A two-strain SARS-COV-2 model for Germany -- Evidence from a Linearization
Thomas Götz, Wolfgang Bock, Robert Rockenfeller, Moritz Schäfer
aa r X i v : . [ q - b i o . P E ] F e b A two–strain SARS–COV–2 model for Germany -Evidence from a Linearization
Thomas G¨otz a, ∗ , Wolfgang Bock b , Robert Rockenfeller a , Moritz Sch¨afer a a Mathematical Institute, University Koblenz, 56070 Koblenz, Germany b Department of Mathematics, TU Kaiserslautern, 67663 Kaiserslautern, Germany
Abstract
Currently, due to the COVID–19 pandemic the public life in most Europeancountries stopped almost completely due to measures against the spread of thevirus. Efforts to limit the number of new infections are threatened by the adventof new variants of the SARS–COV–2 virus, most prominent the B.1.1.7 strainwith higher infectivity. In this article we consider a basic two–strain SIR modelto explain the spread of those variants in Germany on small time scales. For alinearized version of the model we calculate relevant variables like the time ofminimal infections or the dynamics of the share of variants analytically. Theseanalytical approximations and numerical simulations are in a good agreementto data reported by the Robert–Koch–Institute (RKI) in Germany.
Keywords:
COVID–19, Epidemiology, Disease dynamics, Multi–strain model.
1. Introduction
The current COVID–19 pandemic is striking across the world and has putEurope at the dawn of its third wave. In Germany due to the rising numbers atthe end of the year 2020, the non-pharmaceutical intervention (NPI) measureshave been strengthened, leading to a severe lockdown with closing of the mainparts of the daily life. With the reestablishment of the NPIs the reports of newstrains of the SARS–COV–2 virus throughout Europe were rising [1]. Especiallya variant called B.1.1.7 that was first reported in Great Britain [4, 2], showedan increased infectivity[3] with a higher attack rate especially in the youngerage groups. In the last days (February 21, 2021) the incidences are stagnat-ing or slowly increasing, although Germany has not eased the lockdown. Oneexplanation among experts and media is the rising of incidences with the newmutations.In various countries B.1.1.7. is rapidly spreading, see [5] for an overview. In ∗ Corresponding author.
Email addresses: [email protected] (Thomas G¨otz), [email protected] (Wolfgang Bock), [email protected] (Robert Rockenfeller), [email protected] (Moritz Sch¨afer) B . . . v a r i a n t i n % EnglandDenmarkPortugalNetherlandsGermany
Figure 1: Share of the B.1.1.7 variant of the SARS–COV–2 virus in five European countries.Week zero corresponds to the week when the share is approximately 1%.
Figure 1 we show the share of this new strain with respect to analyzed SARS–COV–2–positive tests in a given week in five European countries. Week zero isdefined as the week, when this share was approximately 1%. All five countriesfollow a general logistic trend. The curves for England (blue), Netherlands(orange) and Germany (black) are rather similar, where as Denmark (red) andPortugal (green) also behave similar but slower than the first three. Withinthis paper we will formulate a SIR–based model that predicts these curves andexplains well the observed data in Germany on a small time horizon.
2. Mathematical Model
We consider an SIR–model for the spread of two strains of the SARS–COV–2virus within a constant and serologically naive population. The two competingstrains 1 and 2 are assumed to have different transmission rates β , β > γ >
0. Assuming no secondary infections, the model isbased on the four compartments: susceptible S , infected I and I , indicatingstrain 1 or 2,respectively, and removed R . Neglecting demographic effects like2irth and death, we get S ′ = − ( β I + β I ) SN , (1a) I ′ = β I N S − γI , (1b) I ′ = β I N S − γI , (1c) R ′ = γ ( I + I ) . (1d)For the following analysis and simulations we assume a situation that modelsthe competition between the original SARS–COV–2 virus and mutated variantslike B.1.1.7 that is currently observed in many countries throughout Europe.The second (mutated) strain has a higher infection rate, i.e. β > β . However,at the initial time, the original strain 1 is still dominant in the populationi.e. I (0) > I (0). Current non–pharmaceutical interventions are strict enoughto suppress the original strain, i.e. to force its reproduction number below theepidemic threshold R := β γ <
1. However the mutated strain 2, due to itshigher infectivity, might reach a reproduction number R > • When will strain 2 dominate the dynamics? After what time T ∗ do weobserve I ( T ∗ ) > I ( T ∗ )? • How does the total number of infected I = I + I evolve in time? Atwhat time ˇ T do we observe a local minimum of the infections?In our model, we neglect the effect of possible vaccinations, that might havedifferent efficiency with respect to the two strains.
3. Analysis
Let N denote the constant total population. We rescale the populations s = S/N , x = I /N , y = I /N and r = R/N and introduce a non–dimensionaltime γt . Then we get s ′ = − ( R x + R y ) s (2a) x ′ = ( R s − x (2b) y ′ = ( R s − y (2c) r ′ = x + y (2d)where R i = β i /γ . Setting y = zx , where z denotes the ratio between infectedwith strains 1 and 2, we get z ′ = ( R − R ) sz (3a)3ith the solution z ( t ) = z exp (cid:20) ( R − R ) Z t s ( t ) dt (cid:21) . (3b)In case of R > R , the ratio between the two strains is going to tend towardsstrain 2, i.e. z > In case of dominating susceptibles, i.e. s ≈ x ′ = ( R − x (4a) y ′ = ( R − y , (4b)and we are able to solve them explicitly for the infected compartments x, y . Forboth compartments we observe an exponential behavior; however since R < x is dying out and compartment y is exponentially growingdue to R >
1. The ratio z of the two strains exhibits an exponential increase z ( t ) = z e ( R −R ) t . (5)In this setting we can easily answer the initial questions posed in section 2:1. Strain 2 will ”overtake” strain 1 at time T ∗ , i.e. z ( T ∗ ) = 1. In thelinearized model (4) this time T ∗ is given by T ∗ = − ln z R − R > z < R > R .2. The total infected attain a local minimum at time ˇ T when ( x + y ) ′ ( ˇ T ) = 0.In the linearized model (4) it holds that x ′ + y ′ = ( R − x + ( R − zx and hence z ( ˇ T ) = z e ( R −R ) ˇ T = −R R − >
0. So we arrive atˇ T = 1 R − R ln 1 − R z ( R −
1) = T ∗ · ln 1 − R R − z = z ( ˇ T ) = 1 − R R − . (7b)The minimal number of infected is given by( x + y ) min = x e ( R −
1) ˇ T (1 + ˇ z )= x (cid:20) − R z ( R − (cid:21) ( R − / ( R −R ) · R − R R − p = y/ ( x + y ) = z/ (1 + z ) of the second strain y withrespect to the total infected x + y satisfies in the linearized setting the followinglogistic relation p = z e ( R −R ) t z e ( R −R ) t = z z + e − ( R −R ) t ∈ [0 , . (8)In the linear, single strain model x ′ = ( R − x the reproduction numbersatisfies the relation R = 1 + ddt ln x . Hence we may define analogously the current reproduction number R ( t ) for thetotal infected as R ( t ) := 1 + ddt ln( x + y ) . (9)Using the solution of the linear model x + y = x e ( R − t + x z e ( R − t weobtain the convex combination R ( t ) := (1 − p ) R + p R = R + R − R z e − ( R −R ) t , (10)i.e. a logistic behavior switching between R for t → −∞ and R for t ≫ T , when the total number of infected attains its minimum, the currentreproduction number crosses the stability threshold, i.e. R ( ˇ T ) = 1. For thenon–linear model, the overall behavior of the reproduction number is similar,despite the saturation effect due to the decreasing pool of susceptibles.
4. Simulations
For our simulations, we assume the following data roughly resembling thesituation in Germany by mid of January to mid of February:1. The total population equals to N = 83 millions including 3 millions ofrecovered or vaccinated and x ≃ .
000 infected with variant 1 and y = z · x infected with strain 2.2. The recovery period is assumed to be 1 /γ = 5 days.3. Strain 1 has a reproduction number of R = 0 .
85, i.e. the current lock-down measures are strict enough to mitigate the original strain.4. The mutated strain 2 is assumed to be 50% more infectious, i.e. R =1 . · R = 1 .
275 and hence spreads in time.5. At the initial time (25 January) we assume that only 3% of cases belongto strain 2, i.e. z = 0 . .
6% and in week 6 alreadyaround 22% of infections with strain 2.5 d a y i n c i d e n c e RKI-DataIncidence 35Strain 1Strain 1+2R ± 0.1 Figure 2: Incidence (per 100 .
000 inhabitants in 7 days) of strain 1 (violet) and of both strainscombined (blue) as predicted by the SIR–model (2). Parameters are R = 0 . R = 1 . γ = 1 /
5. The green dots indicate incidences for entire Germany. The shaded area indicatesthe simulation range, if R = 1 . ± .
1. The orange dash line indicates an incidence of 35that is viewed in Germany as a limit for relaxing the current lockdown.
Using the approximations (6) and (7) from the linearized model, strain 2will dominate strain 1 at T ∗ = − ln z R −R = 8 . ≡
41 days. The minimal numberof infected is expected to be ( x + y ) min = 0 . · x at time ˇ T = 6 . ≡
34 days.The non–linear model (2) cannot be solved analytically; hence we performnumerical simulations based on the parameter given above. Figures 2, 3 showthe dynamics of both strains and the current reproduction number based onEqn. (9). The weekly incidences (new infections per 100 .
000 inhabitants within7 days) are shown in Figure 2. The green dots indicate the reported data,see. [10]. The violet curve shows the decay of the original strain 1 with isreproduction number R = 0 . <
1. The blue curve shows the total incidenceof both strains combined. At around 8 March, i.e. 6 weeks after the starting timeof the simulation (25 January), the total number of infected reaches its minimumwith an incidence of about 44 per 100 . R ± .
1) of the reproduction number of the second strain.Based on this simulation, the political target to push the infections below thethreshold of 35 before introducing relaxation measures seems questionable; atleast on a short time horizon. 6 R ( t ) current R-valueR ± 0.1RKI-data for R-interval Figure 3: Current value of the effective reproduction number (9) for both strains. The greendots indicate the RKI–data for the 7day–R0 together with the reported confidence interval.The shaded area indicates the simulation range, if R = 1 . ± . Figure 3 shows the current reproduction number for both strains combinedas defined in Eqn. (9). Again, the green dots and error bars show the 7–dayreproduction number as reported by RKI in its daily situation reports [11].The blue curve shows our simulation results based on the non–linear model (2)and again the shaded area indicates the uncertainty due to variation of thereproduction number of the second strain ( R ± . R ( t ) = 1 as predicted by our model. The non–pharmaceutical interventionsimposed by the government have not been altered in during the time spancovered by the simulations, hence we may explain the increase of the overallreproduction number by growing influence of the second strain.Figure 4 shows the relative share p of strain 2 with respect to the totalinfections with SARS–COV–2 in Germany. The green dots indicate the datareported by RKI for week 4 and week 6, see [9]. The blue curves show our pre-dictions; the dashed one corresponds to the approximation (10) in the linearizedsetting and the solid one corresponds to the non-linear SIR model. Both resultsdo not differ significantly and the linearized model already predicts quite wellthe dynamics of the relative share of the second strain compared to all infec-tions. Both results are within the range of the given data. Again, the shadedarea indicates the uncertainty caused by variations in the reproduction numberfor the second strain. For the beginning of March we expect more than 40% ofinfections with the second strain. 7 S t r a i n i n % SIR-modellinearized modelRKI DataR ± 0.1 Figure 4: Simulation for of the relative share p = y/ ( x + y ) of the second strain. Reproductionnumbers R = 0 . R = 1 . R = 1 . ± .
5. Conclusions and Outlook
In this work we have presented a two–strain SIR model to explain the spreadof SARS–COV–2 variants like B.1.1.7 in Germany. For a linearized version ofthe model we were able to calculate relevant variables like the time of minimalinfections or the dynamics of the share of variants analytically. These analyt-ical approximations as well as simulations for the non–linear SIR model arecompared to infection data reported by RKI. Our model shows a good level ofagreement and gives rise to some concern regarding the near term future of thedynamics. For mid of March we expect to see in Germany a share of at least40% of variants. Moreover, the figure of an incidence of 35 per 100 .
000 and 7days, which was introduced by politics as limit for easing the current lockdownmeasures, seems out of reach.In a follow–up study we will try to investigate the effect of the currentramping up of mass vaccinations. One might expect, that vaccinations will helpto slow down the spread of the disease and hence force the level of incidencebelow a threshold that allows contact tracing by public health authorities.
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Daily Situation Reports , .9 R _ - R _ ( e s t i m a t e d ) EnglandDenmarkPortugalNetherlandsGermany .25 1.3 1.35 1.4 1.45 1.5 r T =0.7 =0.75 =0.8 Time t
Strain 1Strain 2 V a r i a n t i n %%