Discontinuous transitions of social distancing in SIR model
aa r X i v : . [ phy s i c s . s o c - ph ] A ug Discontinuous transitions of social distancing.
R. Arazi and A. Feigel ∗ Racah Institute of Physics, The Hebrew University, 9190401 Jerusalem, Israel
The 1st wave of COVID-19 changed social distancing around the globe: severe lockdowns to stoppandemics at the cost of state economies preceded a series of lockdown lifts. To understand socialdistancing dynamics it is important to combine basic epidemiology models for viral unfold (like SIR)with game theory tools, such as a utility function that quantifies individual or government forecastfor epidemic damage and economy cost as the functions of social distancing. Here we present amodel that predicts a series of discontinuous transitions in social distancing after pandemics climax.Each transition resembles a phase transition and, so, maybe a general phenomenon. Data analysisof the first wave in Austria, Israel, and Germany corroborates the soundness of the model. Besides,this work presents analytical tools to analyze pandemic waves.
I. INTRODUCTION
Pandemics are complex medical and socioeconomicphenomena[1, 2]: near social interactions benefit bothspread of disease and significant part of moderneconomy[3, 4]. During COVID-19 most government andindividuals accepted significant limitation on interper-sonal contacts (aka social distancing) to reduce pan-demics at the cost of the economy. Can the dynam-ics of social distancing be explained as a self-containedsocio-epidemiological model[5–7] or completely subjectedto extrinsic effects[8] is a crucial question for sociophysics,a field that tries to describe social human behavior as aphysical system.Social distancing changes as pandemic unfolds, reflect-ing a change in personal and government beliefs in thefuture. The spread of disease increases the individualprobability to become sick and may collapse the nationalhealth system. Social distancing reduces interpersonalinteractions and complicates some individual production.The level of social distancing corresponds to equilibriumbetween future beliefs on epidemic size and economycost[9, 10].To address social distancing classical epidemiologi-cal modeling is extended with economy or game the-ory tools[5–7, 11, 12]. SIR model[13] addresses diseasespread as gas or network-like[14] interaction of suscep-tible, infected, and recovered individuals. An infectedtransmit the disease to susceptible from the moment ofinfection until he/she recovers. Social distancing reducesinter-person interaction and thus reduces the effectivityof transmission, but at the same time claims significanteconomic price. To find an equilibrium level of social dis-tancing a utility function quantifies the negative weightof epidemics, social distancing itself, and the economiccost of social distancing. Utility function extension ofthe SIR model is in the focus of game theory treatmentof vaccine policies[15] and recent estimates of COVID-19economy damage due to social distancing[4, 9, 10, 16–18]. ∗ [email protected] Here we show that social distancing during a pandemicmay have discontinuous changes that to some extent re-semble phase-transitions. In this work, epidemy startsto unfold according to the SIR model. At each momentsocial distancing depends on utility functions that quan-tify the final epidemic size and social distancing economycost. Optimal values of social distancing correspond tothe maxima of the utility function. Discontinuous tran-sitions correspond to the abrupt changes in maxima lo-cations. This work presents an analytical expression ofthe general utility function in a polynomial form. Utilityfunction, so, reminds the free energy of a system duringGinzburg-Landau phase transition[19–22] and may havesome level of universality.The first component of the utility function is economicgain or loss from a decrease or increase of social distanc-ing. Thus social distancing parameter should addresschanges in the number of individuals that produce ortake place in productive mingling[9, 23].This work associates social distancing with s th = 1 /R ,where R is a basic reproductive number. R is a major pa-rameter of the SIR model - the average number of suscep-tible that an infected person infects. Epidemy breaks outwhen R > . Network interpretation maps SIR model ona network with edge occupancy T = 1 − exp( − s th ) [14]. Autility function, as all other parameters of the model, de-pends on network topology and s th . This work considersonly changes in s th . Association of social distancing tran-sitions only with s th is supported by the fit of COVID-19data.The second component of the utility function com-prises epidemy cost that favors social distancing. In thiswork, this cost corresponds to individual future proba-bility to become sick. This probability is proportional tothe forward amount of infected till the end of epidemicwave - Final Epidemy Size (FES). An important contri-bution of this work is an analytical presentation of FESderivatives due to basic reproductive number with thehelp of Lambert W function[24–26].The work proceeds with the presentation of the SIRmodel with induced transitions (SIRIT), almost analyt-ical treatment of this model, the calibration of an epi-demic and economy parameters of the model using timeseries of active cases and causalities during the 1st waveof COVID-19 in Austria, Israel, and Germany, followedby a discussion of obtained results and their implications. II. SIRIT MODEL
Classical SIR model separates the population in threecompartments: S susceptible, I infected and R recov-ered. The flux between compartments goes in the order S → I → R since susceptible becomes infected at fre-quency β − during an encounter with an infected one.Infected becomes recovered after time γ − on average.The population is well mixed and sustains the gas-likeinteraction of its members.During COVID-19 pandemics, daily worldwidereports[27] include active cases and coronavirus deaths.Active cases are detected infected, which are a fractionof the total infected. Thus in this work, we redefine I as active cases, and instead of recovered R will usedeceased D = M R , where M is Infected Fatality Rate(IFR). IFR is an average probability of an infected todie (for instance, reported Covid-19 IFR in Germany M ≈ . [28]).Here are SIR equations modified to this work: ∂s∂t = − βAsI, A = A ′ N ,∂I∂t = βI ( s − s th ) , s th = γβ = R (1) ∂D∂t = βs th N AIM where I is reported absolute amount of active cases[27], A ′ is a ratio between actual and reported active cases, N is the population size, < s < is the normalized( s = SN ) amount of susceptible and D are reported deathsdue to epidemy. When population size N and A ′ remainconstant it is convenient to unite them into a single pa-rameter A = A ′ N .The parameter s th represents social distancing in thiswork. It also represents a threshold value to ratio ofsusceptible in population: number of infected growthswhen s > s th and reduces when s < s th . Besides s th =1 /R , where R = β/γ is basic reproduction number.In this work s th changes with time according to utilityfunction U ( s th ) . At each moment t the value of s currentth changes if there exists s newth such that: U ( s newth ) > U ( s currentth ) . (2)The utility function consists of two parts that representpandemics cost C and economy gain G : U = C + G, (3) where C is proportion to epidemy size FES[15, 29]: C = − Aβ Z ∞ t Idt. (4)Epidemy cost changes with time as population advancesin ( s, I ) space.Economy gain is some general function G ( s th ) . Let usexpand economy gain in Taylor series around s currentth : G (∆ s th ) = G ( s currentth ) − A open ∆ s − th − A close ∆ s + th + B open (cid:0) ∆ s − th (cid:1) , (5)where ∆ s th = s th − s currentth , ∆ s − th and ∆ s + th are neg-ative and positive value of ∆ s th . The parameters A open , A close , B open are some positive economy weightsthat are independent of epidemy clinics. Utility functionis asymmetric in ∆ s − th and ∆ s + th because of higher eco-nomic price of a lockdown than price of later release. Ma-jor assumption of this work is that A open , A close , B open remain constant at least from epidemy climax (greatestnumber of infected) until the first major change of socialdistancing.Epidemy cost C is expanded in Taylor series of thethird order: C = C ( s currentth ) − a ∆ s th − b ∆ s th − c ∆ s th . (6)Coefficients a, b, c (as epidemy cost itself (4)) depend on s ht , s, I . Expansion of the third order, unlike the secondorder in (5), required due to non-linear behavior of FES(4) and because pandemic cost prevents significant lift ofsocial distancing constraints (the third term in (6)).This work considers only a decrease in social distanc-ing ∆ s th < which corresponds to the reopening ofthe economy. It is a consequence of an assumption that A close ≫ A open - even small constraint on social interac-tions bring significant economy damage. Transition occurif utility function (4), taking into account (5) and (6): U = − ( a + A open )∆ s th − ( b − B open )∆ s th − c ∆ s th , (7)possesses any positive values for ∆ s th < . To consideronly relative changes in s th , we set U ( s currentth ) = 0 .Utility function (7) together with condition (2) pos-sesses Ginzburg-Landau like instability, see Figure 1.First, no transition occurs if U < for all s th . Sec-ond, discontinuous change in s th take place if there issingle value U ( s th ) > . Third, s th changes contin-uously when derivatives of (7) vanish near ∆ s th = 0 ( a + A open = b − B open = 0 ).Discontinuous transition occurs when there exists asingle ∆ s th < root for U = 0 (7). This conditionrequires the determinant of quadratic function U/ ∆ s th (7) to vanish: c ( a + A open ) = ( b − B open ) . (8) -0.21 -0.16 -0.11 -0.06 0.00 0.04 s th s th U t ili t y [ a r b . u . ] Figure 1. Utility as a function of social distancing before,during and after transition takes place. A measure of socialdistancing is < s th < . Transition occur between cur-rent value of s currentth (black square) to its new value s newth (red circle) when U ( s newth ) > U ( s currentth ) ( U ( s currentth ) = 0 ).Before transition U ( s currentth ) is the higest value of the util-ity function. Then there are two possibilities: Discontinouschange s newth = s currentth (solid green line) or continuous chage s newth ≈ s currentth (dashed red line). Two cases (dotted lines)that make possible change of s th to many values do not ex-ist becasue either continuous or discontinuous transition occurbefore. This work consideres only opening of population, thatcorresponds to reduction of s th . Utility function is approxi-mately a cubic fuction of ∆ s th (solid green). The transitionpredicted by exact calculation (solid blue) predict insignifi-cant for this work changes in time and strength of the transi-tion. The corresponding ∆ s th : ∆ s th = b − B open c . (9)The new s newth is: s newth = s currentth + ∆ s th . (10)Coefficients a, b, c possess an analytic approximation,while condition (8) reduces to a polynome of the 4th or-der. III. MAIN ANALYTICAL PROPERTIES
The first two equations of (1) have a solution in theform of Lambert W function[24–26]: s = − s th W (cid:18) − s s th exp (cid:20) A ( I − I ) − s s th (cid:21)(cid:19) , (11)where ( s , I ) are initial values for ratio of susceptible s and amount of infected I correspondingly, see AppendixA. Eq. (11) is valid for any ( s, I ) on the trajectory intime that initiates at ( s , I ) . The parameter s reaches Figure 2. Epidemy wave dynamics with social distancing tran-sitions. SIR predicts wave-like behavior of infected individuals(dash-dotted blue) accompanied by reduction of susceptibleones (dotted blue). An utility finction caused changes in so-cial distancing parameter s th (solid green). Transition (redbars) occur when there exits new value of s newth such that util-ity function U ( s newth ) > . Form of utility function before andduring transition are show in subplots. Region of continuoustransitions follows the series of discontinuous ones. Changesin s th cause time derivative discontinuites in dynamics of in-fected and susceptibles. it smallest value s min when there are no more infected( I = 0 ) at t = ∞ : s min = − s th W (cid:18) − s s th exp (cid:20) − AI + s s th (cid:21)(cid:19) . (12)Following (1), FES at any time t is: Aβ Z ∞ t Idt = − log s | s min s ( t ) , (13)and parameters a, b, c in (6) are the Taylor coefficients: a = − ∂ log s min ∂s th , b = − ∂ log s min ∂s th , c = − ∂ log s min ∂s th . (14)The parameters (14) are polynomials of log s of the or-der 1,2 and 3 correspondingly, see Appendix B. Thuscondition (8) is a polynomial of the 4th order (quatricfunction) of log s , with coefficients that are the functionsof ( s , I , s, I, s th , β, A ) .Consider population at state ( s , I , s, I, s th , β, A ) .To calculate time to transition as a function of A open , B open one need to solve 4th order equation (8) tofind s tr of transition (closes to real root of (8)), calcu-late ∆ s th of transition using (9) then calculate the timeto transition: Z s tr s ds dsdt = Z s tr s ds − βA h I + A h s − s + s th log h ss iii s (15)between current value s and s tr .A function T : s tr , s newth = T ( s , I , s, I, s th , β, A, A open , B open ) , (16)summaries results of the previous paragraph. T returnsfuture value of s = s tr where transition of s th → s newth willtake place. The follow up transition follow T , after up-date of ( s , I , s, I, s th ) → ( s tr , I ( s tr ) , s tr , I ( s tr ) , s newth ) .Many transitions (16) result in Zeno phenomenon - thesystem making an infinite number of transition in a finiteamount of time, see Figure 2. Transitions of s th takeplace until (16) predicts ∆ s th = 0 . Following (8) and (9)it happens when the first two derivatives of (7) vanish: ( a − A open = b − B open = 0) . The time to these infinitenumber of transitions to take place remain finite becausetime is finite to pass between and two values of s , see(15).After the limit of transition, utility function preservescontinuous transition state. Otherwise if s th remain con-stant the utility function would make possible many val-ues of s th with U ( s th ) > , see Figure 1. At the regionof continuous transitions, at each moment equation: a = − A open (17)defines s th , and (1) is solved numerically.Transitions of s th result in time derivative discontinu-ities of s and I . These derivative discontinuities can bedetected, see Figure 2. IV. FIT
SIRIT model provided a successful fit of COVID-19data in Austria, Germany and Israel. Austria and Is-rael are countries with similar population sizes and withsimilar policies during the initial stages of the 1st wave.Germany is a country with about × population sizethat still demonstrates SIR like behavior during the firstwave.The main purpose of the fit is to show that self-contained model SIRIT is capable of description dynam-ics of the 1st COVID-19 wave. Full optimization of the COVID-19 data fit and its validation is out of the scopeof this work.The fit proceeds in the following steps: First, a smallregion around the greatest number of infected, see Fig-ure 3, calibrates s , I , β, A . In this work γ = 0 . , thischoice can be quite arbitrary in the boundaries of re-ported COVID-19 / < γ [ day − ] < values[30].Second, one can see discontinuous changes in timederivatives of reported active cases, especially as devi-ations from initial SIR model fit, , see the outermost leftred bar in Figure 3. This work associates these changeswith transitions of social distancing. The time t and s tr of the first transition are detected. New values of β, s th were fitted for an active case during about 20 days afterthe transition. In all case, fit predicted β to remainedunchanged while s th to accept a new value s newth . Third,eq. (16) is solved for A open and B open (e.g. using Nelder-Mead method), when s tr and s newth are the values fromthe previous step. Fourth, next transitions s ith , where i is the index of the transition, are calculated using (16)and (17).Complete dynamics of the first wave active cases andsusceptible, see Figure 3, follows eqs. (1) and the fitted s , I , s th , β, A, N together with the transitions locations t i , s itr and strengths s ith . See Figure 3 for changes in s th . To fit casualties, an effective population size N is cho-sen to fit reported coronavirus death at the 100th day ofthe first wave. It predicted quite small N less than / Austrian population. This result to be addressed dur-ing the discussion. Besides, there exists some time shiftbetween calculated and reported coronavirus death.Two alternative fits of Austria COVID-19 data arebrought in Figure 4. The first demonstrates the sensi-tivity of the fit to the choice of the first transition. Thesecond demonstrates that for any first transition the en-tire curve can be fitted if A open , B open are fitted separatelyfor any transition.The results for Germany and Israel summarized in Fig-ures 5 and 6 together with Table I. Table I summarizesthe results for all three countries. All countries demon-strated a low size of the effective population. One of theassumptions that s ≈ . In the case of Israel, it re-quired to be constrained. Neither of deviations from thefit refutes the main results of this work. V. DISCUSSION
The work introduces SIRIT, a standard Susceptible-Infected-Recovered (SIR) model extended with a utilityfunction that predicts induced transitions (IT) of socialdistancing. The model provides almost analytical treat-ment and reasonable but an ambiguous fit of COVID-191st wave active cases and casualties. Let’s summarize anddiscuss the main assumptions, results, and implicationsof this work together with some alternative approaches.The validity of the predicted discontinuous dynamicsof social distancing and results depend on the specific choice of the social distancing parameter and the choiceof a utility function.The choice of s th = 1 /R as a social distancing pa-rameter is not unique. In the framework of SIR, for in-stance, another valid candidate is β - the probability ofdisease transmission per unit time[6, 7]. This parameterdepends both on clinics of infection and the mechanismof interpersonal interactions. The fit of the first socialdistancing transitions (deviations from SIR model) afterCOVID-19 climax in Austria, Germany, and Israel, but,demonstrates that β remains constant during the transi-tions. Figure 4. Alternative fits of Austria COVID-19 1st wave A) Fit with different date of the first transition. There is a significantdeviation between reported and calculated active cases. It demonstrates that the fit techniques are sensitive to the choice of thefirst transition. This sensitivity provides a hope that the fit may reveal something real parameters of the population or economyof the state. B) Fit with two different distance transitions. By adjusting utility function weights A open , B open separately forevery transition a better fit can be achieved. In the case of Austria, only two transitions are required. This method is lesssensitive to the choice of the first transition.Figure 3. Fit of Austria COVID-19 1st wave with multiple social distancing transitions. The purpose of the fit is to show thatthere is a possibility to fit the real data using many transitions. A) Active cases - reported (dashed green) and calculated withmany transitions (solid blue). Before transitions (red bars) classical SIR fits well the reported active cases. There is a significantdeviation of SIR from reported cases after the first transition (dashed blue). SIRIT model with many transitions fir well entirerange of the first wave, though the fit was obtained using a small range around the greatest of active cases and characteristicsof the first transition (horizontal error bars). B) Susceptibles and Socal distance parameter s th . At each transition s th changesits value. Series of discontinuous transitions is followed by a continuous change region. C) Coronavirus deaths. There exists atime delay between reported and calculated deaths. This delay can be explained by the long course of COVID-19. An interesting conclusion of this work is that changein social distancing corresponds to γ - a rate of becomingimmune. Transition corresponds to changes in s th = γ/β while β remains constant. It seems a fallacy because γ seems to depend only on epidemy clinics. Even so, so-cial distancing affects γ - society on alert removes con-tagious individuals by distancing from confirmed sick oreven from asymptomatic cases that had contact with asick person.In an alternative way, /s th = R can serve as a socialdistancing parameter. This choice does not change major predictions or analytical developments of this work. Theparameter < s th < that can be compared with thefraction of susceptible in a population serves better thepurpose of this work.The price of pandemics can go beyond the final epi-demy size (FES). For instance, one may include deriva-tives of infected in time as a psychological factor that af-fects individual decision making. There is a lot of room tomake utility function more complicated. Relative weightsof different pandemic characteristics on individual or gov-ernment decision making is an important question forfuture investigations and out of the scope of this work.The first wave of COVID-19 in Austria, Germany,and Israel was fitted using SIRIT in two different ways:The first, series of discontinuous transitions with con- stant economy parameters. The parameters A open , B open are fitted by the first transition. The second, economicweights are fitted for every candidate transition (devia- Figure 5. Fit of Israel COVID-19 1st wave. The results are similar to the case of Austria. The fit is valid until the beginningof the 2nd wave about at the 70th day of the first one A) Active cases. A significant deviation exists between reported andcalculated active cases even before the start if the second wave. B) Susceptible and s th . The social distancing parameter s th remains a bit higher in Israel rather than in Austria or Germany. C) Coronavirus deaths. The time delay between reportedand calculated cases is smaller than in the case of Austria. It can be explained either by late or early reports of coronavirustests or reported deaths in Israel or Austria. D) Alternative fit with two transitions. tion from SIR model). Both these scenarios include dis-continuous changes in social influence and have the samefirst transition. All fit attempts of this work need pa-rameters of the SIR model to be constant from pandemicclimax (greatest number of infected) till the first tran-sition. Deviations from the fit may reflect a change ofregulation and test policies during the first wave.Analysis of the first wave predicts small, less than / , effective population size in all three tested coun-tries, see Table I. An estimate of effective populationsize depends on the choice of M - Infected Fatality Rate(IFR). Increase/reduction in M causes proportional re-duction/increase in effective population size N and pre-dicted ration A ′ between reported and real numbers of in-fected. These values in Table I correspond to M = 0 . .The reported value of M for Germany is . [28]. Thus N and A ′ maybe about × lower than in Table I. Never-theless M as low as . were reported[31]. All otherprediction or results of this work, including the graphs,are independent of M . Mortality rate M > . causesnon-physical A ′ < in case of Israel.An explanation of low N may be that the ini- tial lockdown separated the population in disconnecteddomains[32, 33] and the wave of epidemy occurred ina limited number of domains. The other possible ex-planation is that a significant part of the population isimmune to COVID-19[34]. Finally, SIR approach withquasi-constant parameters may be an oversimplified pre-sentation of reality.Small effective population size during the first wavemay indicate a danger of abrupt transition to a big-ger population size when s th reduces below some crit-ical value. It may result in a significant second waveof the epidemy. The critical value of the basic re-production number was reported for some interactionnetworks[33, 35], while the majority of the networks lackit[32, 36].To conclude, this work predicts observable transitionsof social distancing and provides tools for quantitativeanalysis of pandemic waves. Observable phenomena areessential to test the validity of human behavior model-ing. The tools, as SIR model itself, contribute to socialepidemiology[37] and, to spread of non-contagious but“going-viral” phenomena[38]. Appendix A: Analytic solution of (1)
The first two equations of (1) may be rewritten as: β ∂z∂x z = − βA (cid:18) s th + 1 β z (cid:19) exp ( x ) , (A1) using transformation ∂ log I∂t = z, ∂z∂t = ∂z∂ log I ∂ log
I∂t = ∂z∂ log I z, x = log I . Integration of (A1) results in: I = I + 1 A (cid:20) s − s + s th log (cid:20) ss (cid:21)(cid:21) , (A2) Figure 6. Fit of Germany COVID-19 1st wave. The results are very similar to the case of Austria. A) Active cases B)Susceptible and s th C) Coronavirus deaths. There is a time delay between calculated and reported deaths, like in the case ofAustria. D) Alternative fit with two transitions. s i γ s th β s th β s th β A N A ′ M ( IF R ) D D A open B open Austria, Fig. 3 0.99 242 0.26 0.56 0.46 0.45 0.46 1.26e-05 8.2e5 10 −
32 1.31 5.81Austria, Fig. 3 A 0.99 242 0.26 0.56 0.46 0.48 0.46 1.26e-05 8.2e5 10 −
29 1.84 6.30Austria, Fig. 3 B 0.99 242 0.26 0.56 0.46 0.48 0.46 0.32 0.46 1.26e-05 8.2e5 10 −
29 46 1.84 6.30Germany, Fig. 6 A,B,C 0.99 578 0.26 0.60 0.43 1.24e-6 1.1e7 14 −
45 1.15 5.93Germany, Fig. 6 D 0.99 578 0.26 0.60 0.43 0.47 0.43 0.33 0.52 1.24e-6 1.1e7 14 −
45 65 1.15 5.93Israel, Fig. 5 A,B,C 1.0 535 0.26 0.68 0.38 0.60 0.38 6.34e-06 4.5e5 3 −
40 1.33 4.92Israel, Fig. 5 D 1.0 535 0.26 0.68 0.38 0.60 0.38 0.50 0.51 6.34e-06 4.5e5 3 −
40 56 1.33 4.92Table I. Parameters of SIRIT model that fit the 1st COVID-19 wave of Austria, Germany and Israel. The parameters may beused to reproduce the figures of the article. For definition of parameters see eq. (1). Besides, D and D are the days of the firstand the second (if required) transitions. A open , B open are the weigths of utility function calibrated for the first transition. Aninteresting result is really small effective population size N , less than / of the state population. Greater values of mortalityrate M predict even lower values of population size N (increase in M causes proportional reduction in effective population size N and A ′ ratio between real and reported number of infected). Mortality rate M > . causes non-physical A ′ < in caseof Israel. The second transition in case of Germany and Israel brings change in β . The resutls of all three countries are quitesimilar except A ′ - ratio between real and reported number of infected. where ( s , I ) are initial values of s and I . To derive(11) for s in the form of Lambert W function (defined as W ( x ) , where W exp W = x ) one should rewrite (A2) as: − ss th exp (cid:20) − ss th (cid:21) = − s s th exp h A ( I − I ) − s s th i . (A3)The expressions (11) and (A2) provide connection be-tween s and I along population trajectory in ( s, I ) spacethat initiates at ( s , I ) . Let us define by ( s t , I t ) values of susceptible ratio s and number of infected I along population trajectory in ( s, I ) space that initiates at ( s , I ) . Then let us calculatethe coefficients a, b, c (14) and show that along trajectory ( s t , I t ) they are polynomials of log [ s t ] . Derivatives of log [ s min ] : log [ s min ] = log [ s th ] + log[ W ( f ( s th ))] , (A4)due to s th include derivatives of log [ s th ] and derivativesof log [ W ( f ( s th ))] , where: f ( s th ) = − s s th exp (cid:20) − s + AI s th (cid:21) . (A5)Let us notice that s th , f ( s th ) and W ( f ( s th )) are con-stant along trajectory ( s t , I t ) until value of s th changes bya transition. The values f ( s th ) and W ( f ( s th )) : f = − s s th exp h − s s th − AI s th i , (A6) W ( f ) = − s min s th , (A7)are constant until a transition takes place, because s min (A4) is constant along trajectory unless s th changes itsvalue. Besides: − s s th exp (cid:20) − AI + s s th (cid:21) = (cid:18) − s t s th exp (cid:20) − s t s th − AI t s th (cid:21)(cid:19) = const (A8)because any point ( s t , I t ) can serve as initial value ( s , I ) for continuation of the trajectory.Derivatives of log [ s th ] depend only on s th and, so, areconstant until s th changes.Derivatives of log [ W ( f ( s th ))] are: d log W ( f ) ds th = d log W ( f ) df dfds th d log W ( f ) ds th = d log W ( f ) df (cid:18) dfds th (cid:19) + d log W ( f ) df d fds th d log W ( f ) ds th = d log W ( f ) df (cid:18) dfds th (cid:19) +3 d log W ( f ) df dfds th d fds th + d log W ( f ) df d fds th (A9)Derivatives of log W ( f ) due to f are invariant along ( s t , I t ) trajectory, see Appendix B.Derivatives of f due to s th are polynomials of log [ s t ] .Consider the first derivative of (A5) taking into account(A8): dfds th = gf, (A10)where: g = (cid:18) − s th (cid:19) + (cid:18) s t + AI t s th (cid:19) . (A11)Expression (A2) can be rewritten in the form: AI t + s t = AI + s + s th log (cid:20) s t s (cid:21) (A12)for trajectory ( s t , I t ) . Plugging (A12) in (A11) resultsin: g = (cid:18) − s th (cid:19) + AI + s + s th log h s t s i s th . (A13) Thus g and its derivatives due to s th are linear functionof log [ s t ] . The second and the third derivatives then: d fds th = f g + f dgds th (A14) d fds th = f g + 3 f g dgds th + f d gds th (A15)are the quadratic and cubic polynomials of log [ s t ] . Appendix B: Derivatives of Lambert W function All derivatives of W due to f are constant along anytrajectory in ( s, i ) space: W exp [ W ] = f, (B1) d log Wdf = 1 f (1 + W ) , (B2) d log Wdf = − f d log Wdf − f W h d log Wdf i , (B3) d log Wdf = 1 f d log Wdf − f d log Wdf − W (cid:20) d log Wdf (cid:21) − f W (cid:20) d log Wdf (cid:21) − f W (cid:20) d log Wdf (cid:21) d log Wdf . (B4)The final expressions are invariant until s th changes be-cause they depend on f and W only, see (A7). Appendix C: Final expressions for a,b,c
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