideration of the influence of the immune system's state on the course of the disease and the selection of therapy. Therefore, the task arises to develop a mathematical model of the immune response in COVID-19 and analyze, through it, possible mechanisms of imbalance between different components of the immune system leading to a cytokine storm. This task will be addressed in the following section.
In mathematical modeling of the immune system response to antigenic determinants of various natures (viruses, allergens, tumor antigens, etc.), typically, the components facilitating antigen presentation, its destruction, regulation of proliferative processes, and modulation of the immune response are considered. These include helper, killer, macrophage, and suppressor cells [9].
In each of these links, there is a group of precursor cells, including cytotoxic T-cells – , helper T-cells – , normal macrophages – , suppressor T-cells – ; as well as a group of mature cells forming the immune response, including effector T-cells – , helper T-cells – , activated macrophages – and suppressor T-cells – . The listed cellular populations are responsible for the recognition and destruction virus-infected cells (VIC) – . VIC debris – contribute to the emergence of antigen-presenting cells (AK).
When infected with the coronavirus, the infection affects gene expression and protein synthesis. The virus imposes its protein synthesis algorithms on cells, necessary for its replication. Within hours, it neutralizes the cell's antiviral signaling, delaying and confusing the immune response. The virus reduces the ability of infected cells to translate genes into proteins, thereby decreasing overall protein synthesis. Additionally, it actively degrades cellular messenger RNA (mRNA), while its own mRNA remains protected. Finally, the virus can also prevent the export of mRNA from the cell nucleus, where they are synthesized. Based on the analysis of data [10, 11] on the peculiarities of the immune response in COVID-19, the following assumptions can be made, which should be considered in the model.
1. VIC accelerates the transition of precursor cytotoxic T-cells to a mature form and enhances the proliferation of effector cells. Moreover, they promote the production of lymphoid factors (F) by helper cells: interleukins, growth factors, and others that activate the proliferation of mature T-cells, the transition of normal macrophages into the activated form, and the inflammatory process.
2. Increased inflammation (I) enhances the proliferation rate of mature effector, helper, suppressor T cells, and activated macrophages, as well as accelerates the influx of all precursors. 3. VIC are recognized and destroyed by cytotoxic T-lymphocytes and activated macrophages. 4. The proliferation of helper cells is activated after contact with antigen-presenting cells (AK). 5. The increase in the level of effector cells activates the process of maturation of suppressors. Suppressor cells inhibit the processes of proliferation and maturation of helper and effector cells, as well as the processes of VIC destruction by effectors and macrophages.
6. There is a temporary hierarchy that allows identifying a group of fast variables that manage to reach a stationary state: antigen-presenting bed, lymphoid factors, and the inflammatory reaction.
7. The coronavirus can evade immune system recognition through suppression at the precursor cell level, thereby reducing the levels of killer and helper T-cells. Additionally, it may affect the efficiency of killer T-cells and activated macrophages in destroying cells infected with the coronavirus.
8. The coronavirus can also infect activated macrophages. This increases the level of VIC due to the action of virus-infected macrophages .
9. One of the possible paths of COVID-19 immunopathogenesis is the induction of damage and death to vessel endothelium due to virus replication in the infection focus. Uncontrolled inflammatory reactions lead to significant tissue damage and the development of severe forms of the disease.
10. Tissue damage results in an energy imbalance, disrupting the interrelationships between
energy function and mitochondrial activity for energy synthesis in the cell. A significant
indicator of this imbalance, which increases with infection growth, is the number of structural
disruptions in the cellular energy system. An increase in leads to a decrease in the activity
of effector cells to destroy VIC. In [12], the Lorenz model was used to investigate these
relationships. We will use these results to describe the energy block of the model for cells affected
by the coronavirus. Taking these assumptions into account, the model has the following form:
(
3 ( 1 + 2 )( 1 + 2 )
7 ( 3 + 4 )( 3 + 4 ) − 4 , − 8 , = 9(1 + 10 ) − 11
F − 12 , = 13(1 + 14 ) − 15
− 16 , 3 ( 1 + 2 )( 1 + 2 ) − 17 +
18 ( 5 + 6 )
, 7 20 ( 3 + 4 )( 3 + 4 ) − 19 ( 7 + 8 )
, = 11
F − 21 − 22
, = 15
− 23 + 24 , = 22
− 25 , 26 ( 9 + 10 ) 27 ( 10 + 5 − 29 , − ), = ( + 6 − ) − =
− , , =
12 + , = where are constants that characterize: i = 1, 5, 9, 13 – inflows of respectively; i = 2, 6, 10, 14 – delay mode for activation of , , inflows , respectively; i = 3, 7, 11, 15 – speed of , , , , cells maturation, respectively; i = 4, 8, 12, 16, 17, 19, 21, 23, 25 – death rates of ,
, reproduction rates of , , , , , , , , , respectively; i = 18, 20, 24, 26 – population
, respectively; i = 27, 29 – destruction rates of respectively; i =22 – this rate quantifies how quickly viruses infect macrophages this rate represent how infected macrophages stimulate the reproduction or replication of viruses; ( = 1,14)– are constants that characterize the nonlinear effects of the interaction between different chains of the immune system; ( = 1,4)– are constants characterizing immunosuppression from the coronavirus; 5 – is a constant characterizing a decrease in the efficiency of virus destruction due to structural disruptions in the energy system , , constant characterizing the influence of inflammation on the energy system; – is a constant characterizing the activity of stimulating virus reproduction from the side of macrophages ; are parameters of the Lorenz model, which are functions of energy synthesis and ; 6 – is a
and , ; i =28 – expenditure processes in the cell [12].
infection of macrophages ( 22 = 0).
The mathematical formulation of the problem is as follows: given the model (
Let’s explore what changes in the immune response will result from considering the interactions between the immune and energy systems. In figure 9, the results of modeling are presented with variations in the parameter 5. This parameter characterizes the degree of inhibition of the effector function of killer cells and activated macrophages by the coronavirus. The modeling results indicate that an increase in the degree of inhibition (increasing of 5) leads to an increase in the time to reach the peak in lymphoid factors, inflammation, and . The level of disruptions in the cellular energy system also increases. (S - the area under the curve of disruptions).
The results of the simulation for the case where the VIC infects macrophages are shown in figures 10 and 11. As evident from figure 10, when the value of parameter 28 is reaches 0.07, the total amounts of the viruses, lymphoid factors, inflammation, and disruptions sharply increase, corresponding to the cytokine release syndrome.
Further growth leads to uncontrolled reproduction of the virus, which is incompatible with life. As seen from this figure, the system is highly sensitive to small changes in the parameter 28 within the interval [0.07, 0.0761].
If we consider the variables of model (
A model of the COVID-19 pandemic spread and associated changes in individual economic sectors has been developed. An approach to studying the interrelation within the food production system, food transportation, and the socio-economic sphere has been proposed using the three-sector Lorenz model. The model integrates economic sectors, uniformly described within a single framework, each of which is considered in terms of productivity, employment, and structural disruptions. The model studies were conducted with aim of: 1) assessing the influence of initial conditions and the state of economic sectors on the dynamics of the epidemic; 2) analyzing the impact of the epidemic process on interconnected sectors of the economy. The existence of deterministic and stochastic operational modes of interconnected economic sectors during a pandemic has been demonstrated. The transition from one mode to another is accompanied by an increase in the level of structural disruptions. The dynamics of these disruptions have been determined, along with the dependence of their total quantity on the epidemic process.
A mathematical model of T-cell immune response has been developed considering the peculiarities of its course in COVID-19, where excessive inflammation and increased levels of proinflammatory cytokines lead to cytokine release syndrome, causing damage to vital organs and disruption of immune system. The model considers the influence of cellular energetics, particularly mitochondria, on the regulation of the immune response and the amplification of inflammation. The simulation results have allowed investigation of the following: 1) how increases in the initial levels of virus-infected cells affect the average recovery period of severe symptomatic patients; 2) how inhibition of effector functions of killer cells and the level of inflammation affect the dynamics of the immune response; 3) how changes in the balance of energy synthesis and expenditure within the cell affect the occurrence of periodic and turbulent trajectories of the immune response.
The scientific novelty of the work lies in the following: 1) a proposed approach to the development of complex systems based on the use of multi-sector Lorenz models, allowing the exploration of interrelationships between the state of economy and the dynamics of epidemic; 2) the development of a mathematical model that accounts for the role of herd immunity in the dynamics of pandemic spread; 3) the developed models consider the superposition of two types of random processes - "deterministic" Lorenz stochasticity and traditional Wiener stochasticity. Further development of the work will be associated with solving optimization tasks for immune response management aimed at minimizing virus-induced damage to the organism and substantiating various therapeutic regimens used in treatment.
The work was carried out as part of the scientific project PK № 0122Ш00552 “Comprehensive analysis of robust preventive and adaptive measures for managing food, energy, water, and the social sphere in conditions of systemic risks and consequences of COVID-19”.