Passive and active field theories for disease spreading

arXiv:2307.00437 · cond-mat.soft, cond-mat.stat-mech, physics.soc-ph, q-bio.PE · Submitted 2023-07-01 · Read on arXiv

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Introduction to the show: ident: Genomics Radio. Generated commentary on the latest computational biology and genomics papers.

Ines: Hi, I'm Ines.

Marcus: And I'm Marcus, here with Yuki.

Ines: We're looking at the paper: "Passive and active field theories for disease spreading".

Marcus: Here's the gist of the paper. The SIR-DDFT model is an extension combining epidemiological compartmental models with dynamical density functional theory to explicitly account for social distancing in disease spreading, which allows researchers to investigate how non-reciprocal interactions affect epidemic dynamics in soft matter physics.

Ines: The gist The SIR-DDFT model is a combination of the susceptible-infected-recovered (SIR) model from epidemiology with dynamical density functional theory (DDFT) from nonequilibrium soft matter physics, which allows for an explicit modeling of social distancing Model Foundation The starting point is the very well-known SIR model developed in Ref.

22: (based on earlier work

51, 52: ) which describes the time evolution of the total number of susceptible (S¯), infected (¯I), and recovered (R¯) persons Incorporating Social Distancing via DDFT To tackle the problem where all particles diffuse freely without affecting each other in a human society due to social distancing being a repulsive interaction This leads to an RDDFT model capable of describing epidemic spreading The free energy functional F is given by F = Fid + Fexc + Fext (8) In the present context, repulsive interactions represent the effects of social distancing and isolation This is represented by including a contribution for social distancing Fsd and a contribution for self-isolation Fsi The excess free energy then reads Fexc = Fsd + Fsi (11) Epidemiological Extensions The SIR-DDFT model allows for several extensions motivated by epidemiological considerations 1.

Ines: Governmental interventions: This involves adding dynamic equations for the coefficients Csd and Csi based on the rectangular hysteresis model The physical idea is that the start and end of a shutdown is triggered if the number of infected persons ¯I passes threshold values ¯Istart and ¯Istop, respectively 2.

Paper summary: Ines: So we're looking at this paper "Passive and active field theories for disease spreading". It combines epidemiology with soft matter physics to model things like social distancing.

Marcus: Right. The main point is using a SIR-DDFT model, which takes the standard SIR model and adds dynamical density functional theory to explicitly include social distancing.

Yuki: What this means for us is that we can look at how non-reciprocal interactions affect disease spread through the lens of soft matter physics, which is usually about colloids and polymers.

Ines: The paper claims they extend this SIR-DDFT model both epidemiologically by adding things like vaccines, asymptomatic cases, reinfections, and mutations.

Marcus: And from the soft matter side, they incorporate noise and self-propulsion to make it a more active system. They even derive a phase field crystal model for a simpler description.

Ines: The simulation part is interesting because they look at how these epidemiological models are affected by non-reciprocal interactions using computer simulations of a zombie outbreak.

Marcus: That's what they did in Section V, showing the results of numerical simulations on that zombie outbreak scenario described by equations sixty and sixty-one <ref:2307.00437#pg2,the results of numerical simulations>.

Yuki: I wonder if this framework helps us understand how population genetics or community structure influences these spreading dynamics in a more physical way.

Ines: The paper focuses on modeling the dynamics of susceptible, infected, and recovered populations spatially rather than just total numbers. They use density fields S, I, and R to capture that distribution.

Marcus: They start with the standard SIR model where susceptibles get infected at a rate ceff times I with an effective contact rate ceff, and infected people recover at a rate w and die at a rate m <ref:2307.00437#pg2>.

Yuki: But the spatial distribution matters because it dictates how interactions happen locally. How does this spatial aspect change what we can infer about disease spread in a real population?

Ines: It means we're not just looking at aggregate numbers anymore; we're modeling where the people are, which directly affects contact rates. The paper introduces the free energy functional F, which includes terms for social distancing and self-isolation.

Marcus: They define this excess free energy as Fexc = Fsd plus Fsi, which represents the effects of social distancing and self-isolation in the system <ref:2307.00437#pg2>.

Yuki: So, when we talk about social distancing as a repulsive interaction in soft matter physics, what are we really trying to capture about human behavior?

Ines: They incorporate governmental interventions by adding dynamic equations for coefficients Csd and Csi based on the rectangular hysteresis model. This means contact restrictions aren't constant; they switch on and off depending on whether the infection number crosses certain thresholds <ref:2307.00437#pg3>.

Marcus: That transition logic, using those threshold values like Istart and Istop, is a physical way to model how interventions are imposed and lifted over time. It's not just a simple switch.

Paper summary: Yuki: And then there's the extension for vaccination, which moves people from the susceptible compartment to the recovered one in this model <ref:2307.00437#pg3>. How does that fit into tracking population immunity?

Ines: It also allows for reinfections by introducing a constant reinfection rate cR, where recovered persons can get infected again when they encounter an infected individual <ref:2307.00437#pg3>.

Marcus: They also extend this with fields for exposed and asymptomatic persons, E and A, which adds the incubation period and accounts for people who are infected but not showing symptoms <ref:2307.00437#pg3>.

Yuki: If we look at the mutation extension they include, IM and RM fields. Does this suggest that genetic drift or selective pressures play a role in how these new variants spread through the modeled environment?

Ines: Yes, they add equations for those fields like ∂tIM = ΓIM ∇ · ⃗ IM∇⃗ δF/δIM + cMSIM + cMRRIM − wMIM − mMIM <ref:2307.00437#pg3>.

Marcus: And then they layer in the soft matter physics stuff, adding noise terms to make it stochastic, and active matter where motion depends on an orientation vector that describes self-propulsion <ref:2307.00437#pg2>.

Yuki: That active matter part is interesting because it treats humans as self-propelled particles, which connects the epidemiology directly to physical properties of the material.

Ines: They even simplify things by proposing an SIR-PFC model, replacing the complex DDFT equations with a reference density ρ hat for a phase field crystal <ref:2307.00437#pg3>.

Marcus: So, when they ran those computer simulations on the zombie outbreak described by SZ-DDFT model equations sixty and sixty-one what did they find about the influence of Csz <ref:2307.00437#pg2>?

Ines: They found that Csz has no influence on the final number of susceptibles or zombies at all. What really matters in that simulation is only the kill rate kappa <ref:2307.00437#pg2>.

Yuki: That's a bit surprising, since they built up such a complex model involving many interactions and fields before finding that one parameter dominates the outcome so strongly.

Marcus: Yeah, it means for ending the outbreak, whether kappa is smaller or larger than five determines the result <ref:2307.00437#pg2>. It simplifies the massive model down to a very basic comparison between two numbers.

Ines: The paper shows that passive and active field theories can provide a way to see how these different physical concepts interact in disease spreading, even when we look at complex scenarios like zombie outbreaks <ref:2307.00437#pg2>.

Marcus: It suggests that the mathematical structure of the model itself, incorporating these non-reciprocal interactions, is what drives the outcome more than some of the finer details they added.

Yuki: For someone who only listens to this show, it means we can use physics concepts from soft matter to analyze complex epidemiological dynamics in a way that goes beyond just counting individuals in compartments.

Ines: This work, "Passive and active field theories for disease spreading", suggests that bridging the gap between epidemiology and soft matter physics opens up new ways to model dynamic spreading processes.

Conclusion: Ines: So this paper, "Passive and active field theories for disease spreading," takes the idea of describing how things spread through space, like germs, and puts it into a framework using soft matter physics concepts.

Marcus: Yeah, they're combining epidemiological models with density functional theory to explicitly model social distancing and other complex behaviors in disease dynamics.

Yuki: From a population genetics standpoint, I’m interested in how this spatial modeling of interactions fits into the larger picture of how disease spreads through human communities over time.

Ines: What the authors are doing is extending these models by adding things like vaccines, asymptomatic cases, reinfections, and even mutations to see how those factors play out spatially.

Marcus: And on the physics side they’re adding noise and self-propulsion to treat humans like active particles moving around, which makes the model more dynamic than just a static density field.

Yuki: So it's not just about counting people in compartments anymore; it's about modeling how movement and physical interactions shape the spread of an infection across a landscape.

Ines: They even simplify things by using a phase field crystal model, which is like taking a complex set of equations and turning them into something easier to compute for simulations.

Marcus: The simulation results on that zombie outbreak were pretty telling because they showed that the final outcome really depends on one thing—the kill rate kappa—not all those other interaction terms.

Yuki: That’s interesting because it suggests that for a massive event, the basic killing power is more important than the fine details of how people are interacting or what specific social distancing rules you have.

Ines: It points toward a focus on fundamental parameters like transmission rates and mortality in these complex systems, which is something we need to keep in mind when analyzing any disease model.

Marcus: It also shows that bridging those two fields—epidemiology and soft matter—can give us new ways to look at the underlying physical mechanisms driving these spreading processes.

Yuki: That connection between how particles move and how diseases spread could open up new avenues for understanding population health dynamics in a much more detailed way.

Ines: So this work suggests that using these field theories allows us to study disease spreading not just as a sequence of events, but as a physical process happening in space and time.

Michael te Vrugt, *Julian Jeggle, *Raphael Wittkowski

Institut f¨ur Theoretische Physik, Center for Soft Nanoscience, Westf¨alische Wilhelms-Universit¨at M¨unster

cond-mat.soft, cond-mat.stat-mech, physics.soc-ph, q-bio.PE

Submitted: 2023-07-01

Updated: 2023-07-01

Comments: 15 pages, 2 figures

Journal ref: Journal of Physics A: Mathematical and Theoretical 57, 315003 (2024)

DOI: 10.1088/1751-8121/ad4c31

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 74/100

The gist: The SIR-DDFT model is an extension combining epidemiological compartmental models with dynamical density functional theory to explicitly account for social distancing in disease spreading, which

Key concepts

SIR-DDFT Model: This is a hybrid model that merges the standard SIR epidemiological model with Dynamical Density Functional Theory (DDFT) from soft matter physics. It allows scientists to explicitly simulate how social distancing, treated as repulsive interactions, affects disease spread in a system of interacting individuals.
Social Distancing via DDFT: In this context, social distancing is modeled as a repulsive interaction between individuals. This repulsion is mathematically represented by adding specific free energy contributions ($F_{sd}$ and $F_{si}$) to the total free energy functional. This framework allows for the explicit simulation of how isolation changes infection rates.
Phase Field Crystal (PFC) Model: The complex DDFT equations are simplified into a Phase Field Crystal (PFC) model. This approximation uses a reference density ($ar{ ho}$) to replace complex spatial gradients, allowing for a more manageable mathematical description of the disease spread dynamics.

Summary

The SIR-DDFT model is an extension combining epidemiological compartmental models with dynamical density functional theory to explicitly account for social distancing in disease spreading, which allows researchers to investigate how non-reciprocal interactions affect epidemic dynamics in soft matter physics.

The gist The SIR-DDFT model is a combination of the susceptible-infected-recovered (SIR) model from epidemiology with dynamical density functional theory (DDFT) from nonequilibrium soft matter physics, which allows for an explicit modeling of social distancing

Model Foundation

The starting point is the very well-known SIR model developed in Ref. [22] (based on earlier work [51, 52]) which describes the time evolution of the total number of susceptible (S¯), infected (¯I), and recovered (R¯) persons

Incorporating Social Distancing via DDFT

To tackle the problem where all particles diffuse freely without affecting each other in a human society due to social distancing being a repulsive interaction This leads to an RDDFT model capable of describing epidemic spreading The free energy functional F is given by F = Fid + Fexc + Fext (8) In the present context, repulsive interactions represent the effects of social distancing and isolation This is represented by including a contribution for social distancing Fsd and a contribution for self-isolation Fsi The excess free energy then reads Fexc = Fsd + Fsi (11)

Epidemiological Extensions

The SIR-DDFT model allows for several extensions motivated by epidemiological considerations

Governmental interventions: This involves adding dynamic equations for the coefficients Csd and Csi based on the rectangular hysteresis model The physical idea is that the start and end of a shutdown is triggered if the number of infected persons ¯I passes threshold values ¯Istart and ¯Istop, respectively
Vaccination: Vaccination can be incorporated by assuming it transfers persons from S- to R-compartment, leading to ∂tS = ΓS∇ · ⃗ S∇⃗ δF/δS − cSI − vS If the vaccine does not lead to full immunity, this model can be combined with an extension allowing for reinfections
Exposed and asymptomatic persons: This introduces fields E (exposed) and A (asymptomatic infected) to account for the incubation period and asymptomatic infection The dynamic equations are given by ∂tS = ΓS∇ · ⃗ S∇⃗ δF/δS − cS(A + I), (21) ∂tE = ΓI∇ · ⃗ I∇⃗ δF/δI + cS(A + I) − k1E, (22) ∂tA = ΓI∇ · ⃗ I∇⃗ δF/δI + k1E − (w1 + k2)A, (23) ∂tI = ΓI∇ · ⃗ I∇⃗ δF/δI + k2A − (w + m)I, (24)
Reinfections: A constant reinfection rate cR is introduced such that recovered persons can get infected when encountering infected individuals This leads to the equations ∂tS = ΓS∇ · ⃗ S∇⃗ δF/δS − cSI, (26) ∂tI = ΓI∇ · ⃗ I∇⃗ δF/δI + cSI + cRRI − wI − mI, (27) ∂tR = ΓR∇ · ⃗ R∇⃗ δF/δR + wI − cRRI. (28)
Mutations: This is incorporated by adding fields IM and RM for persons infected with a mutation and recovered from it, leading to equations like ∂tIM = ΓIM ∇ · ⃗ IM∇⃗ δF/δIM + cMSIM + cMRRIM − wMIM − mMIM

Soft Matter Physics Extensions

The model can be extended by incorporating physics from soft matter through several avenues

Noise: The deterministic theory is replaced by stochastic variants where noise terms are added, such as ∂tS = ΓS∇ · ⃗ S∇⃗ δF/δS + ∇ · ⃗ (p 2ΓSkBTSSvecηS) − cSI These stochastic equations describe the actual (possibly spatially averaged) densities, whereas the deterministic model gives the ensemble-averaged densities
Active matter: Humans are treated as active particles, leading to an active DDFT equation where motion depends on orientation vector ⃗u that describes self-propulsion The governing equation for an active DDFT reads ∂tS = ΓS∇ · ⃗ S∇⃗ δF/δS + DR,SβS∂ϕ/∂ϕ − vS∇ · ⃗ (Σu) The infection term is modified to account for orientation dependence, leading to ∂tI = (c⋆ϕ I)S
Phase field crystal model: To simplify the complex DDFT equations, the SIR-DDFT model can be approximated by an SIR-PFC model using standard approximations like replacing ∇·⃗ ϕ∇⃗ with a reference density ρˆ The resulting SIR-PFC model is given by ∂tS = ΓSρˆ∇⃗ 2δF/δS − cSI

Zombie Outbreak Scenario

The paper investigates a zombie outbreak using the SZ-DDFT model, which describes non-reciprocal interactions The SZ-DDFT model is given by ∂tS = DS∇⃗ 2S − ΓS∇ · ⃗ S∇⃗ (CszKsz⋆ Z) − bSZ, (60) ∂tZ = DZ∇⃗ 2Z + ΓZ∇ · ⃗ Z∇⃗ (CzsKzsS + (b − κ)SZ, (61) This model is obtained from the SIR-DDFT model by dropping the field R, changing the reaction terms, and flipping the sign of the interaction term in the dynamic equation for Z while keeping the sign in the dynamic equation for I

Simulation Results

Numerical simulations of a zombie outbreak based on Eqs. (60) and (61) were performed using finite difference schemes The simulation results show that Csz has no influence on the final number of susceptibles and zombies, and all that matters is the kill rate κ For ending a zombie apocalypse, the determining factor is whether κ is smaller or lager than 5.

Improvements for AI systems

Active Matter Simulation for Disease Dynamics (SZ-DDFT Model): The improved system can simulate a zombie outbreak using a zombie outbreak model governed by non-reciprocal interactions, which is described by the SZ-DDFT model where zombies exert a repulsive force on susceptibles and humans exert an attractive force on zombies.
Incorporation of Non-Reciprocal Interactions: The system can explicitly model non-reciprocal couplings by using the SZ-DDFT model derived from the SIR-DDFT model by flipping the sign of the interaction term in the dynamic equation for Z while keeping the sign in the dynamic equation for I.
Modeling Spatial and Temporal Heterogeneity: The system can incorporate non-pharmaceutical interventions by adding dynamic equations for the coefficients Csd and Csi based on a rectangular hysteresis model, allowing interactions to be timedependent based on infection thresholds.
Handling Complex Infection Dynamics: The system can account for the incubation period and asymptomatic spread by introducing fields E (exposed) and A (asymptomatic infected persons), leading to dynamic equations that describe transitions from exposed to infectious states at rate k1 and from asymptomatic to recovered at rate w1.
Modeling Immune Escape: The system can incorporate mutations by adding fields IM and RM, allowing for different infection rates based on the variant encountered, specifically where Susceptibles are infected with a rate c when encountering someone infected with the wild type and with a rate cM when encountering someone infected with a mutation.
Adopting Stochastic Dynamics: The system can transition from deterministic to stochastic dynamics by replacing the deterministic DDFT terms in Eqs. (14)-(16) by the ones known from stochastic DDFT to model fluctuations of the density fields will in practice also induce fluctuations on the SIR level.
Integrating Active Particle Motion: The system can model human motion as active particles by replacing the passive DDFT diffusion term with an active DDFT equation that includes terms for self-propulsion velocity v0∇ · ⃗ (ρ⃗u) and orientation dependence, where the infection probability depends on the orientation difference (c⋆ϕ I)S.
Applying Phase Field Crystal Approximation: The system can use a simplified SIR-PFC model by making approximations such as replacing ∇· ⃗ ϕ∇⃗ (with ϕ = S, I, R) in front of δF/δϕ by ˆρ∇⃗ 2 with a reference density ρˆ and using a Taylor expansion for the ideal gas free energy.