momentum managing epidemic spread and bessel functions ... · managing epidemics, naturally mostly...

MOMENTUM MANAGING EPIDEMIC SPREADAND BESSEL FUNCTIONS

IVAN CHEREDNIK, UNC CHAPEL HILL †

Abstract. Starting with the power law for the total number ofinfections during the middle stages of epidemics, we propose dif-ferential equations describing the process of momentum epidemicmanagement, which is a set of measures aimed at reducing theepidemic spread via timely response to the dynamic of the num-ber of infections. In the most aggressive mode, the saturation ofthe number of infections can be achieved sufficiently quickly. Our2-phase formula matches very well the total numbers of infectionsfor Covid-19 in many countries; modeling the first phase is basedon Bessel functions. We discuss Austria, Germany, Japan, Israel,Italy, the Netherlands, Sweden, Switzerland, UK, and the USA.

Key words: epidemic spread, epidemic psychology, Bessel functions

MSC 2010: 92B05, 33C10

1. Our approach and findings. A system of differential equationsis proposed describing the effects of momentum management of epi-demics , which can be defined as a system of measures of any natureaimed at reducing the epidemic spread by regulating the intensity ofthese measures on the basis of the latest total numbers of infections.The ”hard measures” are the key, the detection and isolation of infectedpeople and closing the places where the spread is the most likely. Wedemonstrate mathematically that the epidemic will reach some satura-tion, followed by a period of essentially linear growth. The two-phaseformula , which is the main output of this paper, is tested well for thespread of Covid-19 in many countries. The first phase is describedby Bessel functions with high accuracy. The exactness of our formulafor the second phase is even more surprising taking into considera-tion many factors influencing the management of the epidemic at laterstages. A ”soft” version of our system of differential equations is used.The readers mostly interested in the final formulas and applications inparticular countries can go directly to the figures from Section 11.

† May 29, 2020. Partially supported by NSF grant DMS–1901796 and theSimons Foundation.

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The second phase is generally when ”hard measures” are relaxed oreven abandoned. Wearing the protective masks and social distancingbecome the key. The self-isolation and prompt requesting medical helpare of this kind too. The impact of these and similar soft measures canbe significant for reducing the transmission rate, but the first ”hardphase” are the key for reaching the saturation reasonably quickly.

The formula for the second phase is Ctc/2 cos(d log(t)), where t istime, d reflects the intensity of ”soft” measures, and c is the initialtransmission rate. Here c/2 ≈ 1, so the number of new daily detectedcases is essentially constant. If it is small enough, the spread can becontrolled: new clusters of the disease can be then promptly detected,an so on. Also, asymptomatic (mild) cases begin to dominate at thesecond phase, which is a positive development too. Mathematically,the first phase based on ”hard” measures is necessary for phase 2.

Focus on risk-management. We attribute the surprising similarity ofthe curves of the total numbers of detected cases in many countriesto the uniformity of the measures employed and the ways we reactto the data. Other numbers can be used too, but they are generallymore complex. It is not really important in our approach that thesenumbers mostly reflect symptomatic cases and can be underreportedbecause of various reasons. As far as they influence the decisions of theauthorities in charge and our own behavior, they can be used. Such a”sociological” approach focused on the epidemic management explainswell the similarities of the curves from so many places.

We see 3 basic types of management. The countries in the first groupare determined to reach ”double digit numbers” of new daily infections.The second group is when the reduction of hard measures begins uponthe first signs of the stabilization of the daily numbers, which can bestill very high; this is the switch to the (AB)–mode or (B)-mode inour model. The third group of countries is where ”hard” measures arenot employed systematically, which can be due to a variety of reasons,including insufficient capacities or political decisions. Some measuresare always in place, for instance, self-isolation of those who think thatthey can be infected. Their prompt requests for medical help can workwell in the countries with solid health-care systems, like Sweden. Also,travel restrictions, common everywhere, definitely reduce the spread.

Testing our theory. Our research was organized as follows. The ap-plications of our theory to mode (A), the most aggressive respond tothe spread, was essentially done around April 15 of 2020. Not toomany countries reached the ”saturation” by then, but the middle stagesmatched this new theory almost with an accuracy of physics laws. Thechallenge was to understand the latter stages.

MOMENTUM MANAGING EPIDEMIC SPREAD 3

The parameter c, the initial transmission rate, is determined duringthe early stages of the epidemic, as well as a, the intensity of hard mea-sures. They appeared coinciding for the UK and the Netherlands. Thelatter reached the saturation, the end of the first phase, after about45 days (counting from March 13). For UK, it appeared a significantlyslower process, which we attribute to general relaxation of ”hard” mea-sures after the ”turning point”, i.e. before the saturation was reached.It is even more visible in the USA, with the same a and somewhatsmaller c ; the middle stage perfectly matched our Bessel-type formula,but then the response to the epidemic changed.

At the end of April we saw some signs of the switch from mode (A) tomode (AB) in the USA and UK. This is a transitional mode, where themain measures are ”hard”, but the response to the current total numberof infections is as in (B), not really ”aggressive”. The corresponding(AB)-curves, called w-curves in the paper, were calculated for the USA,UK on May 5. The expected dates of ”technical saturations” underthe (AB)–mode were correspondingly May 30 and June 10 for thesecountries. This did not materialize for the USA due to the significantreduction of the hard measures starting approximately in the middleof May, which is the switch to mode (B) in our theory; the curve forUK is still within the (AB) forecast cone.

Accordingly, a period of linear growth of the total number of (de-tected) infections is likely to last for some time in the USA with rel-atively high numbers of new daily cases. Mathematically, the corre-sponding curve becomes similar to that for Sweden, multiplied by 50or so. The distance to the saturation under the (B)-mode is uncertainwhen the phase-two curve is almost linear. The switch from (A) to(AB) generally delays the saturation 1.5-2 times, assuming that thecurve remains in the forecast cone.

We cannot of course evaluate the actual cost of ”hard” measures.Their economic and societal impact is huge. However long periods ofessentially constant but high daily numbers of new infections involvesignificant risks; for instance, the detection and isolation of new clustersof disease obviously becomes more difficult.

The general theory remains essentially unchanged from its first postedvariant (April 13). However only now all its main features, includingthe least aggressive mode (B), the transitional mode (AB), and the us-age of both Bessel-type solutions of our system of differential equationsare confirmed to occur practically. The earlier variants were focused onthe dominating Bessel-type solution; it was found that both solutionsare really needed to model the spread in Italy, Germany, Japan andother countries. Also, the log(t)–saturation of solutions of type (B),predicted theoretically, appeared the key during the second phase .

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Classical theory. The simplest equation for the spread of communica-ble diseases results in exponential growth of the number of infections,which is mostly applicable to the initial stages of epidemics. We focuson the middle stages, where the growth is no greater than some powerfunctions in time, which requires different equations.

The equally classical logistic equations for the spread, as well as theSIR and SID models, assume that the number of infections is compa-rable with the whole population, which we do not impose. The spreadof this kind was not really the case with the epidemics we faced duringthe last 100 years, even when the numbers of infections were big. Thisis obviously due to better disease control worldwide. The herd immu-nity was and is present, but the epidemics were generally significantlyreduced or terminated before it had full effect.

The reality now is the power-type growth of epidemics after a shortperiods of exponential growth (if any), which then changes to suffi-ciently long periods of essentially linear growth. These stages can beclearly seen with Covid-19 . The saturation process does occur, but thishappens well before the number of infected people becomes comparablewith the whole population.

Anyway, the power law of epidemics must be the starting point ofany analysis if we want our mathematical models to be up to date, thechallenges with Covid-19 included. Our approach is entirely based onthis assumption. The absence of satisfactory mathematical modeling ofCovid-19 based on SIR-SID-type equations is hardly surprising. Andour formulas are not only accurate: they are simple, algebraic anddepend on very few parameters.

Beyond epidemics. There is a strong connection of our approach withbehavioral science, including behavioral finance. We actually closelyfollow paper [Ch] devoted to momentum risk-taking with momentuminvesting as the main application. For instance, the aggressive man-agement of type (A) from Section 2 is almost a direct counterpart ofprofit taking from Section 2.6 of this paper. The measures of type(B) are parallel to the investing regimes discussed in Section 2.4. Thecorresponding differential equations are essentially the same. The keylink to financial mathematics is that the price function from [Ch] is acounterpart of the protection function in the present paper.

Paper [Ch] can be considered as part of a program toward generalpurpose artificial intelligence, the most difficult and ambitious amongall AI–related research directions. Momentum risk-taking is a verygeneral concept. We even argue in [Ch] that the mathematical mecha-nisms we propose can be present in neural processes in our brain, whichwould explain their universality. See Section 1.4, including an instruc-tional example of self-driving cars. Generally, this is about the ways


our brain processes information and manages risks real-time, which iscertainly applicable to behavioral aspects of epidemics.

2. Two kinds of management. There is a long history and manyaspects of mathematical modeling epidemic spread; see e.g. [He] for areview. The present paper seems the first one where the momentummanagement of the epidemic is considered the main factor determiningits spread. The parameter c, the initial transmission coefficient, reflectsthe strength of the virus and the ”normal” number of contacts in theinfected areas. However, the intensity a of the measures, the secondparameter of the theory, and their type, hard or soft, are up to us. Thetwo basic approaches we consider are essentially as follows:

(A) aggressive enforcement of the measures of immediate impactreacting to the current absolute numbers of infections and equally ag-gressive reduction of these measures when these numbers decrease;

(B) a more balanced and more defensive approach when mathemat-ically we react to the average numbers of infections to date and theemployed measures are mostly of indirect and palliative nature.

Hard and soft measures. To clarify, an action of type (A) can be aprompt detection and isolation of infected people and those of high riskto be infected, closing factories and other places where the spread isthe most likely and so on. The measures like wearing protective masks,recommended self-isolation, restrictions on the size of events and socialdistancing are typical for (B). This distinction heavily impact thedifferential equations we propose. However the main difference betweenthe modes, (A) vs. (B), is the way the number of infections is treated:the absolute number of infections is the trigger for (A), whereas theaverage number of infections to date is what we monitor under (B).

Generally, the (A)–type approach provides the fastest possible and”hard” response to the changes with the number of infections, whereaswe somewhat postpone with our actions until the averages reach properlevels under (B), and the measures we implement are ”softer”. Mathe-matically, the latter way is slower but better protected against stochas-tic fluctuations, but (B) alone will significantly delay the terminationof the epidemic, which we justify mathematically within our approach.

There is almost a direct analogy with investing , especially with profittaking. Practically, a trader either directly uses the price targets orprefers to rely on the so-called technical analysis , based on some ofchart averages. Let us emphasize that by ”actions”, we means notonly those by the authorities in charge of the epidemic. Our own indi-vidual ways of reacting to epidemic figures are essentially no different.Epidemiology has very strong roots in behavioral science, psychology,sociology, mass and collective behavior; see e.g. [St].

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Focus on ”periodicity”. Both modes, (A) and (B), are momentum,entirely based on the current trend, but they are quite different qual-itatively and quantitatively. As we will see, both provide essentiallythe same kind of power growth of the spread in the beginning, approx-imately tc in terms of time t for the initial transmission coefficient c.The main difference is that the solutions of the differential equations oftype (A) are quasi-periodic , namely asymptotically periodic functionsin t multiplied by power functions. They can be used only when theyincrease; we model the total number of infections. For (B), we havelog(t)-quasi-periodicity, but it can reveal itself only at late stages. Thistype of periodicity is not related to the periodicity of the models basedon seasonal factors, various delays and other mechanisms of this na-ture; see e.g. [HL]. Our periodicity and the corresponding saturationare entirely due to active momentum management.

The main objective of any managing epidemics is to end them. How-ever the termination based on such (any) aggressive ”interference” canlead to the recurrence of the epidemic too: the ”quasi-periodicity” maybegin working ”against us”. This can be avoided only if we continueto stick to the prevention measures as much as possible even when thenumber of new infections goes down significantly. Reducing them toomuch on the first signs of improvement is a way to the recurrence ofthe epidemic, which we see mathematically within our approach.

3. Power law of epidemics. Here and below we will assume that thenumber of people perceptive to the virus is unlimited, i.e. we do nottake into consideration in this paper any kind of saturation when thenumber of infected people is comparable with the whole population.Accordingly, herd immunity and similar factors are not considered.Also, we disregard the average duration of the disease and that for thequarantine periods imposed; the total number of infections regardlessof the output is what we are going to model, which is commonly used.

For any choice of units, days are the most common, let Un be thenumber of infected individuals at the moments n = 0, 1, 2 . . .. Thesimplest equation of the epidemic spread and its solution are:

Un−Un−1 =σ Un−1 for n = 1, 2, . . . , and Un = C(1+σ)n(1)

for some constant C and the intensity σ. Here σ is the transmissionrate , which is the number of infections transmitted by an average in-fected individual during 1 day (or other time-unit), assuming that the”pool” of non-infected perceptive people is unlimited. So this modelresults in exponential growth of the number of infections, which canbe practically present only at relatively early stages of epidemics, es-pecially with the epidemics we faced during the last 100 years.


The main factor ignored in (1) is that Un −Un−1 is actually propor-tional to Un−1−Un−p, where p is the period when the infected peoplespread the virus in the most intensive way. We are going to replacethis difference by Un−1/(n−1) up to some coefficient of proportionality.This is of course correct if Un grows essentially linearly, but we claimthat this is more general, attributing it to the reduction of our contactsover time during the epidemics. Let us try to clarify this.

Transmission rate sociologically. If the total number of cases growthslinearly, i.e. when the number of new daily infections is constant, thisis not really sufficient for us psychologically to change our behavior.However, we react strongly if the growth is parabolic or so, of course notonly with epidemics. The greater the exponent the stronger response.

Practically Un−1/(n−1) seems almost a perfect way to see the trend .Generally, (Un−1 − Un−2) gives this, but it may be not what we really”calculate” in this and similar decision-making situations.

First of all, such difference or the corresponding derivatives are poorlyprotected against the random fluctuations of Un, which can be extreme.Second, momentum decision-making is a very much intuitive process;storing and processing the prior U -numbers in our brain is more in-volved than ”keeping” Un−1/(n−1). Our brain can perfectly processthe input like ”yesterday number is Un−1 and this took (n−1) days”,though the mechanisms here are complicated. Third, the actions ofinfected people or those who think they can be infected become lesschaotic over time, which reduces the transmission of the virus. Also,the medical help they receive is of course better at later stages.

Some biological aspects. The viral fitness is an obvious component ofσ. Its diminishing over time can be expected, but this is involved. Thiscan happen because of the virus replication errors, especially typicalfor RNA viruses, which are of highly variable and adaptable nature.The RNA viruses, Covid-19 included, replicate with fidelity that isclose to error catastrophe. See e.g. [CJLP, Co] for some review, per-spectives and interesting predictions. Such matters are well beyondthis paper, but one biological aspect must be mentioned, concerningthe asymptomatic cases.

The viruses mutate at very high rates. They can ”soften” over timeto better coexist with the hosts, though fast and efficient spread is ofcourse the ”prime objective” of any virus. Such softening can resultin an increase of asymptomatic cases, difficult to detect. So this cancontribute to diminishing σ we observe, though this is not becauseof the decrease of the spread of the disease. We model the available(posted) numbers Un, which mostly reflect the symptomatic cases.

To summarize, it is not impossible that the replication errors and”softening the virus” may result in diminishing σ at later stages of the

8 IVAN CHEREDNIK

epidemic, but we think that the reduction of the contacts of infectedpeople with the others dominates here, which is directly linked to be-havioral science, sociology and psychology. This is different for animalepidemics, though there are some behavioral aspects there too.

Master equation revisited. We conclude that a reasonable equationdescribing the middle stages of epidemics and the corresponding as-ymptotic growth of its solutions are as follows:

Un − Un−1 =σ

n− 1Un−1, n = 1, 2, . . . , Un ≈ Cnσ(2)

for some constant C. When σ = 1, Un is exactly Cn. As we outlined,the main mechanism for this power law is likely to be of behavioralnature, but it can include biological factors too. Needless to say thatpower laws are fundamental everywhere in natural sciences.

Power-law models for epidemics and the spread of infectious are notreally something new; see e.g. [MH]. However, our approach is verydifferent; it describes our response to the epidemic information avail-able. Actually it is not that important in this paper what are the exactreasons for dividing σ by the time, though we suggest some underlyingprinciples. What is the key for us is that the switch from (1) to (2) isa mathematical necessity; the exponential growth is unsustainable.

The power growth can be of course unsustainable long-term too,but the quadratic and then linear growth can last during epidemics,including Covid-19 , where both types are clearly present. Let us men-tion that there are some indications that the ratio between all infectedpeople and those ”seriously or and critically ill” during Covid-19 has atendency to be proportional to t during the middle periods, but this isfar from solid. Thus the ”σ/(n−1)” can be due to some microbiologicalprocesses: the virus is probably getting ”softer” inversely proportion-ally to time, but this is very preliminary and qualitative.

Anyway, our focus is on the middle periods, when the precise man-agement is of key importance. The management depends mostly onprojections and considerations of global nature in the beginning; andit is relatively straightforward : ”restrict what can be restricted”.

4. Hard measures. In the realm of differential equations, our start-ing equation (2) becomes as follows:

dU(t)

dt= c

U(t)

t, where t is time and c replaces σ.(3)

We apply Taylor formula to Un − Un−1, so c is ”essentially” σ; we willuse the most relevant one below depending on the context, mostly c.

The aim is to reduce the ”free” spread of the epidemic modeledin (3) by aggressive imposing protection measures. Accordingly, the


protection function P (t) is the output of a measure or measures. Itis a counterpart of the price function in [Ch]. Essentially we havetwo types of measures: (a) isolating infected people, and (b) generaldiminishing the number of contacts. We will call them ”hard” and”soft”. They will correspond to the management modes (A) and (B);see below and Section 2. Let us begin with the ”hard” ones.

Protection function. The main ”hard” measure is testing and thendetection of infected individuals, followed by their isolation. This ishard by any standards, especially if there are many asymptomatic casesand the treatment is unknown as with Covid-19.

Generally, P (t) is the key numerical characteristic of the output ofa given measure; its derivative is then the productivity. For ”detection& isolation”, we are going to use the following natural definition:

P (t) = the total number of infected individuals detected till t.

The primary measure here is testing ; the number T (t) of tests till tis what we can really implement and control. The detection of infectedpeople is its main purpose, but the number of tests is obviously notdirectly related to the number of detections, i.e. to the number ofpositive tests . The efficiency of testing requires solid priorities, focuson the groups with main risks, and solving quite a few problems.

Even simple mentioning problems with testing, detection and isola-tion is well beyond our paper. However, numerically we can use thefollowing. During the epidemics, essentially during the stages of lineargrowth, which are the key for us, the number of positive tests can bemostly assumed a stable fraction of the total number of tests. WhenP (t) = γT (t) for some γ, finding γ experimentally is not a problem.

The middle stages of epidemics mostly consist of such linear periods,though it depends. For instance, preliminary data indicate that theinitial c for Covid-19 is about 2, i.e. the spread is essentially quadraticbefore the active management begin. The measures we discus reducethe growth to linear almost everywhere, which perfectly matches ourformula tc/2 under type (A) management or that of type (B) for suffi-ciently large intensity ”a”. As we have already mentioned, ”momen-tum managing” is mainly present during the middle stages; not much”mathematics” is used in the beginning or at the end of epidemics.

Thus, we are going to assume that the effect of isolating infectedpeople is approximately linear. I.e. an isolated infected individual willnot transmit the disease to (t − t•) people till the moment t (now)after the isolation at t• < t. It can be C(t − t•), but we make C = 1;this is a normalization matter (see below). This kind of ”linearization”is very common when composing differential equations. We use thatwe are ”near” the stage of linear growth of epidemics, though this

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does not mean of course that the solutions of our equations will growlinearly. Even if c > 1, this assumption makes sense : the number of”preventions” due to one isolation at t• can be smaller than (t− t•)c.

Let the isolations occur at the moments {ti} from some t0 till t(now). Then these individuals would infect

∑i(t− ti), which sum must

be therefore subtracted from U(t). Its t–derivative is exactly P (t).

Other P -functions can be used here, but this particular one has avery important feature: it does not depend on the moments of timewhen the infected individuals were detected. If P (t) depends on {ti},the required mathematical tools can be much more involved.

The relation resulting from our analysis is as follows:

dU(t)/dt = c U(t)/t− P (t) for type (A).(4)

More systematically, the minimum of (t− ti) and the average isolationperiod must be taken instead of (t − ti), but we disregard the averageduration of the disease and quarantine terms throughout this paper.

With closing factories, we count the numbers of workers there inthe definition of P (t). Of course only a fraction of the workers canbe infected when a factory is closed, but the assumption is that if itcontinues to operate, then almost everyone there can be expected tobecome infected. This is an obvious simplification. The exact effectof closing (and then reopening) factories and other places where fastspread of the disease can be expected is almost impossible to estimate.

So simplifying assumptions are necessary here; there are too manyfactors involved with any employed measures, and they are mostly ofstochastic nature. In (i), (ii) below, we will address this as follows: in-stead of using a specific P (t) we will ”postulate” its natural properties.

We emphasize that the measures of type (A) impact U(t) in complexways: isolating one individual prevents a ramified sequence of trans-missions. The ”soft” measures of type (B), to be considered next, aresimpler: they result in a kind of reduction of the c–coefficient.

5. Soft measures. These measures are different from ”hard” ones.Wearing protective masks is a key preventive measure of this kind.Social distancing considered mathematically is of similar type. Now:

P (t) = the number of infected people who began wearing the masksbefore t multiplied by the efficiency of the mask and the c–coefficient.

Using the masks for the whole population, infected or not, ”simply”changes c ; this is not ”momentum management”. Let V (t) be thenumber of infected people wearing the masks. Then V (t) ≤ U(t) and


dU(t)

dt= c

U(t)−V (t)

t+ c′

V (t)

t= c

U(t)

t− (c− c′) V (t)

t(5)

for c−c′ = c(1−c′/c) = cκ, where κ is the mask efficiency. For instance,κ = 1 if c′ = 0, i.e. if the mask is fully efficient for infected people whowear it. The same consideration works for social distancing , with κbeing the efficiency of the corresponding distance.

An instructional example is when we assume that the fraction ofthose wearing the masks among all infected people is fixed. If V (t) =νU(t) for some 0 ≤ ν ≤ 1, then we have:

dU(t)/dt = c (U(t)− νU(t))/t+ c′ν U(t)/t = (c− ν(c− c′))U(t)/t.

I.e. this measure results in fact in a recalculation of the c–coefficientunder the proportionality assumption.

We see that the output of ”soft” measures is heavily based on prob-abilities. However, we only need the following: the greater intensity ofany measure the greater the reduction of new infections. So we practi-cally control the efficiency of a measure by increasing or decreasing itsintensity , even without knowing the exact mechanisms of its impact.This is what we are going to formalize now.

General approach. From now on, the intensity of a measure or severalof them will be the main control parameter. As we already discussed,the greater the intensity, the greater the number of isolated infectedindividuals in (A) or the number of infected people who use the masksin (B). If this dependence is of linear type, which can be expected,the corresponding coefficient of proportionality is sufficient to know.Respectively, the exact definition of the protection function P (t) isnot really needed to compose the corresponding differential equations.What we really need is as follows:

(i) the usage of P reduces dU(t)/dt, possibly with some coefficientof proportionality, by P (t)/t for mode (B), the average of P taken fromt = 0, or directly by P (t) under the most aggressive mode (A);

(ii) the derivative dP (t)/dt, the intensity, is proportional to U(t)/tunder (B), the average number of infections from t = 0, or directly toU(t) in the most aggressive variant, which is (A) considered above.

Item (i) has been already discussed. Let us clarify (ii), which deter-mines the intensity of the measures dP (t)/dt we employ. It must beproportional to the current number of infections for (A) or its average,U(t)/t, for (B). These are the most natural choices, practically andtheoretically. Indeed, the effect of the current number of infections toour actions can be either direct or via some averages. If the averages

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are used, then actually there are not many mathematical choices. Thekey objective is of course: to end the epidemic. A mathematical an-swer is: this is granted under mode (A) or the transitional mode (AB)defined as follows.

Let us employ ”hard” measures as in (A), however following lessaggressive management formula for dP (t)/dt from (B). I.e. this is acombination of (i) for (A) with (ii) for (B), some transitional mode.As we will see, the epidemic will end then, but the time to ”saturation”can be significantly longer than under (A).

6. Type (B) management. Let t0 be the starting point of the man-agement; so we can assume that P (t0) = 0. The following normal-

ization is somewhat convenient: u(t)def== U(t)/U(t0) and p(t)

def==

P (t)/U(t0), i.e. u(t0) = 1 and p(t0) = 0. Recall that U(t) and P (t) arethe number of infections at the moment t and the corresponding valueof the P -function. Here P is naturally the total of all measures in theconsidered mode, the sum of the corresponding P .

Recall that mode (A) has no protection against stochastic fluctua-tions and is significantly more aggressive than (B). Philosophically,the smaller interference in natural processes the better: the results willlast longer. But with epidemics, we cannot afford waiting too long.

Type (B) equations. As it was stated in Section 4, we couple (i) oftype (B), which is relation (5), with (ii) for the same mode. I.e. thederivative of u(t) will be ”adjusted” by −p(t)/t and, correspondingly,the rate of change of p will be taken proportional to u(t)/t. This meansthat the impact of u(t) to p(t) and vice versa goes through the averages ;i.e. the response to u(t) is not ”immediate” as in (A).

Note that the exact −p(t)/t or, later, −p(t) in the equation fordu(t)/dt is a matter of normalization. A possible proportionality coef-ficient here, if relevant, can and will be moved to the second equation.

The equations under (B) become as follows:

du(t)

dt= c

u(t)

t− p(t)

t,(6)

dp(t)

dt=

a

tu(t).(7)

Here c is the original intensity of the spread, the transmission coeffi-cient, and a is the intensity of the protection measure(s). This systemcan be readily integrated. Substituting u(t) = tr, the roots of the

characteristic equation are r1,2 = c/2 ±√D, where D = c2/4 − a.


Accordingly, when D 6= 0, t > 0:

u(t) = C1tr1 + C2t

r2 if D > 0 for constants C1, C2 , and(8)

u(t) = tc2 (C1 sin(

√−D log(t))+C2 cos(

√−D log(t))) if D < 0,(9)

where the constants are adjusted to ensure that p(t0)=0 and u(t0)=1.Note that a proper branch of log must be chosen for t near zero.

So for a < c2/4, the initial tc will be reduced up to tr1 with c/2 <r1 < c. When any a > c2/4, the power growth is always of type tc/2.

The periodicity with respect to log(t) has the period 2π/(√a−c2/4),

which will result in the ”saturation”, which is at the first t such thatdu(t)/dt = 0. Comparing with (A) or (B), this is generally for large t.

Positivity of du/dt. The control parameter a, the intensity, is actuallynot arbitrary. To see this let us invoke v(t) = V (t)/U(t0), where V (t)is the number of infected people wearing the protective masks. It wasused in the definition of P : P (t) = κcV (t), where κ is the efficiency ofthe mask (from 0 to 1). We then have:

du(t)

dt= c

u(t)

t− κcv(t)

t,dv(t)

dt=

a

κc

u(t)

t.(10)

Here v(t) ≤ u(t) by the definition. This inequality generally providesno restriction on the derivatives. However, if v(t) is essentially pro-portional to u(t), there are some consequences. Let us assume thatv(t) ∼ γu(t). Then we obtain the relation: a = c2κγ(1 − κγ). Ac-

cordingly, the maximal value of a is amax = c2

4in this case, which is at

κ = 12γ

provided that 1≥γ≥ 12. This gives D = 0 for γ = 1, i.e. this is

the case when all wear the masks of efficiency 1/2; then u(t) = Ctc/2,which is obvious from to the definition of κ.

Generally, du(t)/dt ≥ c(1 − κ)u(t)/t, and du(t)/dt ≥ 0 for u(t) =u(t)tc(κ−1). Thus the solutions u(t) from (8) and (9) qualitatively growfaster than tc(1−κ). The positivity of du(t)/dt is obvious a priori.

Due to the log(t)–periodicity (if present), the first maxima of u(t)are (B)–type saturation points. We must stop using u(t) after this.This log(t)-saturation can be seen and is an important factor duringthe second phases of our 2-phase model. The hard measures are sig-nificantly reduced then and this kind of periodicity remains the mainmathematical source of the saturation. See Section 11.

7. Type (A) management. The most ”aggressive” model of mo-mentum management is of type (A). Mathematically, we replace the

14 IVAN CHEREDNIK

average p(t)/t in (6) by p(t), and u(t)/t by u(t) in (7). Actually, thefirst change affects the solutions greater than the second. One has:

du(t)

dt= c

u(t)

t− p(t),(11)

dp(t)

dt= a u(t).(12)

Solving system (11)&(12) goes as follows:

t2d2p

dt2− ctdp

dt+ et2p = 0 = t2

d2u

dt2− ctdu

dt+ et2u+ cu,(13)

u=A1u1+A2u

2, u1,2(t)= t1+c2 Jα1,2(

√at) for α1,2 =±c−1

2.(14)

Here the parameters a, c are assumed generic, A1,2 are underminedconstants, and we use the Bessel functions of the first kind:

Jα(x) =∞∑m=0

(−1)m(x/2)2m+α

m!Γ(m+ α + 1).

See [Wa] (Ch.3, S 3.1). We will also need the asymptotic formula fromS 7.21 there:

Jα(x) ∼√

2

πxcos(x− πα

2− π

4) for x >> α2 − 1/4.

It gives that u1,2(t) are approximately C tc/2 cos(√at − φ1,2) for some

constant C and φ1,2 = ± c−12π + π

4.

Quasi-periodicity. We conclude that for sufficiently big t, the functionu(t) is basically:

u(t) ≈ tc/2(A sin(

√at+ πc/2) +B cos(

√at− πc/2)

),(15)

for some constants A,B. So it is quasi-periodic , which means thatthe periodicity is up to a power function and only asymptotically; theasymptotic t-period is 2π√

a.

In our setting, u(t) always grows, so du(t)/dt ≥ 0. The techni-cal end of an epidemic is when u(t) reaches its first maximum; π

2√a

is a reasonable estimate, but not too exact. For instance, u(t) =tc/2+1/2J(c−1)/2(

√at) for c = 2.2, a = 1/5 reaches its first maximum

at about t = 4.85, not at π2√a

= 3.51; see Figure 1.

Transitional mode. Presumably it is applicable to model epidemicspread in the places where hard measures are applied cautiously (if


any) and at later stages of epidemics, when hard measures are reducedor even abandoned. This mode is transitional mode between (A) and(B): one replaces (12) by dp(t)/dt = au(t)/t, but at the same timetakes the equation for du(t)/dt from mode (B), which is (6):

dw(t)

dt= c

w(t)

t− p(t), dp(t)

dt= b

w(t)

t,(16)

where we replaced u(t), a by w(t), b; the c remains unchanged. It wascalled transitional (AB)–mode at the end of Section 4.

This mode basically means that we become cautious with our re-sponse to u(t) by choosing the ”control formula” for dp(t)/dt from(B), better protected against fluctuations. However the measures weare going to employ are still of ”hard” type, as under mode (A). Thismakes sense practically; let us see what this gives theoretically.

This system is still integrable in terms of Bessel functions; see for-mulas (2.20), (2.21) in [Ch]. The leading fundamental solution is

w1(t) = t(c+1)/2Jc−1(2√bt), where c > 1.(17)

The second one is w2(t) = t(c+1)/2J1−c(2√bt). One has:

u1(t) ≈ tc(√a/2)(c−1)/2

Γ((c− 1)/2), w1(t) ≈ tc

(√b/2)c−1

Γ(c− 1),(18)

when t is sufficiently close to zero. Practically, they ”almost” coincideupon the coefficient of proportionality from (18) in sufficiently largeareas. This is important for the ”forecasting tool” below.

The quasi-periodicity will be now with respect to t1/2 and the corre-sponding asymptotic period will be π/

√a. I.e. the process of reaching

the saturation is ”slower” than (A), but still significantly faster than(B), where the ”mathematical periodicity” (if any!) is in terms of log(t)in (9). This perfectly matches the qualitative description of (AB) as atransition from (B) to (A).

Actually we have a family (AB)µ for −1 ≤ µ < 1 of such modes,described by the system

dw(t)

dt= c

w(t)

t− p(t)/tµ, dp(t)

dt= b

w(t)

t.(19)

Here the term p(t)/tµ means that the impact of one isolation of aninfected individuals grows non-linearly over time, which makes sense.Assuming that c > 2, the dominant solution is

16 IVAN CHEREDNIK

w(t) = tc+1−µ

2 J c1−µ−1

( 2√b

1− µt(1−µ)/2

), w ∼ tc for t ≈ 0.(20)

The second one is for 1− c1−µ ; we follow [Ch]. When µ = −1, the

argument of J is const·t , and we arrive at some counterpart of u(t).

8. Finding u-curves. In the range till April 14 the graphs of oursolutions u(t) presented in this section match surprisingly well the to-tal number of infections for these countries starting when these num-bers begin to grow ”significantly”. These moments are approximatelyaround March 16 for the USA and UK, March 13 for Switzerland, andMarch 7 for Austria.

Let us mention that https://ourworldindata.org/coronavirusis mostly used for the data, updated at 11:30 London time. We alsouse "worldometers" We will always take x = days/10.

The USA data. The scaling coefficient 1.7 in Figure 1 is adjustedto match Figure 2 for the United States. For the USA, we set y =infections/100K, and take March 17 the beginning of the period of”significant growth”. Also, c = 2.2, a = 0.2 appeared perfect.

Concerning c, the transmission rate, it reflects of course the virustransmission strength, but it also depends very significantly on the waypeople respond to the current numbers of infections, which includes allthe related information provided by the authorities in charge and massmedia. It is basically around c = 2 for Covid-19, but there are somevariations; say, c = 2.4 appeared more appropriate mathematically forUK and Austria, and c = 2.8 for ”the world”.

The parameter a, the intensity of type (A) measures, is 0.2 for theUSA and UK, and 1/3 for Austria. The scaling coefficient, 1.7 forthe USA, is adjusted to make the y as the total number of infectionsdivided by a proper power of 10.

The red dots in all figures show the corresponding actual total num-bers of infections. They perfectly match u(t) = 1.7t1.6J0.6(t

√0.2) in

Figure 1. Accordingly, if this curves continues, i.e. the intensity ofhard measures remains essentially unchanged, the number of cases inthe USA would reach its preliminary saturation at ttop = 4.85 (48.5days from 03/17 to May 5) with utop = 10.3482, i.e. with 1034820infections (it was 609516 at 04/15). This did not happen, which willbe address below.


1 2 3 4

2

4

6

8

10

Figure 1. u(t)=1.7 t(c+1)/2J(c−1)/2(√at) for c=2.2, a=0.2

Figure 2. Covid-19 in the United States

Of course the saturation t = ttop is a technical one, not the end of thespread of infection even if it is reached. The data from South Koreaand quite a few other countries demonstrate that some linear growth ofthe total number of cases can be expected around and after ttop. Thisis due to reducing the ”hard” measures, new clusters of infection canemerge, and of course no country is isolated from infections from otherplaces.

With these reservations, the graphs we provide and similar ones weconsidered for Covid-19 demonstrate that Bessel functions model verywell the period of ”aggressive management”.

18 IVAN CHEREDNIK

The mode (A), resulting in Bessel functions, is of course not the onlyforce. It is always combined with mode (B), and there are some otherphases, before and after. We thoroughly discussed this above, and seeFigure 5 below.

Assuming that Bessel-type functions model well the period of theintensive growth, one can try to ”capture” the parameters c, a evenearly stages, before or near the turning point of the epidemic. If a, care known, the first local maximum of the corresponding Bessel func-tion times tc/2+1/2 gives an estimate for the possible ”saturation”, thetechnical end of the epidemic. This is of course under the steady (A)-management, which appeared the case in sufficiently many countries.

In the data we provide, a, the intensity (A)–measures, is 0.2 forthe USA, UK, Italy and the Netherlands. It is 0.3 − 0.35 for Israel,Austria, Japan, Germany; 0.1 for Sweden. The parameter c, the initialtransmission rate, is 2.2 for the USA, 2.4 for UK, Austria, Sweden, theNetherlands, 2.6 for Italy, Germany, Japan. Here c can be clearly seenat early stages; a depends on the management, but it is stable anduniform during the middle stages in quite a few countries.

Covid-19 in UK. The superb match of red dots and our u(t) for theUSA can be because this is an average over 50 states. Let us considernow UK for the period from March 16 till April 15.

1 2 3 4 5

5

10

15

Figure 3. u(t)=2.2 t(c+1)/2J(c−1)/2(√at) for c=2.4, a=0.2

Then c = 2.4, a = 0.2 work fine, and the scaling coefficient is 2.2.The total number of cases will be now divided by 10K, not by 100K asfor the USA. The increase of c to c = 2.4 (very) qualitatively indicatesthat the ”response” of the population to this kind of thread was slowerin UK than in the USA during the considered period; by 9%, not much.


Recall that if the spread of disease is not actively managed, thenthe growth ≈ tc can be expected, so c reflects the strength of the virusand how we, especially the infected people and those who think thatthey are infected, react to the numbers of infections before the activemanagement begins. Posting these numbers, discussing them by theauthorities in charge and in the media is of course some kind of man-agement, but passive. By ”active”, we mean the (A), (B)–measures.

The estimate for the ”saturation moment” was 5.17, i.e about 51days after March 16, somewhere around May 6 with the correspondingnumber of infections about 170000, assuming that the ”hard” measureswould continue as before April 15. The latter did not materialize.

Austria: 3/07-4/15. This is the case of almost a complete ”cycle”.There is some switch to a linear growth closer to and after the ”satura-tion”. common in all countries that reached the saturation. Modelingthis whole period by Bessel function appeared quite doable, so themanagement was really of type (A) and steady.

1 2 3 4

5

10

15

Figure 4. u(t)=3 t(c+1)/2J(c−1)/2(√at) for c=2.4, a=1/3

Here we start with March 07; Austria began earlier with the protec-tion measures than UK and the USA. The parameters a = 0.33, c = 2.4and the scaling coefficient 3 appeared the best for Austria; we dividenow the number of infections by 1000, i.e. 14 in the graph correspondsto 14000 infections.

Reaching ttop, as we discussed, is not really the end. The epidemiccontinues after this moment. Moreover, the period before such ”sat-uration” is generally a union of segments corresponding to differentstages. Let us comment on it using Switzerland as an example.

20 IVAN CHEREDNIK

Switzerland: different stages. We focus on the period March 13- April12. As we have already discussed, (A) is supposed to dominate mostlyduring the final ”half” of this period toward ttop, i.e. around the turn-ing point and later. Since we do not discuss the combination of modes(A), (B) in one system of differential equations, practically: the mini-mum of the u–functions obtained for these two modes dominates andis supposed to be seen in real charts.

Recall that we consider only ”total cases”, the numbers of all de-tected infections till t, and begin at the moment when the power growthbegins, which was around March 12 for Switzerland in Figure 5. I.e.t = 0 is March 12. This moment is essentially when the active measuresbegin; t0 is when the first results of these measures can be clearly seen,around March 20. Practically, the initial (mostly) parabolic growthwill start switching to the linear one at t0.

The ”free” spread of epidemics of quadratic type can be clearly seenin quite a few countries at early stages of Covid-19; say, in Brazil,Mexico, Sweden, Switzerland are examples. Here we use that c ≈ 2.

Generally, mode (B) ensures a continuous transition from the ”free”growth u(t) ∼ tc to u(t) ∼ tc/2. This is by making a for (B) from a = 0to a = c2/4 ≈ 1. We assuming that the initial c is about 2, whichseems to be quite universal for Covid-19. After a = 1 the exponentwill stabilize: u(t) ∼ tc/2 for sufficiently large t, even if a continues toincrease. For a > 1, the u(t) given by (9) behaves somewhat differently;cos(·) and sin(·) contribute.

Total Cases

(Linear Scale)

30k

25k

20k

10k

0

• Cases

Quadratic

growth

Type B:

Linear

growthType A leads:

fnct

Initial period, some exp growth Type B transition:

from a=0 a=1

Exponential growth changes to quadratic,

when people begin to reduce the contacts

Intensity of type B measures (masks etc.)

increases from a=0; after a=1 the growth

remains essentially linear for whatever a

Type A measures (hard) begin

with B, but will dominate later

Figure 5. Covid-19 in Switzerland


Mode (A), aggressive detection and isolation of infected people andso on, begins somewhat later than (B). It takes time to increase thetesting capacities and solve related problems. However, eventually itwill be the main reason for u(t) to ”saturate”.

Let us briefly comment here on changing the starting point from theabsolute start of the epidemic to some t•. The equations will remainthe same, but c/t must be replaced by c•/(t − t•) for the current c•.There can be different phases of epidemics, so this can be necessary,and not only once. Such a split of the total investment period intointervals is very common in stock markets.

9. Comparing the modes. Under any type of momentum manage-ment, the most aggressive for (A) or the least aggressive for (B), asymp-totically the u–function, describing the total number of infections, isa power function, times cos(·) in terms of t or log(t) for (A) or (B)correspondingly. Assuming that ”free” u(t) is proportional to tc forsmall t, i.e. before the active management begins, this power functionis tc/2 for u(t) under (A) for any a, c or under (B) provided a > c2/4.See (11)&(12) and (6)&(7). If a < c2/4 for (B), the leading solution

becomes tc/2+√c2/4−a

, i.e. the exponent here can be from c/2 to c; thereis no log(t)–periodicity for such a.

Diminishing the exponent here is of course of importance. The orig-inal coefficient c will eventually drop without any management and”forcing” c to change to c/2 will bring us closer to the end of the epi-demic. This can be related to the following feature of Covid-19. Aftera relatively short periods of parabolic growth of the number of infec-tions, which corresponds to c ≈ 2, the growth becomes linear almosteverywhere and then it lasts. See Figure 5. This passage is fully incor-porated in our formula for u(t) in terms of Bessel functions. Withinmode (B), it can be interpreted as increasing the intensity a of softmeasures from a = 0 to c2/4.

Periodicity is the key. The main difference between (A) and (B) is thequasi-periodicity of the corresponding solutions of type (A); see (14).For type (B), the periodicity is with respect to log(t) when a > c2/4,which is a significantly slower process than for (A). Asymptotically,u(t) always becomes quasi-periodic under (A) for a 6= 0 due to (15) andthe epidemic can be expected to ”end” at about π

2√a, more precisely,

at ttop of the corresponding u(t). After ttop, the usage of u(t) mustbe stopped; u(t) cannot decrease. The log(t)–quasi-periodicity for (B)provides the second (final) saturation point during the second ”linear-type” phase of our 2-phase model; see Section 11.

22 IVAN CHEREDNIK

We note that the quasi-periodicity of our u–functions under momen-tum management of type (A) is not connected at all with the period-icity of epidemics associated with seasonal factors, biological reasons,or various delays; see e.g. [HL]. It entirely results from the activemanagement, which is a combination of the measures employed, theireffect, and our general response to the threat.

With this important reservation, the approximate reflection symme-try of du(t)/dt for u(t) = tc/2+1/2J(c−1)/2(

√at) in the range from t = 0

to ttop, corresponding to the first local maximum of u(t), can be inter-preted as a variant of Farr’s law of epidemics. Generally, the portionsof the corresponding graph before and after the turning point are sup-posed to be essentially symmetric to each other. This is not exactlytrue for du(t)/dt, but close enough. See e.g. Figure 1; the turning pointis at max{du(t)/dt}. This holds for w(t), describing the (AB)–mode,when the second ”half” becomes somewhat longer than the first. Thisis only within the first phase of our 2-phase model.

Main findings. The ”power law of epidemics” presented as a differen-tial (initially difference) equation is the starting point of our approachto momentum management. This is different from other power lawsfor infectious diseases; compare e.g. with [MH]. We deduce it fromthe response of individuals to the epidemic data. The switch to thepower growth after a very short period of exponential growth (if any)is mainly attributed to our response to the epidemic data, a variant ofgeneral ”momentum risk-taking”.

This alone is insufficient to model the spread; the main problemis to ”add” some mechanisms ending epidemics quickly and then pre-vent their possible recurrence. These are major challenges, biologically,psychologically, sociologically and mathematically. One can expect thisproblem to be well beyond the power law of epidemics , but we demon-strate that mathematically there is a path: Bessel functions appeareda natural link from the power law of epidemics to the ”saturation”.

We provide justifications of our differential equations; they are basedon various simplifications, but the match with real data appeared al-most perfect. The ”megaproblem” of understanding and managing epi-demics is very ramified and interdisciplinary; significant assumptionsare inevitable in any models. We model only the period when the totalnumber of infections is significantly smaller than the whole population,but this is the essence of our approach. We also disregard the recoveryprocess and the quarantine durations.

Making our justifications more ”rigorous” is a challenge, but we thinkthat the experimental confirmations of our two-phase solution are thekey by now. We are of course fully aware of the statistical nature ofthe problem. The usage of random processes, like Bessel processes, is


reasonable here to address the randomness around epidemics. Somecombination with the traditional SIR and SID models can be possibletoo, but our approach seems quite different so far.

The main outcome of our modeling and this paper is that the mea-sures of ”hard type”, like detecting infected people, followed their iso-lation, and closing the places where the spread is very likely, are thekey for ending an epidemic. This must be done at least during the ini-tial and middle stages of the epidemic. Moreover, such measures mustbe employed strictly proportionally to the current numbers u(t) of in-fections (not its derivative of any kind), which is the most aggressivemomentum way to react. We call this kind of management mode (A)in the paper, and see it in many countries, at least during the middlestages of the epidemic.

Then the point of the first maximum of the corresponding Besselfunction times tc/2+1/2 is a good estimate for the duration of the epi-demic, and the value at this point is an expected top number of infec-tions. Surprisingly, this formula works very well for almost the wholeperiod of extensive growth of the spread, assuming that ”hard” mea-sures of type (A) play the leading role. It seems a real discovery. Ifthe saturation is successfully reached at ttop, the second phase beginsaround this moment: the switch to mode (B). This is significantlymore challenging mathematically. It is heavily influenced by economic,political and other factors. So it is even more surprising, that it can bemodeled with high accuracy using our system of differential equationsof type (B). See our graphs for Israel, Italy, Germany, Japan, and theNetherlands in Section 11.

Mode (B) is the usage of the average u(t)/t here instead of u(t) and,especially, relying mostly on ”soft” measures, like wearing protectivemasks. When used alone, it can significantly slow down the process of”reaching zero”; this results from our model. Though a combinationof ”hard measures” with the ”slow response”, based on u(t)/t insteadof u(t), can be seen too, which is our mode (AB), especially at laterstages of the epidemic.

The best way to address the late stages appeared our ”two-phasemodel”. Since the hard measures are reduced or even abandoned, weswitch from mode (A) to (B), which can be seen in quite a few countriesthat reached the ”saturation”. There is a clear linear-type patternaround the ”technical saturation”, which is ttop, the point of maximumof our Bessel-type u(t). Then the total number of detected infectionsu(t) closely follows Ctc/2 cos(d log(t) for some constant C and d =√−D from (7). Here c is the exponent of the initial power growth of

24 IVAN CHEREDNIK

u(t); it remains the same for any modes, (A), (B) or (AB). As for d, itmust be adjusted numerically to match the actual number of total casesnear and after the saturation; C is simply the rescaling coefficient.

This formula describes the late stages with surprising accuracy inmany countries that reached the saturation; there are already quite afew. The usage of our two-phase model is naturally less relevant for thecountries that begin reducing or abandoning hard measures upon thefirst signs of the stabilization of new daily cases or implement theminsufficiently. This includes the countries where the total number ofinfections matched very well the corresponding Bessel-type u(t) duringthe middle stages. The USA and UK are such; the moments when they”left” the initial u–curves can be clearly detected. See Section 10.

If the period of linear growth becomes long cos(d log(t)) can be dif-ficult to find; here d is mathematically the key for the ”ultimate” sat-uration during the second phase. Such long periods of linear growthare typical for countries that do not employ the ”hard” measures sys-tematically, like Sweden. The growth can be even quadratic if hardmeasures are not employed systematically, or because their health-caresystems are not strong enough, or due to various other factors.

The risks of recurrence. This is a clear challenge for microbiology,epidemiology and population genetics; very much depends on the typeof virus. The ”natural” c–coefficient reflects both, the transmissionstrength of the virus and the ”normal” intensity of our contacts. Thesecond component will return back to normal after the epidemic isannounced finished. If the ”microbiological component” is essentiallythe same as it was before, the recurrence of the epidemic is likely afterthe protection measures are removed. The recurrence of the epidemicis a sort of ”cost” of our aggressive interference in its natural course.

The whole purpose of aggressive management, mode (A) in our ap-proach, is to keep the new daily infections at reasonably small levels,”to reach double digit numbers of new infections”. Then it becomespossible to promptly detect them and act correspondingly. The satura-tion in one place is of course not the end of the epidemic. New clustersof infections are quite likely to emerge, and the epidemic suppressed inone area can still continue in other places.

The detection and prompt isolation of infected people is obviouslymuch more difficult when the numbers of new daily cases are high,even if they are essentially constant. After the turning point of theepidemic, the protection measures are likely to begin being reduced oreven abandoned. This must be done cautiously to avoid the recurrenceof the epidemic, especially if the ”hard measures” are abandoned. Oth-erwise the epidemic may resume; mathematically, mode (B) alone isgenerally insufficient to terminate the epidemic.


These are real risks. The economic and other ”costs” of hard mea-sures are very high. It is understandably easy to ”forget” that the turn-ing point and the saturation were achieved reasonably quickly mostlydue to the hard measures imposed. The corresponding underlyingmathematical processes have a strong tendency to become periodic,which can ”play against us” as we approach the end of the epidemic.

Unless the herd immunity is reached or the virus lost its strength,”restarting” the epidemic with few infected individuals left after theprevious cycle is standard. Significant numbers of those recovered andthe asymptomatic cases provide some protection, but it can be insuffi-cient. For the next cycle(s), our better preparedness can be expected,which will diminish c. At least, the soft and hard measures can berestarted without any delays if necessary.

Finally, let us emphasize that our differential equations describe theoptimal momentum response aimed at diminishing the new infectionsto zero in the following sense. The main our assumption is that peo-ple and the authorities in charge constantly monitor the number ofinfections and respond to it in the most ”aggressive” way.

This is actually similar to the ways stock markets work. Professionaltraders simply cannot afford not to react to any news and any change ofthe stock prices, even if they seem random, temporary or insignificant.Indeed, any particular event or a change of the share-price can be thebeginning of a new trend. Applying this to managing epidemics, weare supposed to closely follow the data. The ”flexibility” here is theusage of some average numbers as the triggers, to ”protect” yourselfagainst random fluctuations. This is exactly the ”defensive” approachfrom (B) or (AB) versus that from mode (A).

The table: (A) vs. (B). For convenience of the readers, let us providea basic table presenting the (A)-mode and the (B)-mode with somesimplifications: Figure 6. Recall that c is the initial transmission rate,a the control parameter, which is the intensity of the management, κthe efficiency of protective masks.

By ”Detections”, we mean P (t): the total number of detected andisolated infected individuals till t. Practically, if the number of in-fections doubles, then the (A)–type response is to double the rate ofincrease of tests (not just the number of tests). Recall that we do notsubtract the closed cases and disregard similar factors.

Accordingly, by ”Masks used”, we mean the total number of infectedpeople who began using the masks before t. If the number of infectionsdoubles, than the total number of masks must be doubled too under(B), not the rate of change as for (A). The same intensity of applyingthe measures as for (B) is for mode (AB), but now to be used forthe ”hard” measures like testing & detection. This will provide the

26 IVAN CHEREDNIK

U(t) : total #

of all infections

till moment “t” CONTROL GROWTH ENDING

Natural Course:

no active (gov)

management

None, though

“c” diminishes

over time “t ”

Due to virus

mutations, … ,

herd immunity

Under type A:

the detection

and isolation

Under type B:

masks, social

distancing etc.

No fast ending

due to log(t)-

“periodicity”

Firstly,

Approximately

after a)

for intensity a

cos(q log(t))

and

~ / cos( )

cos( log )

= 4

t >0

Figure 6. Two types of momentum management

periodicity with respect to√t, which is not too different practically

from the t–periodicity for (A). See Sections 4 and 5.

10. Forecast cones. Our key finding is that the total numbers ofdetected infections can be described with a high accuracy by u(t) =C tc/2+1/2Jc/2−1/2(

√at), as it was demonstrated in Figure 1 and many

others. However it models mostly the initial and middle stages of theperiod of intensive growth of the spread, when the hard measures arecoupled with the most aggressive response to the current number of to-tal cases. It seems essentially sufficient for forecasting if the (A)–modeis employed all the way until the number of new detected infectionsdrop almost to zero. Then ttop, the first zero of du(t)/dt, is a rea-sonable estimate for the ”technical end of epidemics”. South Korea,Austria, Israel, Italy, Germany, Japan, the Netherlands and quite a fewother countries did exactly this.

Some linear growth is common after ttop, to be exact Ctc/2 cos(d log(t))for some C, d. The fluctuations, new clusters of infection and so onare quite possible during such ”linear periods”. However ttop, if it is


reached, can be considered as some saturation moment, mathematicallyand practically,

To examine the usage of type (A) curves for forecasting we focused onthe USA, UK, and Israel, where the initial data, the red dots matchedwell the corresponding Bessel-type functions. Without changing theparameters a, c, C, we compare them with the test periods; the blackdots were added after the parameters of u(t) were fixed. The expectedttop appeared good for Israel. With the USA and UK, the match withu–curves was good enough for sufficiently long periods, but the curvesin both countries remained essentially linear significantly longer thanfor u(t). Even now there no clear trend toward the mathematicallyexpected saturation.

We note that the parameters a, c were the same for UK and theNetherlands, and not too different for the USA. Generally, mode (A)works well for many countries, however not for these particular two.Actually, the graphs show the moments of the switch from mode (A)to what looked like the (AB)-mode. Accordingly, we determined andemployed the corresponding w(t) curves of type (AB). By construction,they are supposed to match u(t) well for the periods of red dots .

The role of the (AB)-mode. Generally the later stages are difficultto forecast if the reduction of hard measures (if any) begins almostafter the turning point, or what looks like a turning point. The growthcan be expected linear then, but the number of daily new infectionscan be very high. From our perspective, this means a partial switchfrom mode (A) to mode (B). We need to be more exact here. If thehard measures are still obviously present, but the response to the totalnumber of infections becomes softer, this is our mode (AB). Let us tryto explain the practical difference.

For instance, if the number of new cases is essentially a constant,even uncomfortably high, the (AB) or (B)–response is to keep thetesting-detection constant too. The same kind of management is whenplaces with potentially high risks of the spread of Covid-19 are allowedto reopen. Definitely the improvements with testing and better capac-ities for isolation and treatment are important factors here, but thisis ”soft way”. Another argument in favor of reopening is that peoplewho suspect that they are infected begin more actively request help atthis stage. Mathematically, it appeared that the complex process ofreducing and abandoning hard measures can be uniformly described asthe (B)–type response.

Considering the (AB) mode does not seem necessary for the countriesthat reached stable relatively small numbers of new daily infections.For such countries, the passage is from (A) directly to mode (B). Thisis the core of our two-phase approach, to be considered next. However

28 IVAN CHEREDNIK

if the new daily cases are constant but high, the mode (AB) is a natural”next” stage after (A).

The blue dots till 05/27 were added to check this hypothesis, afterthe w(t)–function was determined. Actually with Sweden, it was notsupposed to hold since no aggressive approach was implemented there.Though mathematically the curves were produced ok for this coun-try, but the expectations were that the actual u(t) would be greaterthan u(t), w(t) we produced. This ”anti-prediction” successfully wentthrough. It appeared that the actual total number of the detected casesfor the USA followed the same ”Sweden pattern”, which is connectedwith the significant reduction of ”hard” measures there approximatelyin the middle of May.

Currently both countries are in their ”essentially linear” late periodswith constant, but high, numbers of new daily cases. This makes itimpossible to determine by now the parameter d from C cos(d log(t)),which describes well the second (late) phase in quite a few countriesthat went through the saturation, i.e. reach relatively small numbersof daily cases. Concerning UK, it remains within the forecast conebetween u and w; its ”expiration” is June 10.

Thus, the main assumption for (AB) is that the hard measures arestill in place, but the response to the current total number of detectedinfections is via u(t)/t instead of u(t). The saturation is expectedto occur significantly later, but not too much later. This mode isgoverned by w(t). The point twtop, where the (first) maximum of w(t)occurs, seems a reasonable upper bound for the ”technical saturation”,with our usual reservation about the continued usage of the ”hard”measures. The prior tutop = ttop then is a lower bound; we obtain someforecast cone .

Actually even relatively minor deviations with a, c can lead to signif-icant changes of u(t) over time, so some ”cones” are inevitable. How-ever, we have something more fundamental here. The switch to w(t)is due to a different kind of management.

When the daily numbers of new cases are high, there are significantchances of new clusters and fluctuations of the data of all kinds. So weneed to model a process with many uncertainties. However, we thinkthat this is basically no different from what we did for the middlestages, where u(t) was surprisingly efficient.

Let u(t) = Ctc+12 J(c−1)/2(t

√a), w(t) = Dt

c+12 Jc−1(2

√tb). Using

(18), the match u(t) ≈ w(t) near t = 0 gives:

C

D≈( 2b√

a

)c−12 Γ((c+ 1)/2)

Γ(c)for the Γ-function.(21)


Blue dots:

05/06-05/27

Figure 7. USA: c=2.2, a=0.2, C=1.7; b=0.35, D=1.9.

The cone for the USA. The prior ttop = tutop for u(t) was May 5, witha, c, C calculated on the bases of the data till April 16, marked by reddots. It was under the expectations that the ”hard” measures wouldapplied as in (A) (as before April 16). The black dots represent thecontrol period. The challenge was to understand how far u(t) can beused toward the ”later stages”. The match was very good for quitea long period of time (the same with UK), but then there were somechanges, which we mostly attribute to the change of the managementmode. This of course includes (everywhere) the actions of the people,not only those by the authorities in charge.

The 3 major spikes in the daily cases and other factors like de factorelaxing hard measures obviously delayed the saturation. Whatever thereasons, the switch to the (AB)-mode and w(t) appeared necessary forthe purpose of modeling.

The current trend seems toward w(t). Generally, the black dots canbe expected to stay within the forecast cone , subject to all standardwarnings. The cone is defined as the area between u(t) extended by aconstant u(ttop) for t > tutop and the graph of w(u) till twtop, which is ap-proximately May 30, 2020. The parameters b,D of w(t) are calculatedto ensure good match with red dots; c, the initial transmission rate, isthe same for u(t) and w(t). The graphs of u(t) and w(t) are very closeto each other in the range of red dots. The above relation for C/Dholds with the accuracy about 20%.

The w–saturation value is around 1.6M, but mostly we monitor thetrend , the ”derivative” of the graph of black dots, which is supposedto be close to the derivative of w(t) or u(t). See Figure 7. Some spikes

30 IVAN CHEREDNIK

with number of cases are inevitable; it is acceptable if the dots continueto be mainly ”parallel” to u(t) or w(t) (or in between).

2 4 6 8

5

10

15

20

25

30

Blue dots:

05/06-05/27

Figure 8. UK: c=2.4, a=0.2, C=2.2; b=0.35, D=2.5.

UK: till June 10. The graphs of u(t), w(t) with all red-black dotsavailable by now are in Figure 8. The expected twtop is around June 9.The total number of detected infections is expected 330K. These ”pre-dictions” will of course depend on many developments. However, thereis an important argument in favor of the stability of our model: anyspikes with the numbers of infections are supposed to trigger actions ofauthorities in charge and influence of own protection measures. Thisis a rationale for relatively uniform patterns of the spread of Covid-19.

Sweden: anti-forecast. Sweden is actually expected not to followour u(t) and w(t); this country does not follow hard ways with fightingCovid-19. Nevertheless, we calculated w(t); see Figure 9. Its expirationis approximately after 100 days (x = 10); i.e. the w-saturation is aboutJune 15. We do not expect this saturation to occur. Definitely somemeasure are in place in Sweden; for instance, people readily contactthe authorities with any symptoms of Covid-19. However this did notchange much so the linear spread so far.

Israel: u(t) worked. Here the usage of u(t) was sufficient to modelthe total number of detected cases till the ”saturation”, which oc-curred almost exactly when u(t) reached its maximum. The blackdots demonstrate well the ”linear period” after the saturation, whichwas essentially of the same type in South Korea, Austria and quite afew countries that went through the saturation. However we providethe forecast cone calculated as above on the basis of w(t). In contrast


2 4 6 8 10

5

10

15

20

25

30

35

Blue dots:

05/06-05/27

Figure 9. w(t)=2.1t(c+1)/2Jc−1(2√bt); b=0.3, c=2.4.

to the USA, UK and especially Sweden, the black dots are much closerto the flat line started at the u-top, which was at t = 4.5 (April 26).

1 2 3 4 5 6 7

5

10

15

20

25

30

Black dots :

5 05

Figure 10. u(t)=2.2 t(c+1)/2J(c−1)/2(√at), c=2.6, a=0.3

.

11. Two-phase solution. Let us provide a two-phase model for thenumber of total infections: from the beginning of the intensive growththrough the saturation ttop of the (A)–phase and till the ”final satu-ration” of the (B)–phase. It works surprisingly well for the countriesthat reached reasonably small numbers of new daily infections duringthe first phase, so the (AB)–mode is mostly unnecessary for them.

32 IVAN CHEREDNIK

We determined the type (A) parameters a, c, d, and the scaling co-efficient C, using the whole period till May 22 (2020). So all dots arered unless for Israel, where we used a, c, C from the middle of April.

Here ddef==√−D; it equals very approximately to

√a for considered

countries; so phase 2 is roughly governed by (15) where t inside cos(·)is replaced by log(t).

The epidemic is of course far from over, but the first cycle hopefullyapproaches the end in these countries; so no test periods, ”black dots”,seem necessary. Recall that the parameters a, c, C for the first phaseare mostly obtained on the basis of the period before or around the”turning points”; d can be determined only later. Also, the same c,the initial transmission coefficient, is supposed to be used for both, (A)and (B), according to our theory.

Israel: 03/13-05/22. We already closely considered the first phase, tillttop. Let us now focus on the second phase. The leading solution from(7) is uB(t) = CBt

c/2 cos(d log(Max(1, t))), where c must be the one weused in u(t) = 2.2 t(c+1)/2J(c−1)/2(

√at), c = 2.6, a = 0.3. However, now

d =√aB − c2/4 = 0.6 and the scaling coefficient CB = 3.4 must be

adjusted independently. The intensity parameters a = aA and aB for(B) have different meaning and anyway they are control parameters,i.e. up to us. At later stages, we can assume that aB > c2/4.

Here Max(1, t) is to avoid some ambiguity at t = 0; and we use uBanyway for t > 1. Importantly, we start uB at t = 0, not at someintermediate point. Practically, this means that we fine type (B) for-mula that provide the correct saturation point ttop and the rest of thecurve. Thus, some sharp usage of ”soft” measures can be mathemat-ically sufficient to end the epidemic, but d can be determined only inthe vicinity of ttop, if it is reached.

Italy: 2/22-5/22. See Figure 12. The starting point is 2/12, whenthe total number of infections was 17; in this paper, we always subtractthis initial value when calculating our dots. One has:

u1,2(t) = 0.8 t(c+1)/2J± c−12

(√at), u(t) = u1(t)− u2(t), and

uB(t) =2.85 tc/2 cos(d log(Max(1, t))), c=2.6, a=0.2, d=0.5.

For the first time in this paper, we use the second solution u2(t) from(14). For t ≈ 0, it is approximately ∼ t, i.e. smaller than ∼ tc for thedominating solution u1, and max{u2(t)} occurs significantly earlier.This maximum is the reason for a well-visible bent in the middle of thesequence of red dots. Actually, we can see some kind of the bent forIsrael too, but it was short-lived and did not prevented us from using


0 1 2 3 4 5 6 7

5

10

15

du(t)/dt

Figure 11. Israel: c=2.6, a=0.3, d=0.6.

u1 only. This is actually common; see below. The linear combinationu1(t)− u2(t) was found numerically; the coefficient of u2 is not always−1. Here and further in this section, u1 = Cu1 and u2 = Cu2 for u1,2

from (14) with the same rescaling coefficient C, found numerically.

Figure 12. Italy: c=2.6, a=0.2, d=0.5.

34 IVAN CHEREDNIK

Germany: 3/07-5/22. See Figure 13. We begin here with the initialnumber of total infections 684, which must be subtracted when calcu-lating the red dots. This is the moment when the curve began to lookstable, i.e. a systematic management began. One has:

u1,2(t) =1.3 t(c+1)/2J± c−12

(√at), u(t) = u1(t)− 0.7u2(t), and


0 1 2 3 4 5 6

5

10

15

20

25

du(t)/dt

u =

Figure 13. Germany: c=2.6, a=0.35, d=0.56.

It makes sense to provide the u–curve for the first phase in Ger-many using only u1; Figure 14. We disregard the bent in the mid-dle of phase one , actually similarly to what we did for Israel. Theparameters will be different: a = 0.2, c = 2.6, C = 1.7, i.e. nowu(t) = 1.7t1.8J0.8(t

√0.2). Note that when doing forecasting, only u1(t)

can be used before the bent occurs. After the bent, we ”basically”return to the curve in terms of u1 only, similarly to what we observedabove with Israel. This pattern remains essentially the same for Italyand Japan. Let us give the corresponding parameters for u = u1: Italy(c = 2.5, a = 0.1, C = 1), Japan (c = 2.6, a = 0.1, C = 0.9).

When using only u1, the match with the actual numbers of totalcases during phase 1 will be weaker, but such u can be still used forforecasting. Practically, we obtain a, c, C before or around the turningpoint, when the bent (if any) can be not really visible. Then they mustbe adjusted constantly, especially if the bent is detected or significantfluctuations occur. The initial a, c, C are of course subject to quite afew uncertainties. However, if u(t) match well the ”red dots”.


Japan: 3/20- 5/22. See Figure 15. It was a sort of the second wavein Japan, with already 950 total infections on March 20. The curve isdiscontinuous, but manageable by our 2-phase solution :

u1,2(t) =1.5 t(c+1)/2J± c−12

(√at), u(t) = u1(t)− 0.4u2(t), and


The Netherlands: 03/13-5/22. Figure 16. The response to Covid-19was relatively late in the Netherlands; the number of the total case was383 on 3/13, the beginning of intensive spread from our perspective.However, the country perfectly reached the saturation at ttop with asingle u(t) = u1(t), and then smoothly switched to phase 2:

u(t) = 0.5 t(c+1)/2 J c−12

(√at), c=2.4, a=0.2,

uB(t) =0.86 tc/2 cos(d log(Max(1, t))), d=0.54.

0 2 4 6 8

5

10

15

20 Germany: 03/07- 05/22; single u(t)

(a=0.2; c=2.6; C=1.7)

du(t)/dt

u(t) = u (t)

(mode A)

1

Figure 14. Germany: u = u1 = 1.7t1.8J0.8(t√

0.2).

To finalize, the best ways to use our curves seem as follows:

(1): determine a, c, C when the spread looks essentially linear;(2): update them constantly till the turning point and beyond;(3): expect the ”bents” to appear and add the u2(t) if needed;(4): try to adjust the intensity of the measures to match u(t) ;(5): at the turning point, determine b,D and the bound w(t);(6): after the saturation at ttop, find d and switch to phase 2 ,(7): ”detection - isolation” must be continued after saturation.

36 IVAN CHEREDNIK

0 1 2 3 4 5 6

5

10

15

20

Figure 15. Japan: u = 1.5t1.8(J0.8 − 0.4J−0.8)(t√

0.3).

0 1 2 3 4 5 6 7

1

2

3

4

5

Figure 16. The Netherlands: u = 0.5t1.7J0.7(t√

0.2).

Generally, our forecasting tool can serve the best if the data and themeasures are as uniform and ”stable” as possible. Then underreportingthe number of infections, focusing on symptomatic cases, and inevitablefluctuations with the data may not influence too much the applicabilityof the u,w–curves and uB.


Acknowledgements. I’d like to thank ETH-ITS for outstandinghospitality. My special thanks are to Giovanni Felder, Rahul Pandhari-pande and the management of ETH-ITS. I also thank very much DavidKazhdan for his valuable comments and suggestions; a good portion ofthis paper presents my attempts to answer his questions.

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[HL] H. Hethcote, and S. Levin, Periodicity in Epidemiological Models, In:Applied Mathematical Ecology. Biomathematics, 18, 193–211, Springer,Berlin, Heidelberg, S. Levin, T. Hallam, L. Gross (eds), 1989. 6, 22

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(I. Cherednik) Department of Mathematics, UNC Chapel Hill, NorthCarolina 27599, USA, [email protected]

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