The Missing Link in the Kinetic Theory of Chemical Reactions and Epidemics L

The missing link in the kinetic theory of chemical reactions and epidemics L. Vanel1* 1Institut Lumière Matière, Univ Claude Bernard Lyon 1, Univ Lyon, CNRS; F-69622, Villeurbanne, France. *Corresponding author. Email: [email protected]

In thermodynamics, affinity is the driving force of chemical reactions. At equilibrium, affinity is zero, and the reaction quotient equals the constant of equilibrium. Thermodynamics predicts the timescale of chemical reactions but not rate laws that are purely phenomenological relations. Here, we derive an affinity relaxation theory (ART) that predicts chemical rate laws. Strikingly, the simplest chemical reaction in the ART model has a close resemblance with epidemic models instead of the mass-action rate law, revealing that epidemic models are actually more accurate chemical rate laws than the mass-action law. However, epidemic models appear in turn as an approximation of the ART model, where conceptual incoherencies of epidemic models are exposed and resolved. Furthermore, the mathematical Gompertz solution of the model links chemical rate laws with empirical biological and cancer growth laws. The ART model provides an entirely new framework to describe dynamic phenomena in chemical reactions, epidemics and biology.

Introduction The mass-action law introduced in 1864 by Guldberg and Waage (1) was the first equation linking the rate of a chemical reaction to the concentration of reactants. A main idea that quickly followed is that a reaction can occur in two opposite directions and that equilibrium corresponds to the point where the speeds of the forward and backward reaction cancel each other (2). Chemical thermodynamics established a long time ago that mass-action rate laws are compatible with the thermodynamic constant of equilibrium of a chemical reaction (3) and that rate constants depend on the reaction activation energy (4). However, the kinetic form of the rate laws does not follow first principles in thermodynamics (5). These rate laws remain phenomenological relations requiring experimental validation (6). Prigogine et al. have shown that close to equilibrium, the speed of a chemical reaction should be proportional to affinity (7), a quantity defined as the negative derivative of the Gibbs energy with respect to the extent of the reaction (3). Affinity represents the thermodynamic driving force of a chemical reaction. Prigogine and Defay derived an equation of evolution for affinity, which predicts an exponential relaxation towards equilibrium when the rate of variation of the other state parameters (pressure, temperature) is small enough (3). However, they used a linear dependence of the reaction rate on affinity, an assumption expected to be valid only close to equilibrium. Indeed, far from equilibrium, compatibility with mass-action rate laws implies a nonlinear dependence of the reaction rate on affinity (7). Predicting the reaction rate from affinity was attempted, but a purely empirical decay rate law was used (8). In this article, we postulate that exponential relaxation of affinity provides a general mechanism of convergence towards equilibrium and derive a new theory for the rate of chemical reactions as a replacement for phenomenological rate laws. This convergence mechanism is analogous to stress relaxation in a Maxwell fluid suddenly put out-of-equilibrium (9,10). In the affinity relaxation theory (ART) of chemical reactions, the reaction rate depends linearly on affinity, even far away from equilibrium. The theory removes the need to consider separate forward and backward reactions and predicts a significant time delay in the peak reaction rate that is nonexistent in conventional rate laws. We did not find chemical reaction data that could directly confirm the existence of this peak. Instead, we find an unforeseen application to the peak rate of an epidemic wave, reinstating the historical link between the rate laws of chemistry and mathematical epidemiology (11). We also find that the mathematical solution for the evolution of the reaction rate has the same generic form as the growth rate laws used in biology (12,13).

Results and discussion To illustrate the consequences of this new theory, we consider here the simplest chemical reaction AB. We note xA and xB as the number fractions of molecules A and B and assume they form an ideal solution. The equilibrium reaction constant is Ke=xB,e/xA,e. We note K=xB/xA the reaction quotient corresponding to the out-of-equilibrium value and the value at an initial reference time Ko=xB,o/xA,o. The affinity of the chemical reaction at any time is then A=kBT ln(Ke/K) and at the initial reference time Ao=kBT ln(Ke/Ko). Assuming an exponential relaxation of affinity towards -t/ equilibrium A=Aoe (3), we find that the reaction quotient K verifies:

et/ (1)et/ KtKK   oe (1) This prediction has a mathematical form that falls into the family of Gompertz functions (14). Eq. (1) is a generic result that applies to any chemical reaction provided the ideal solution assumption is valid (15). To find the corresponding reaction rate, we need to express the link between the extent of reaction  and the reaction quotient K. In the simple chemical reaction considered here, the number of molecules are NA=NA,o and NB=NB,o+(No=NA,o+NB,o), allowing us to express number fractions xA=NA/No and xB=NB/No as a function of the extent of reaction . We find that the reaction rate verifies (13):

dξ No  xxABA (2) dtkT  B This relation is valid for all values of Ao, however small or large. Thus, the speed of reaction varies linearly with affinity even very far from equilibrium. Second, there are no forward or backward reactions in this expression. The direction in which the reaction will take place depends only on the sign of affinity. If Ko K. In this approach, there is no longer a need for the concept of forward and backward reactions that cancel each other at equilibrium (2). Third, the mass-action rate law appears to be an approximation of this theoretical prediction. The usual mass-action rate law states that d/dt=kNoxAkNoxB, where k and k are the kinetic constants of the forward and backward reactions. The solution of this rate law is then an exponential variation of the extent of reaction ξξe 1exp(() ) kkt and of the reaction rate d/dt. In the case of a complete forward reaction (k+≫ k_), Eq. (2) shows that k+ is not a constant but evolves as the product of affinity and fraction of B molecules. Since affinity decreases exponentially and fraction xB increases in the course of the reaction, the product xBA has a bell-shaped variation, and the mass-action rate law appears as an approximate expression of the reaction rate.

Here, we consider the time evolution of the rate law Eq. (2) (Fig. 1A, 1B, 1C) for various equilibrium fractions xB,e and initial fractions xB,o

dξ N xxB,, p1 B p   o (3) dt p  12 xBp,  and occurs at time (Fig. 2C):

Ao t p  ln  (4) A p where Ap is the value of affinity at fraction xB,p. The overall shape of the reaction rate depends on the equilibrium fraction (Fig. 2D). Eq. (4) shows that a peak time exists only if Ao > Ap (here, A>0), which means xB,o < xB,p. This is a necessary condition for the appearance of a delayed peak rate. The time delay increases with smaller initial fractions of B molecules (Fig. 2C). In the opposite case where xB,o > xB,p, the maximum reaction rate occurs at the initial time and decreases almost purely exponentially with time. In that case, the mass-action rate law remains a very good approximation.

Fig. 1. Reaction rate of a simple chemical reaction in the ART model. (A), Molecule B number fraction when the initial fraction is smaller than xp. (B), Reaction rate showing a peak at time tp. (C) Reaction rate in logarithmic scale.

Fig. 2. Parameters controlling the reaction rate shape. (A) The fraction xp below which a peak will appear only depends on the equilibrium fraction xe. (B) When the initial fraction xo is smaller than xp, the peak rate value depends only on the equilibrium fraction xe, while (C) the time at which the peak appears depends crucially on the initial fraction xo. (D) The corresponding overall shape of the reaction rate for a given initial fraction xo depends on the equilibrium fraction xe (curves are shifted vertically for better visualization).

We are not aware of any chemical reaction data to compare with the prediction of a delayed peak in the reaction rate of a simple reaction. One may then wonder to what extent Eq. (2) gives a valid prediction for the reaction rate. One problem is that chemical reactions are usually fast, so it would be difficult to measure with enough precision the variation of reaction rate with time. In addition, the initial instant of a chemical reaction is often ill defined experimentally, especially when it involves a mixing step of reactants or catalysts. From a theoretical perspective, rate laws proportional to both the amount of reactant and product exist in the distant but historically related (11) field of mathematical epidemiology (16). A central assumption in most epidemic models is indeed that the rate of infection grows with the product of the number of susceptible persons by the number of infected persons, which is analogous to the product of the A and B molecule fraction that appears in Eq. (2). As an example of epidemic data observations, we show in Fig. 3A and 3B the daily rate of COVID-19 mortality in the United Kingdom (17). We observe that the shape is very close to the reaction rate predicted by Eq. (2) when the equilibrium fraction is -7 xe=0.5 and the initial fraction of B molecules is xo≅1.21 10 . The ART model of chemical reactions appears equivalent to the SIR epidemic model (16), except for the behavior at the tails. Theoretically, the reaction acceleration d2/dt2 has the same mathematical form as the infection equation of the SIR model but with variable rate coefficients (see Methods). Similarly, Eq. (2) leads to the first equation of the SIR model when replacing the product AkBT by Sd/dt, a conceptually confusing approximation mixing up the meaning of  with d/dt (see Methods and Fig. S1). In the ART model, the time-dependent coefficients lead to an inherent decrease in the reproduction number Ro (14) from an initial value of approximately 2 down to zero during the reaction. The important quantity controlling the epidemic dynamics is then the relaxation time  instead of the reproduction number Ro. The relaxation time  is easily determined from the exponential decrease with time of dln/dt, which is a direct consequence of affinity relaxation in Eq. (2) for xB≪ 1. Noting N the total mortality, we indeed found that dln/dt was almost purely exponential (Fig. 3C). The ART model captures this behavior very well but not the SIR model, especially at the start of the epidemic wave, where the SIR model displays a plateau corresponding to a constant rate of exponential growth. While we have found a possible theoretical connection between epidemic dynamics and a new and more accurate prediction of the rate of an elementary chemical reaction, empirical observations have already led some scientists to compare the shape of an epidemic wave with Gompertz functions similar to Eq. (1) and consider ad hoc decreasing exponential terms in the epidemic rate equations (18-23). In our theoretical derivation, the exponential relaxation comes from the decrease in affinity during convergence towards equilibrium of a chemical reaction. This kinetic model also suggests a parallel relationship between herd immunity (24-25) and an equilibrium corresponding to a state of maximum mixing entropy, a step forward from the concept of random mixing in epidemic theory (24). Finally, the presence of Gompertz functions in other out-of-equilibrium problems, such as biological growth or the growth of tumours (12-13,26), invites us to search for possible connections with this new kinetic theory of chemical reactions.

Fig. 3. Comparison between an epidemic wave, the ART model and the SIR model. (A) The reaction rate of a simple chemical reaction with an equilibrium fraction close to 0.5 very well describes the shape of an epidemic wave, here the daily mortality rate in the United Kingdom (15). (B), Same data as in (A), shown in logarithmic scale. (C) The daily rate of the logarithm of total mortality displays almost purely exponential behavior. See Methods for details on the fitting procedure of the ART and SIR models.

For a generic chemical reaction:

11221122RRRPPP iiiiiinn  , where i are the stoichiometric coefficients, and the theory predicts a reaction rate (15) dξ  A N  ξ  T oo (5), 2 dt   i 2   o i xi where NNoio  , , xi are the number fractions of each species and oi  (i >0 for products, i i

i <0 for reactants). Since affinity and number fractions are all known functions of the extent of reaction , Eq. (5) is a self-consistent nonlinear differential equation for . Reactants and products have the same role in this relation, except for the sign of the stoichiometric coefficients that defines the positive direction of the reaction, traditionally from left to right. This is a vector description of chemical kinetics where the one-dimensional force vector is affinity, the position vector is the extent of reaction and the velocity vector is its time derivative. The more general connection between the various forms of phenomenological rate laws that exist in chemistry (6) and the general prediction of the reaction rate, Eq. (5), is a remaining formidable task to undertake. We also note that the formula proposed by Prigogine and Defay for the evolution of affinity (3) includes the effect of pressure and temperature rates. Adding them to the theory could help understand out-of-equilibrium dynamics in systems where temperature or pressure variations are important (27-29).

Materials and Methods Analysis of the first wave in UK Data consolidated by Our World In Data (17) has been used. The reported number of infected persons depends greatly on the testing rate of the population. We instead consider the statistically more reliable reported number of deaths, even though it also suffers from delays between the actual death date and the reported date. Weekly fluctuations in data reporting are evident for all countries when looking at the raw data. In the case of the UK, these fluctuations disappear when considering the actual death date rather than the reported date. For that reason, we are considering here the weekly running average of the number of deaths Nd instead of the daily reported values. Adjustment of the model to the death data is done assuming a death rate of r (1 death per hundred cases) by using a least-square method minimizing simultaneously the difference between the model and the cumulative number of deaths and the daily rate of deaths (in lin and log scale). We use Eq. (1) in the following form: (tt )/ (tt )/ e o 1e o x xx   oe (6) 111xxx  oe where x=N/No, xo=1/No, xe=0.5 and N=Nd/r. Values obtained for the three free parameters are st 58.9 days since Jan. 1 2020, No=8.24 million and =20.3 days when fitting data until day 210. Changing r does not change the shape of the reaction rate; it only changes No in proportion to r. We also compare the first wave in the UK with a prediction from the SIR model14. The equations used have the classical form: dS dI dR  IS;;  SI  I  I (7) dt Noo dt N dt We compare the daily rate of new infections, estimated again from the daily rate of deaths using the death rate parameter r = 0.01, to the value of dR/dt (or equivalently ) predicted by the SIR model with constant parameters. Using the same minimization procedure and starting with one infected case, S(0)= No1 and I(0)=1, the values obtained for the parameters are 34.8 days since st -2 Jan. 1 2020, , = 4.08 10 and No = 4.11 million when fitting data until day 210.

Link between the SIR model and the ART model of chemical reactions The time derivative of Eq. (2) gives the reaction acceleration (15): dd2ξξ1  121 x  A (8) dtdt2  T where T=1/kBT. Using the following identities SNNxoo 1 , I d dt  and

N So S K , we can rewrite Eq. (8) as:

dI I S I I  I (9) d t N o

KSe KSe with I  2ln  and I 1ln . Eq. (9) is equivalent to the infection equation NSo  NSo  nd of the SIR model (2 equation in Eq. (7)). However, contrary to the SIR model, I and I are functions of the time-dependent population S. Two interesting limits are at time t=0, where I/I ≈2 far from equilibrium, and at long times, where I/I ≈0. In other words, the instantaneous 14 reproduction number RI/I evolves from 2 at the beginning of the reaction to 0 at the end. Rewriting similarly the reaction rate Eq. (2), we obtain: dS   I 1 IdtS (10)  2  dtN o

Here, we find a more important difference with the SIR model than the nonconstant value of I in the ART model. Indeed, in the SIR model, the product of S times I appears, while here, S is multiplied by the integral of I (half the integral). However, since I is proportional to affinity and 1 thus decreases exponentially with time, while ∫ 2 Idt is a monotonic growing function, the product I ∫ Idt may easily be mistaken with I. Indeed, Eq. (2) also writes:  I I 1 Idt S (11)  2  No

When xe=0.5 ( Ke 1), there is at most a factor of 2 between I ∫ Idt and I (Fig. S1). The corresponding ratio S (Fig. S1B) would represent the coefficient  in the SIR model if it was a constant, with an expected value between 1 and 2 in the ART model. However, contrary to the SIR model, S is a different quantity from the coefficient I introduced in Eq. (9). Nevertheless, taking -7 the typical values ≈20 days and xo=1/No≈10 obtained for the UK, we find a numerical estimate -1 I =2 ln(Ke/Ko)/ ≈1.61 day at time t=0, going towards zero at long times. Thus, even though S and I are not the same quantities in the ART model and do not even have the same units, their practical numerical values are not so different. This may be the reason why an approximation of the ART model where I ∫ Idt is replaced by SI in Eq. (10) and I is replaced by S in Eq. (9) would lead to the SIR model. Note that for any other equilibrium constant, the error factor at long times would be 1+Ke, a potentially large error when Ke becomes large.

References 1. E. O. Voit, H. A. Martens, S. W. Omholt, 150 years of the mass action law. PLoS Comput. Biol. 11, e1004012 (2015). 2. K. J. Laidler, Chemical kinetics (Harper and Row, New York, 1987). 3. I. Prigogine, R. Defay, Chemical thermodynamics (Longmans, London, 1954). 4. H. Eyring, The activated complex in chemical reactions. J. Chem. Phys. 3, 107-115 (1935). 5. S. R. de Groot, P. Mazur, Non-equilibrium thermodynamics (Dover, New York, 1984). 6. S. M. Walas, Kinetics for chemical engineers (Butterworth Publishers, Stoneham, 1989). 7. I. Prigogine, P. Outer, C. L. Herbo, Affinity and reaction rate close to equilibrium. J. Phys. Colloid Chem. 52, 321-331 (1948). 8. M. Garfinkle, Non-equilibrium thermodynamics of closed-system reactions. Mater. Chem. 7, 359-393 (1982). 9. M. Doi, S. F. Edwards, The theory of polymer dynamics (Clarendon press, Oxford, 1986). 10. J.-P. Hansen, I. R. McDonald, Theory of simple liquids (Elsevier Academic Press, Amsterdam, 2006). 11. H. Heesterbeek, “The law of mass-action in epidemiology: A historical perspective” in Ecological Paradigms Lost, K. Cuddington, B. E. Beisner, Eds. (Academic Press, Burlington, 2005), pp. 81-105. 12. C. Winsor, The Gompertz curve as a growth curve. PNAS 18, 1-8 (1932). 13. G. G. Steel, Growth kinetics of tumours (Clarendon, Oxford, 1974). 14. P. Waliszewski, J. Konarski, “A mystery of the Gompertz function”, in Fractals in biology and medicine, G. A. Losa, D. Merlini, T. F. Nonnenmacher, E. R. Weibel, Eds. (Birkhauser Verlag, Basel, 2005), pp. 277-286. 15. See supplementary materials. 16. J. D. Murray, Mathematical Biology I. An Introduction, 3rd ed.; Interdisciplinary Applied Mathematics Vol. 17. (Springer, New York, 2002). 17. M. Roser, H. Ritchie, E. Ortiz-Ospina, J. Hasell, Coronavirus pandemic (COVID-19). Our World in Data, (2020). https://ourworldindata.org/coronavirus. 18. K. M. C. Tjørve, E. Tjørve, The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the Unified-Richards family. PLoS One 12, e0178691 (2017). 19. F. W. Nutter, Quantifying the temporal dynamics of plant virus epidemics: a review. Crop Prot. 16, 603–618 (1997). 20. R. Burger, G. Chowell, L. Yissedt Lara-Diaz, Comparative analysis of phenomenological growth models applied to epidemic outbreaks. Math. Biosci. Eng. 16, 4250–4273 (2019). 21. Y. T. Utsunomiya et al. Growth rate and acceleration analysis of the COVID-19 pandemic reveals the effect of public health measures in real time. Front. Med. 7, 247 (2020). 22. D. Lanteri, D. Carco, P. Castorina, How macroscopic laws describe complex dynamics: Asymptomatic population and Covid-19 spreading. Int. J. Mod. Phys. C 31, 2050112 (2020). 23. A. Ohnishi, Y. Namekawa, T. Fukui, Universality in COVID-19 spread in view of the Gompertz function. Prog. Theor. Exp. Phys. 2020, 123J01 (2020). 22. J. P. Fox, L. Elveback, W. Scott, L. Gatewood, E. Ackerman, Herd immunity - Basic concept and relevance to public health immunization practices. Am. J. Epidemiol. 94, 179-189 (1971). 23. T. Britton, F. Ball, P. Trapman, A mathematical model reveals the influence of population heterogeneity on herd immunity to SARS-CoV-2. Science 369, 846–849 (2020). 26. P. Castorina, P. P. Delsanto, C. Guiot, Classification scheme for phenomenological universalities in growth problems in physics and other sciences, Phys. Rev. Lett. 96, 188701 (2006) 27. M. E. Cates, J. Vollmer, A. Wagner, D. Vollmer, Phase separation in binary fluid mixtures with continuously ramped temperature. Philos. Trans. A Math. Phys. Eng. Sci. 361(1805), 793-804 (2003). 28. C. Devailly, C. Crauste-Thibierge, A. Petrosyan, S. Ciliberto, Phase-transition oscillations induced by a strongly focused laser beam. Phys. Rev. E 92, 052312 (2015). 29. T. Sawato, Y. Shinozaki, N. Saito, M. Yamaguchi, Chemical CD oscillation and chemical resonance phenomena in a competitive self-catalytic reaction system: a single temperature oscillation induces CD oscillations twice. Chem. Sci. 10, 1735-1740 (2019).

Supplementary Text Full derivation of the reaction rate

We consider the following generic chemical reaction:

11221122RRRPPP iiiiiinn  , where the Ri and Pi are the reactants and products of the forward reaction, and  i are the stoichiometric coefficients. In the following mathematical formulation, is an algebraic quantity with positive sign for products and negative sign for reactants. We note Nio, the initial number of molecule i and ξ the extent of reaction (we use a bold notation for the extent of reaction to remember it can be either positive or negative, depending on the direction of the reaction). The number of molecules of each species varies with the extent of reaction according to:

NNiioi,  ξ

NNNtotioo  ξ where oi  and NNoio  , . The molecule number fraction of each species is then: i i

Ni, oi  ξ xi ()ξ  , Noo ξ related to the usual bulk concentrations Ci=xi(No+o)/V, where V is the total volume. The affinity of this reaction is (5): G A    ii ξ TP, i where G is the Gibbs free energy. For an ideal solution, the affinity becomes: o A  i i k B Tln x i  k B T ln K e  ln K i

i where we have introduced the equilibrium reaction constant Kxeie  , and the reaction quotient i

i i Kx  i . The initial value of affinity is then AoBeok TKKlnln , with Kxo  i, o . i i t/ Assuming that affinity converges exponentially towards equilibrium AA oe (7), we find that: et/ (1et/ ) K t  Koe K , which corresponds to Eq. (1) in the main text.

Expressing the exponential evolution of affinity as a differential equation, we find: ddAA KdK ln AAξ  00   TT  dtdtdtdK d / ξ where we used the fact that K is also a known function of ξ and noted TB1/kT.

1

We also have:

dK  1 dxdxdxjjjjj   xxxKii j ijji  ddxdxdξξξξ jjj i ji jj

dxx  oi oiN ,  ξ  iiio i dNNξξξ 2 oooo  Noo ξ

dKK  i  io i x  dNxξξooi i  Finally, the reaction rate is: Nx ξ dξ AAN  ξ ooi  TTooi dt  2 i  2 22  o ijoixx i xi i j ii

The simplest chemical reaction: reaction rate and its characterisistic values

Consider the simplest chemical reaction where AB, we have  A 1 and B  1 ( o  0 ) and we find:

dξ TTAANo x A x B No x A x B dt xAB x

Noting xxB  and xxA 1 , the time derivative of this reaction rate is: dd2ξξ1  121 x  A dtdt2  T It cancels in a non-trivial way when:

121xpTp A  xe 1 xp 12ln1xp   1 xxep

This equation gives an implicit relation xxpe()between the number fraction at the peak rate and the equilibrium number fraction. The value at the peak is then: xx1 dξ NNoopp   TA pxx pp1  dtxp 12 p and depends only on the equilibrium number fraction not the initial one. The time tp at which this peak occurs is: A ln KK t lno ln eo p A ln KK p  ep Considering the case where xo0, we see that the peak only exists if Ao>Ap, or Ko

2

Fig. S1. Comparison between the SIR and ART models for the rate equation dS/dt. (A) The quantity I of 1 the SIR model and I ∫ 2 Idt of the ART model appearing in the rate equation dS/dt remain close to each other. (B) The ratio S between the two quantities reaches a factor of 2 at long times. These curves are 8 obtained for No=10 , Ke=1 and =20 days.

3