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Modeling and Controlling the Spread of Epidemic with Various Social and Economic Scenarios
S. P. Lukyanets, I. S. Gandzha, O. V. Kliushnichenko
TL;DR
The paper asks how epidemic spread can be modeled and controlled when transmission depends on social environments and recovery depends on economic resources. It extends SIR/SIQR-style dynamics with direct and indirect transmission, social control parameters, and Arrhenius-like resource-dependent recovery. The model demonstrates that quarantine can reduce transmission, while limited resources can sharply increase fatal cases.
Problem
Epidemic dynamics depend on heterogeneous social conditions and control strategies, while the relevant kinetic coefficients are not directly accessible as societal control parameters.
Method
The paper extends epidemic compartment models with direct and fomite-mediated transmission, socially parameterized rates, and activation-type recovery coupled to a collective economic resource.
Results
Nearly 6 times more fatal cases occur with limited resource than with unlimited resource after only a 5% resource drop in the reported example.
Takeaways & Limitations
The model links epidemic control measures and economic losses, showing that resource constraints can aggravate epidemic outcomes within the modeled social group.
Takeaways & Limitations
The general nonlinear epidemic equations have no known analytical solutions; analytical treatment is available only in the special case Γis = Γqs = 0.
Abstract
from arXiv · showhide
We propose a dynamical model for describing the spread of epidemics. This model is an extension of the SIQR (susceptible-infected-quarantined-recovered) and SIRP (susceptible-infected-recovered-pathogen) models used earlier to describe various scenarios of epidemic spreading. As compared to the basic SIR model, our model takes into account two possible routes of contagion transmission: direct from the infected compartment to the susceptible compartment and indirect via some intermediate medium or fomites. Transmission rates are estimated in terms of average distances between the individuals in selected social environments and characteristic time spans for which the individuals stay in each of these environments. We also introduce a collective economic resource associated with the average amount of money or income per individual to describe the socioeconomic interplay between the spreading process and the resource available to infected individuals. The epidemic-resource coupling is supposed to be of activation type, with the recovery rate governed by the Arrhenius-like law. Our model brings an advantage of building various control strategies to mitigate the effect of epidemic and can be applied, in particular, to modeling the spread of COVID-19.
1. Introduction
The paper develops a feedback model of epidemic spreading that connects socially accessible transmission controls and quarantine with a collective economic resource affecting recovery. It extends epidemic modeling to include direct and indirect transmission and explores socioeconomic consequences and control strategies.
- Motivation and modeling context: Epidemic spreading models describe processes governed by kinetic coefficients representing elementary-process probabilities per unit time.More complex models may include stochastic effects, spatial flows, and nontrivial spatial structure.
- Motivation and modeling context: Selecting epidemic-control strategies involves socially accessible parameters such as working duration, public-place density, disinfection frequency, and transport intensity.The paper frames strategy selection as an optimal-control problem for feedback systems or, more generally, a game-theoretic problem.
- Socioeconomic feedback: Epidemic spread and quarantine reduce the collective resource, which can lower medical-service quality and recovery rates, creating a feedback toward further resource reduction.The paper uses a simple dynamical feedback model to examine control measures and their consequences.
- Model extensions: The model represents contagion transmission across environments with different population densities, average distances, and daily residence times.Examples include home, shops, transport, and work.
- Model extensions: It includes indirect transmission through contaminated media or fomites, with rates depending on the same social-control parameters used for direct transmission.The intermediate medium or objects are treated as a transmission channel in each social location.
- Socioeconomic feedback: The epidemic-resource coupling uses an activation-type recovery dependence, with recovery rates proportional to exp(−E/ρ).Here E represents a minimum resource-consumption level, while ρ denotes the collective economic resource.
2. Direct and indirect transmission rates
The paper estimates direct and indirect transmission rates from average social distances and residence times across locations. It uses these parameters to compare social scenarios and quantify how quarantine changes the integral transmission rate.
- Control parameters: Transmission rates are estimated using socially accessible controls: average distance ℓj and time Tj spent in location j.These parameters are applied to both direct infected-to-susceptible transmission and indirect transmission through the environment.
- Direct transmission: The model assumes infection occurs when susceptible and infected individuals approach within a characteristic correlation radius ℓc.The correlation radius can be associated with a socially safe distance.
- Direct transmission: The daily integral transmission rate combines contributions from multiple social locations, each characterized by its own distance and residence time.The locations may include home, shopping, public transport, and work.
- Scenario comparison: Soft quarantine reduces the integral transmission rate β by a factor of 3, while strict quarantine reduces it by a factor of 7.The scenarios differ through social distancing, restricted transport, limited contacts, and partial reductions in economic activity.
- Scenario comparison: Casual-scenario estimates of β fall within approximately 0.6 T −1, matching the range derived from COVID-19 data for Wuhan.The estimate is presented as a rough calculation based on the social-control parameters.
- Model limitations: The direct transmission-rate estimate is explicitly described as very rough because fluctuations in nearby individual counts affect the rate.The paper estimates this uncertainty using packing and occupancy considerations.
- Model limitations: The scattering cross-section may depend on density through a more complex nonlinear incidence function than the simplified form used here.The paper notes an alternative dependence proportional to c^α, with 1 ≤ α ≤ 3.
- Indirect transmission: The indirect-transmission formulation treats contaminated media or objects as a cloud associated with each social location.The cloud's pathogen density and its transmission parameter determine infection growth attributed to the indirect route.
3. Dynamical model
The model extends SIR-like epidemic dynamics with susceptible, infected, quarantined, recovered, and deceased compartments, pathogen clouds, and an economic resource that modulates recovery. It specifies social-location-dependent transmission, resource dynamics, transition processes, equilibria, and simplifying assumptions for analysis.
- Compartment structure: The population is divided into susceptible, infected, quarantined, recovered, and deceased compartments governed by a system of ordinary differential equations.Infected and quarantined individuals can recover with or without acquired immunity; quarantined individuals can also die.
- Transmission processes: Direct transmission rates vary across social locations, while pathogen-density variables represent indirect transmission through environmental clouds.Each cloud receives pathogen shedding from infected individuals and loses pathogen through natural inactivation, decontamination, or other routes.
- Economic coupling: The collective resource ρ represents average money or income per individual and evolves through acquisition by active individuals, expenses or taxes, and external inflow or outflow.Quarantined individuals are assumed not to work, and the acquisition rate is proportional to average working time.
- Economic coupling: Recovery rates depend on the economic resource through an activation-type mechanism, with E defining the minimum resource consumption associated with recovery.The model distinguishes resource-rich and resource-poor conditions for recovery and fatality among quarantined individuals.
- Assumptions: The model assumes a closed, uniformly distributed population without natural demography, latent exposure periods, pre-existing immunity, or immunity loss; quarantined individuals do not infect others.The total population density remains constant: s(t) + i(t) + q(t) + r(t) + f(t) = 1.
- Equilibria and analysis: The system has disease-free and endemic equilibria, and R0 determines outbreak behavior: R0 ≤ 1 yields a stable disease-free equilibrium, whereas R0 > 1 yields an unstable one.The first two equations are strongly nonlinear, and analytical solutions are unavailable in general; a particular no-immunity-loss case is treated separately.
4. Examples
The examples examine epidemic dynamics under different initial infection levels, transmission routes, quarantine strategies, and resource constraints. They show that interventions can delay or reduce epidemic peaks, while limited resources and economically disruptive quarantine can increase fatal outcomes.
- 4.1. Effect of i0: When R0 = 2.5, increasing the initial infected density makes the epidemic reach the same peak intensity faster.The examples use i0 = 10^-7 and i0 = 10^-5; the larger initial density shortens the time to the peak.
- 4.2. Effect of indirect transmission: Including indirect transmission increases R0, potentially producing larger and earlier infected and quarantined peaks.The indirect route is represented by transmission through an environmental cloud.
- 4.3. Quarantine scenario: Reducing transmission through quarantine lowers R0 and epidemic intensity, but ending quarantine can produce a second wave.In the example, quarantine reduces R0 from 3.5 to 1.1, and transmission resumes after quarantine ends.
- 4.3. Quarantine scenario: Quarantine lowers peak heights and buys time, yet cumulative quarantined and fatal cases remain nearly unchanged when mortality stays constant.The result contrasts delayed epidemic progression with little change in eventual total illness and deaths.
- 4.4. Effect of limited resource: A normalized activation level E = 0.1 makes fatal cases nearly 6 times larger than with unlimited resource, despite only a 5% resource decline.Nonzero E reduces recovery rates through the factor exp(−E/ρ).
- 4.4. Effect of limited resource: Strict quarantine initially reduces fatal cases but later makes them larger than without quarantine because the resource acquisition rate falls to k = 0.5.Soft quarantine leaves k = 1 and causes fatal cases to grow more slowly during quarantine before returning toward the no-quarantine value.
5. Conclusion
The proposed epidemic model combines density-dependent direct transmission with indirect fomite transmission and couples infection dynamics to a collective economic resource. It is intended for COVID-19 and other spreading processes, while showing that epidemic control and economic losses can interact negatively.
- The model explicitly incorporates transmission rates that depend on local population density in different social zones and includes an indirect fomite-mediated transmission channel.These features extend the model beyond direct person-to-person transmission.
- A negative feedback links infected-population size with a collective economic resource representing average money or income per individual.The coupling represents socioeconomic interplay within the spreading process.
- Epidemic spread and quarantine use are connected with economic losses that can aggravate the epidemic’s negative outcomes.
- The model can be applied to COVID-19 epidemics in particular social groups and regions, as well as to information, rumors, ideas, and concepts.
Appendix A. Model parameters (extended)
The appendix specifies estimates for direct and indirect transmission parameters and describes how cloud-mediated pathogen transfer can be parameterized. It also identifies fitting to statistical data as a practical route for estimating difficult-to-derive transmission efficiencies.
- Table 1 reports direct transmission rates ωj and β computed from formulas (2) and (3), while Table A.1 lists model parameters and computational estimates.
- Ωj estimates pathogen transmission from cloud j to a susceptible individual using cloud-contact rate, transferred pathogen volume, pathogen quant, and infection probability.
- The pathogen shedding rate σj is estimated from infected-cloud contact rate, transferred pathogen quants per contact, and cloud capacity Vj.
- The transmission-efficiency parameter χj retains difficult-to-estimate quantities and can therefore be fitted to real statistical data.The appendix notes that indirect and direct transmission routes may be assumed to have approximately equal influence when selecting χj.
- The cloud-to-susceptible contact rate Γjs is assumed inversely proportional to the squared average interpersonal distance ℓj.
- Tables A.2 and A.3 provide indirect transmission rates for a casual scenario and available COVID-19 estimates of the basic reproduction number R0.
Appendix B. Extension to multiple groups
The model is extended to multiple groups, with each group governed by its own dynamical equations and direct and indirect transmission rates. The appendix also identifies tables for model parameters, indirect rates, and COVID-19 reproduction-number estimates.
- The model can be extended to a multigroup formulation, including subdivision by age.
- Each group n is governed by dynamical equations, with index m running over all groups.
- Direct and indirect transmission rates are defined for the multigroup dynamics.
- Table A.1 is identified as the source for model parameters, while Table A.2 reports scaled indirect rates and aggregate indirect transmission rate βp.
- Table A.3 lists available estimates of R0 for COVID-19, cholera, and pandemic influenza, with some values depending on medium, temperature, or surface type.
- Pathogen dynamics within cloud j are represented by a separate equation.