Chapter 2

Modeling epidemic outbreaks

An infectionous desease may be either endemic, rooted in the population, ormanifest itself in term of isolated and well defined outbreaks, the subject ofthe present chapter.

2.1 Modeling in epidemiology

There are illnesses, like measles, that come and go recurrently. Looking atthe local statistics of measle outbreaks, see Fig. 2.1, one can observe that out-breaks occur in quite regular time intervals within a given city. Interestinglythough, these outbreaks can be either in phase (synchronized) or out of phasebetween different cities.

The oscillations in the number of infected persons are definitely not har-monic, they share many characteristics with relaxation oscillations, whichtypically have silent and active phases.

The SIRS Model A standard approach to model the dynamics of infectiousdiseases is the SIRS model. At any time an individual can belong to one ofthe three classes:

S : susceptible,I : infected,R : recovered.

The dynamics is governed by the following rules:

(a) Susceptibles pass to the infected state, with a certain probability, aftercoming into contact with one infected individual.

(b) Infected individuals pass to the recovered state after a fixed period of timeτI .

(c) Recovered individuals return to the susceptible state after a recovery timeτR, when immunity is lost, and the S→ I→ R→ S cycle is complete.

1

2 2 Modeling epidemic outbreaks

0

0.5

1

44 46 48 50 52 54 56 580

0.5

1

years

wee

kly

mea

sle

case

s

Fig. 2.1 Observation of the number of infected persons in a study on illnesses. Top:

Weekly cases of measle cases in Birmingham (red line) and Newcastle (blue line). Bottom:Weekly cases of measle cases in Cambridge (green line) and in Norwich (brown line). From

He (2003).

When τR →∞ (lifelong immunity) the model reduces to the SIR-model.

The Discrete Time Model We consider first the discrete time SIRSmodel, with t = 1, 2, 3, . . . and τI = 1: The infected phase is normally shortand we can use it to set the unit of time. The recovery time τR is then amultiple of τI = 1.We define with

xt the fraction of infected individuals at time t,st the percentage of susceptible individuals at time t,

which obey

st = 1− xt −τR∑k=1

xt−k = 1−τR∑k=0

xt−k , (2.1)

as the fraction of susceptible individuals is just 1 minus the number of infectedindividuals minus the number of individuals in the recovery state, compareFig. 2.2.

The Recursion Relation We denote with a the rate of transmitting aninfection when there is a contact between an infected individual and a sus-ceptible individual:

2.1 Modeling in epidemiology 3

S S S I R R R S S state

1 2 3 4 5 6 7 8 9 time

Fig. 2.2 Example of the course of an individual infection within the SIRS model with an

infection time τI = 1 and a recovery time τR = 3. The number of individuals recoveringat time t is just the sum of infected individuals at times t − 1, t − 2 and t − 3, compare

Eq. (2.1)

xt+1 = axtst = a xt

(1−

τR∑k=0

xt−k

). (2.2)

Relation to the Logistic Map For τR = 0 the discrete time SIRS model(2.2) reduces to the logistic map

xt+1 = axt (1− xt) .

For a < 1 it has only the trivial fixpoint xt ≡ 0, the illness dies out. Thenon-trivial steady state is

x(1) = 1− 1

a, for 1 < a < 3 .

For a = 3 there is a Hopf bifurcation and for a > 3 the system oscillates witha period of 2. Eq. (2.2) has a similar behavior, but the resulting oscillationsmay depend on the initial condition and for τR � τI ≡ 1 show featurescharacteristic of relaxation oscillators, see Fig. 2.3.

2.1.1 Two Coupled Epidemic Centers

We consider now two epidemic centers with variables

s(1,2)t , x

(1,2)t ,

denoting the fraction of susceptible/infected individuals in the respectivecities. Different dynamical couplings are conceivable, via exchange or visitsof susceptible or infected individuals. We consider with

x(1)t+1 = a

(x(1)t + e x

(2)t

)s(1)t , x

(2)t+1 = a

(x(2)t + e x

(1)t

)s(2)t (2.3)

the visit of a small fraction e of infected individuals to the other center.Eq. (2.3) determines the time evolution of the epidemics together with

4 2 Modeling epidemic outbreaks

0 20 40 60 80 1000

0.1

0.2

xt+1

= 2.2 xt (1-x

t-x

t-1-x

t-2-x

t-3-x

t-4-x

t-5-x

t-6)

time

xt

Fig. 2.3 Example of a solution to the SIRS model, Eq. (2.2), for τR = 6. The number

of infected individuals might drop to very low values during the silent phase in between

two outbreaks as most of the population is first infected and then immunized during anoutbreak

Eq. (2.1), generalized to both centers. For e = 1 there is no distinctionbetween the two centers anymore and their dynamics can be merged via

xt = x(1)t + x

(2)t and st = s

(1)t + s

(2)t to the one of a single center.

In Phase Versus Out of Phase Synchronization In Fig. 2.4 we presentthe results from a numerical simulation of the coupled model, illustrating thetypical behavior. We see that the outbreaks of epidemics in the SIRS modelindeed occur in phase for a moderate to large coupling constant e. For verysmall coupling e between the two centers of epidemics on the other hand, thesynchronization becomes antiphase, as is sometimes observed in reality, seeFig. 2.1.

Time Scale Separation The reason for the occurrence of out of phasesynchronization is the emergence of two separate time scales in the limittR � 1 and e � 1. A small seed ∼eax(1)s(2) of infections in the second cityneeds substantial time to induce a full-scale outbreak, even via exponentialgrowth, when e is too small. But in order to remain in phase with the currentoutbreak in the first city the outbreak occurring in the second city may notlag too far behind. When the dynamics is symmetric under exchange 1 ↔ 2the system then settles in antiphase cycles.

2.2 Modeling epidemic control

For the SIR (Susceptible- Infected- Recovered) model, an infected personeither dies or recoveres for life. We rewrite the expression (2.2) for the SIRSmodel as

2.2 Modeling epidemic control 5

0

0.1

0.2

0.3

xt

(i)a=2, e=0.005, τ

R=6, x

0

(1)=0.01, x

0

(2)=0

a=2, e=0.100, τR=6, x

0

(1)=0.01, x

0

(2)=0

time0 10 20 30 40 50 60

0

0.1

0.2

0.3

xt

(i)

Fig. 2.4 Time evolution of the fraction of infected individuals x(1)(t) and x(2)(t) within

the SIRS model, Eq. (2.3), for two epidemic centers i = 1, 2 with recovery times τR = 6 and

infection rates a = 2, see Eq. (2.2). For a very weak coupling e = 0.005 (top) the outbreaksoccur out of phase, for a moderate coupling e = 0.1 (bottom) in phase

It+1 = gtIt(1−Xt), Xt =

∞∑k=0

It−k , (2.4)

with gt being the reproduction factor at time t, It the fraction of infected,the actual cases, and Xt the total number of cases (as a fraction of thepopulation).

Contaiment Epidemic models consider in general constant reproductionfactors gt, an assumption that is arguably a valid basis for the modeling of’normal’ diseases, i.e. infectious diseases that do not threaten public healthto a significant extent (e.g. influenza). Larger outbreaks like the COVID-19virus lead however to massive political and social responses. The reproductionfactor evolves in this situation in line with the progressing of the disease. Forthis purpose we investigate the following two functional dependencies:

gt =

g0

1 + αIt(short term)

g01 + αXt

(long term)(2.5)

The parameter α describes the strength of the reaction of the society to thespreading of the disease and g0 corresponds to the ‘intrinsic’ or medical infec-tion growth factor of the disease in the absence of any behavioral reaction.

6 2 Modeling epidemic outbreaks

0.2 0.4 0.6 0.8 1

all cases: X

0

0.1

0.2

0.3

actu

al c

ases

: I

α = 0, uncontrolled

α = 10, history-aware control

start of outbreakend of outbreak: X

tot

g0 = 3

peak

Fig. 2.5 XI representation of an epidemic outbreak. For the continuous-time SIR

model with long term control a closed expression I = I(X) between the fraction I/X ofactual/overall cases exists, see (2.13). Shown is I(X), for α = 0, no control and α = 10,

with g0 = 3. The outbreak, which starts at X = I = 0, is maximal at the inflection point,

the ‘peak’, ending when the number of actual cases has dropped to zero. At this point thenumber of infected reaches Xtot.

The first case in equation (2.5), termed here ‘short term’, describes the sit-uation where society reacts primarily to the percentage It of the populationcurrently infected. Progressively more rigorous social-distancing and relatedmeasures will be enforced when the number of individuals able to infect oth-ers continues to rise. The functional form chosen in Eq. (2.5) implies thatit becomes progressively more difficult to reduce the reproduction factor byincreasing social distancing.

For the second strategy examined, denoted ’long term’, the reaction isbased on the overall history of the epidemic. Society takes the total numberof past and current infected cases, Xt, into account. Not only the present butalso the past situation matters, the reaction is ‘long term’.

2.2.1 Continuous-time SIR model

We denote with S = S(t) the fraction of susceptible (non-affected) people,with I = I(t) the fraction of the population that is currently ill (activecases), and with R = R(t) the fraction of recovered. Normalization demandsS + I + R = 1 at all times. For the continuous-time SIR model, we opt forthe parametrization

τ S = −gSI, τ I =(gS − 1

)I, τR = I , (2.6)

which allows to separate the characteristic time scale τ and infectious spread-ing, with the latter being determined by the dimensionless reproduction factorg. Normalization is conserved, as S+ I + R = 0. Infection and recovery ratesare g/τ and 1/τ . The number of infected grows as long as I > 0, namely when

2.2 Modeling epidemic control 7

gS > 1. The turning point S = 1/g is sometimes called the ‘herd immunity’point, which is however a misleading term. The outbreak continuous evenafter the peak is reached, at I = 0, albeith with progressively diminsish-ing infection rates. Making connection with the discrete case, (2.4), infectionpeaks when

1 = g(1−X) = gS, X = I +R = 1− S , (2.7)

which confirms that g is indeed the infection factor. Epidemic spreading isobserved for g > 1.

Equivalence of continous and discrete time formulation The secondequation of (2.6) may be written as

τ I + I = gIS, S = 1−X . (2.8)

Next we consider the case that the changes of I = I(t) over the course of aperiod τ are small, namely that the substitution I(t) ≈ (I(t + τ) − I(t))/τholds. We then obtain

gI(t)[1−X(t)

]= τ I(t) + I(t) ≈ I(t+ τ) (2.9)

which is equivalent to the discrete SIR model (2.4), with a time period of τ .

SIR model with long term control We examine the long term controlstrategy,

g =g0

1 + α(1− S). (2.10)

A functional relation between S and I is obtained considering I/S, whichresults in

dI = −dS +1

g(S)SdS = −dS +

1

g0

1 + α(1− S)

SdS . (2.11)

Integrating leads to

I = −(α

g0+ 1

)S +

1 + α

g0log(S) + c , (2.12)

where the integration constant c is given by the condition I(S= 1) = 0. Wehence obtain

I =α+ g0g0

X +1 + α

g0log(1−X) , (2.13)

when substituting S = 1−X. The number of actual cases, I, vanishes bothwhen X = 0, the starting point of the outbreak, and when the epidemic stops.The overall number of cases, Xtot, is obtained consequently by the non-trivialroot Xtot of (2.13), as illustrated in Figure 2.5. The relation (2.13) is denotedthe ‘XI representation’ of an epidiemic outbreak.

8 2 Modeling epidemic outbreaks

Weak epidemics expansion Expanding (2.13) in X, which becomes smallwhen α� 1, one obtains

I ≈ 1 + α

2g0X

[2g0 − 1

1 + α−X

]=g0 − 1

g0X

(1− X

Xtot

)(2.14)

which yields

I ≈ g0 − 1

g0X +O(X2), Xtot ≈ 2

g0 − 1

1 + α. (2.15)

The first relation shows that the slope dI/dX at X → 0 has two key proper-ties. Firstly, that the slope, (g0 − 1)/g0, is independent of α. This result wasto be expected, as α incorporates the reaction to the outbreak, which meansthat α contributes only to higher order. Secondly, the effective population sizeNp drops out of (2.14), which implies that the dimensionless growth factorg0 is uniquely determined, modulo the noise inherent in field data, by meas-suring the slope of the daily case numbers with respect to the cummulativecase count.

Peak medical load The maximal number Ipeak of actual cases, the peak

medical load, is attained at the inflection point, when I = 0,

gS = 1, g0S = 1 + α(1− S), Speak =1 + α

g0 + α, (2.16)

see (2.7). For the epidemic state, g > 1, one has Speak < 1, withlimα→∞ Speak = 1. With the XI representation (2.13) we obtain

Ipeak =g0 − 1

g0+

1 + α

g0log

(1 + α

g0 + α

), Xpeak =

g0 − 1

g0 + α, (2.17)

for the maximal number Ipeak of actual cases and for Xpeak = 1− Speak. Forstrong control, α� 1, the expansion

Ipeak∣∣α�1

≈ (g0 − 1)2

g0α, Ipeak

∣∣α=0

= 1− 1 + log(g0)

g0(2.18)

holds. Without control, when α = 0, close to the entire population is infectedsimultaneously when g0 is large. Combining (2.18) with (2.15) one obtainsthe large-α relation

Xtot ≈2g0g0 − 1

Ipeak (2.19)

between the total impact of the epidemic, Xtot and the maximum number ofactual cases Ipeak. This relation can be used to estimate Xtot once the peakhas been reached.

2.2 Modeling epidemic control 9

0.0 0.2 0.4 0.6 0.8 1.0X (total)

0.0

0.1

0.2

0.3

0.4

I (cu

rrent

)

Xpeak

Ipeak

0.0 0.5 1.0 1.5 2.0 2.5Xdata / Xpeak

0.00

0.25

0.50

0.75

1.00

I data

/Ipe

ak

AustraliaAustriaBergamo (ITA)GermanyIcelandIsraelItalyRoma (ITA)S.KoreaSpain

0.0 2.5 5.0 7.5 10.0 12.5total case number in thousands

0

200

400

600

800

new

daily

cas

es

AustraliaAustriaBergamoS.Korea

0.0 2.5 5.0 7.5 10.0total case number in thousands

0

100

200

300

400

500

new

daily

cas

es

a b

c

ddata: n = 5 fit: n = 1 fit: n = 5 fit: n = 7

Fig. 2.6 Modeling of COVID-19 outbreaks. a, Model illustration, as in Fig. 2.5. b,

Model validation for a choice of four countries/regions. The model (lines) fits the five-daycentered averages of COVID-19 case counts well. For South Korea data till March 10 (2020)

has been used for the XI-fit, at which point a transition from long-term overall control to

the tracking of individuals is observable. c, Data collapse shown for ten countries/regions.Rescaling with the peak values Xpeak and Ipeak, obtained from the XI fit, maps COVID-19

case counts approximately onto a universal inverted parabola. d, Robustness test. The oftenstrong daily fluctuations are smoothed by n-day centered averages. Shown are the Bergamo

data (dots, n = 5) and XI-fits to n = 1 (no average), n = 5 and n = 7. Convergence of the

XI-representation is observed. From Gros et al. (2020).

2.2.2 Alarmistic long term control

A possible generalization of the control function (2.10) is

r =r0

1− α log(1−X)=

r01− α log(S)

, (2.20)

10 2 Modeling epidemic outbreaks

0 50000 1e+05 1.5e+05 2e+05

X: total case/death count

0

2000

4000

6000

8000

I: d

aily

new

cas

es /

dea

ths

Italy: infected

Italy: deaths*7

Spain: infected

Spain: deaths*9

Covid-19, 7-day centered averages, till April 26, 2020

Fig. 2.7 For the 2020 Covid-19 outbreak, a comparison of the XI-statistics of the number

of infected (solid circles) and of deceased (open circle), the latter multiplied by a factor

of fdeath. Death- and infected counts trace each other for the cases shown here, Italy(fdeath = 7) and Spain (fdeath = 9, which is however not the case of all countries. For

Germany fdeath = 22.

which models the situation that the response of the socio-political systemdiverges whenX → 1. One could call (2.20) ‘alarmistic’ response. The originalformulation (2.10) is recovered when the outbreak is small, since

− log(1−X)∣∣X�1

∼ X . (2.21)

In equivalence to (2.11) one obtains

dI = −dS +1

g0

1− α log(S)

SdS , (2.22)

which yields with

I = 1− S +log(S)

g0− α(log(S))2

2g0(2.23)

the XI representation

I = X +log(1−X)

g0

(1− α log(1−X)

2

). (2.24)

With fractional infection cases being small, in praxis, it is sufficient to modelfield data using (2.10).

2.3 Modelling field data 11

2.3 Modelling field data

The containment policy enacted by a given country affects the timeline ofthe outbreak, which can be analyzed within the XI representation. Modelingdirectly the timeline of reported infection cases would involve in contrast anon-trivial numerical simulation effort.

Reported outbreak data, the head count for the number of actual and

overall cases, respectively N(data)I and N

(data)X , needs to be normalized with

respect to the population size Np,

I(data) =N

(data)I

Np, X(data) =

N(data)X

Np. (2.25)

Outbreaks are often confined from the beginning to a fraction of the popu-lation, either due to geographical, transportation or social barriers. For thepurpose of data modeling, Np is therefore not necessarily the nominal pop-ulation size, say 80 Million for Germany, but a parameter to be adapted.Technically the optimal parameters are determined via a least-square fit ofI(data) = Np I(X(data)/Np), where I(X) is the XI relation (2.24), or respec-tively (2.12).

Loss function Field data is crowded at low levels of X and I in the XIrepresentation. Our aim is a fitting routine that takes the range X ∈ [0, Xmx]uniformly into account. This is achieve when minimizing the loss function

U =∑t

ut

(I(data)t − I(theory)(X(data)

t ))2, ut = X

(data)t −X(data)

t−1 ,

(2.26)

where I(theory)(X(data)t ) is given by (2.13), respectively by (2.24). The weigh-

ing factor ut ≡ I(data)t attributes to each data point a weight that is propor-tional to the distance of its predecessor. With (2.26) it becomes irrelevantwhere the timeline of field data is truncated, both at the start or at theend. Adding a large number of null measurements after the epidemic stoppedwould not alter the result. Numerically, the minimum of U as a function ofg0 and α is evaluated.

Fig. 2.6 presents the XI representation of selected 2020 COVID-19 out-breaks. Theory and data agree well.

Death count representation An infected person shows up in official casestatistics when he/she has been tested. Undetected cases will not influencethe functional development of an outbreak as long as the ratio detected/un-detected remains constant. An increase in testing activity, which has beentypically for the 2020 Covid-19 outbreak, may however lead to distortions.

The spreading of deathly pathogens may be traced also with respect tothe statistics of the death count. Deaths induced by the pathogen may be

12 2 Modeling epidemic outbreaks

attributed to other causes, but this kind of miss-counting is not as severe asfor the detection of infected. For Covid-19, death counts are maximally offby a factor of two. As shown in Fig. 2.7, death count and infected data traceeach other nicely in most cases. This results indicates, that changes in theamount of testing have only no minor impact on the nominal evolution ofCovid-19 outbreaks.

2.4 Epidemiological forecasting

Given that real-world data follows closely the XI representation, one use theextracted g0 and α to forecast the timeline of expected case counts. In a firststep we reduced the original model (2.4).

Reduced controlled SIR model Considering that τ S = −gSI and thatS = −X, one can use the XI representation (2.13) to reduce the continuoustime SIR model to

τX =1−X

1 + αXg0 I(X) , (2.27)

which yields the time-dependent X(t) = X((t− t0)/τ) as a one dimensionalintegral. The characteristic time τ and the starting time t0 are to be deter-mined by comparing to the known history of the outbreak. In general thishas to be done numericaly.

Integrating the reduced SIR model Fractional case counts are mostof the time small. We then use the weak outbreak expansion (2.14) for thereduced SIR equation, to obtain

τX ≈ 1−X1 + αX

(g0 − 1

)X

(1− X

Xtot

). (2.28)

The partial fraction decomposition

1

X(1−X)(1−X/Xtot)=

Xtot

Xtot − 1

−1

X − 1+

1

X(2.29)

+1

Xtot − 1

1

X −Xtot

implies that

X

X(1−X)(1−X/Xtot)=

Xtot

Xtot − 1

−1

X − 1+

Xtot

Xtot − 1

1

X −Xtot. (2.30)

With this result we find

g0 − 1

τdt =

dX

X− Xtot(1 + α)

1−Xtot

dX

1−X+

1 + αXtot

1−Xtot

dX

Xtot −X(2.31)

2.4 Epidemiological forecasting 13

for (2.28). Note the double sign change and that 0 < X < Xtot < 1. Integra-tion yields

g0 − 1

τ(t− t0) = log(X) +

Xtot(1 + α)

1−Xtotlog(1−X)

− 1 + αXtot

1−Xtotlog(Xtot −X) ,

or

t− t0 =τ

g0 − 1log

(X(1−X)c

(Xtot −X)d

), (2.32)

with

c =Xtot(1 + α)

1−Xtot, d =

1 + αXtot

1−Xtot, . (2.33)

Eq. (2.32) is the desired explicit function t = t(X) for time as a function oftotal case fraction X. One extracts g0, α and Xtot from the XI representationto determine τ and t0 in a second step from a least-square fit of (2.32) tothe observed timeline of the epidemic. Note that the argument of the log()in (2.32) diverges for X → Xtot, which implies that Xtot is reached onlyasymptotically for t→∞.

Contained epidemics forecasting The exact solution (2.32) of the SIRmodel in the weak-outbreak approxmation can be simplified further, usingthat not only X � 1, but also that Xtot � 1:

Xtot ≈ 2Xpeak = 2g0 − 1

g0 + α,

α

g0 + α≈ 1 , (2.34)

which leads toXtot(1 + α) ≈ Xtotα ≈ 2(g0 − 1) (2.35)

andc = 2g0 − 2, d = 2g0 − 1 . (2.36)

Using again 1−Xtot ≈ 1 we find

t− t0 =τ

g0 − 1log

(X/Xtot

Xd−1tot (1−X/Xtot)d

)(2.37)

for the time prediction relation. The time difference t2−t1 between two eventscharacterized respectively by a given X1 and X2 is then

t2 − t1 =τ

g0 − 1

[log

(X2/Xtot

(1−X2/Xtot)2g0−1

)− log

(X1/Xtot

(1−X1/Xtot)2g0−1

)].

(2.38)

Relative peak times The result (2.38) for the time difference of two epi-demiological event can be used to estimate two key time spans:

14 2 Modeling epidemic outbreaks

– run-up time, Tup, defined as the time needed to reach the peak startingfrom an initial case count Xstart = fXtot.

– run-down time, Tdown, defined as the time needed to reach Xend = (1−f)Xtot down from the peak.

Assuming f � 1, we obtain

Tup =τ

g0 − 1log

(22g0−2

f

)(2.39)

and

Tdown =τ

g0 − 1log

(1

22g0−2f2g0−1

). (2.40)

Note that Tdown/Tup → 1 for g0 → 1. For f � 1 we hence find

limf→0

Tdown

Tup= 2g0 − 1 . (2.41)

For Covid-19, g0 is typically of the order 1.2-1.4 (Gros et al. 2020), whichimplies that it takes a substantial amount of time to return to low infectionlevels, viz 40-80% longer than it took to reach the point of maximal medicalload.

2.5 Political reaction dynamics

Say, that politics wants to achieve a certain target infection factor g∞. Polit-ical actors cannot influence gt directly, they can only change their strategyand see what happends. This situation may be described by

It+1 = gt(αt)It(1−Xt), αt+1 = αt + εα(gt − g∞) (2.42)

where is the update rate. The reaction strength α is increased/decreasedwhen the current gt is too large/small.

References

He, D., Stone, L. 2003 Spatio-temporal synchronization of recurrent epidemics. Proceed-ings of the Royal Society London B 270, 1519–1526.

Gros, C., Valenti, R., Schneider, L., Valenti, K., Gros, D. 2020 Containment effi-ciency and control strategies for the Corona pandemic costs. To be published , .