aa r X i v : . [ phy s i c s . s o c - ph ] A ug Exact Properties of SIQR model for COVID-19
Takashi Odagaki ∗ Kyushu UniversityNishiku, Fukuoka 819-0395, JapanandResearch Institute for Science Education, Inc.Kitaku, Kyoto 603-8346, Japan
August 18, 2020
Abstract
The SIQR model is reformulated where compartments for infected and quarantinedare redefined so as to be appropriate to COVID-19, and exact properties of the model arepresented. It is shown that the maximum number of infected at large depends strongly onthe quarantine rate and that the quarantine measure is more effective than the lockdownmeasure in controlling the pandemic. The peak of the number of quarantined patientsis shown to appear some time later than the time that the number of infected becomesmaximum. On the basis of the expected utility theory, a theoretical framework to find out anoptimum strategy in the space of lockdown measure and quarantine measure is proposed forminimizing the maximum number of infected and for controlling the outbreak of pandemicat its early stage.
Since November 2019, the pandemic COVID-19 is still expanding in the world, infected ofwhich totals more than 15 million at 22 nd of July, 2020[1]. It is an important problem of socio-physics to construct a simple model by which one can understand the nature of the outbreak,and to construct a theoretical framework for formulating the optimum strategy to control it.[2]1pidemics can be considered to be a problem of physics concerning reaction and relaxationprocesses and the simplest understanding of its outbreak can be provided by a mean field analy-sis. The SIR model[3] assumes three compartments (or species) in population, susceptible (S),infected (I) and removed (R), and the infection is transferried from an infected individual toa susceptible individual. The number of symptomatic patients, which is identical to the num-ber of infected, decreases in the community by treatment and/or quarantine. The SIR modelis considered a standard model to explain the infection trajectory of ordinary epidemics likeinfluenza.COVID-19 has unusual characteristics: (1) transmission of the virus by presymptomatic pa-tients and (2) existence of asymptomatic infectious patients, and (3) patients, symptomatic orasymptomatic, can be identified by PCR test. Because of these characteristics, the number ofinfected cannot be obtained directly and the number of daily confirmed new cases and its timedependence are the only essential observables. Therefore, COVID-19 showing these character-istics may not be represented properly by the SIR and the SEIR models which assume that thenumber of patients is known and do not treat quarantined patients as a compartment.The SIQR model [5, 6] is a compartmental model which represents a community by fourcompartments, assuming an additional compartment of quarantined (Q), and describes the trans-mission process by a system of ordinary nonlinear differential equations. The SIQR modelseems to be appropriate to COVID-19 and it has been successfully applied in the analysis of theearly stage of the outbreak of COVID-19 in Italy[7], India[8, 9], Sweden[10] and Japan[11].In this paper, I redefine the SIQR model so as to make it appropriate to COVID-19[11].Namely, I classify patients into two groups; (1) infected patients at large (I) who can be in anyof three states, presymptomatic, symptomatic and asymptomatic and (2) quarantined patients(Q) who are in a hospital or self-isolated at home and no longer infectious in the community. Itreat the number of daily confirmed new cases explicitly and consider the quarantine rate or the2raction of infected at large put in a quarantine or self-isolation as a key parameter which can bedetermined from the observation of the daily confirmed new cases. I also discuss the optimumstrategy for controlling the pandemic.This paper is organized as follows. First, I explain in Sec. 2 the SIQR model and discussits relevancy to COVID-19. I also present the basic properties of the model, showing parame-ter dependence of the maximum number of infected. Section 3 considers the exact solution ofthe SIQR model and shows the time dependence of various quantities including the number ofquarantined. A theoretical frame work for optimizing measures to control the outbreak is dis-cussed in Sec. 4, where the expected utility theory[12] is exploited. The frame work is appliedfor optimizing strategy for reducing the epidemic peak and for stamping out the epidemic asfast as possible. Results are discussed in Sec. 5.
The basic concept of the SIQR model is identical to the chemical reaction, which can be de-scribed by rate equations. The dynamics of the SIQR model is given by the following set ofordinary nonlinear differential equations: dxdt = − βx ( t ) y ( t ) , (1) dydt = βx ( t ) y ( t ) − qy ( t ) − γy ( t ) , (2) dwdt = qy ( t ) − γ ′ w ( t ) , (3) dzdt = γy ( t ) + γ ′ w ( t ) , (4)where t is the time, and x ( t ) , y ( t ) , w ( t ) and z ( t ) are the fractions of population ( N ) in each com-partment, susceptible, infected at large, quarantined patients and recovered (and died) patients.3ere, I assumed that new patients immediately after they get infected cannot be quarantinedsince the incubation period is long for COVID-19. The parameters of this model are a transmis-sion coefficient β , quarantine rate of infected at large q , and recovery rates γ and γ ′ of infectedat large and quarantined, respectively. These parameters can be estimated from observations; β , γ and γ ′ from epidemiological survey and q from the time dependence of the daily confirmednew cases. Although time can be scaled by one of parametes, I use one day as a unit of time inthis paper so that ordinary citizens and policy makers can understand results without difficulties.Infected at large, regardless whether they are symptomatic or asymptomatic, are quarantinedat a per capita rate q and become non-infectious in the community. Quarantined patients recoverat a per capita rate γ ′ (where /γ ′ is the average time it takes for recovery) and infected at largebecome non-infectious at a per capita rate γ (where /γ is the average time that an infectedpatient at large is capable of infecting others). It is apparent that Eqs. (1) ∼ (4) guaranteee theconservation of population x ( t ) + y ( t ) + w ( t ) + z ( t ) = 1 .If one considers quarantined and recoverd together as removed, the set of differential equa-tion is the same as the set of equations for the SIR model with removal rate of infected q + γ .Since the value of q depends strongly on government policies and the only observable is thenumber of quarantined patients ∆ w ( t ) = qy ( t ) on each day, it is important to treat the quaran-tined and recovered patients separately in the analysis of the outbreak of COVID-19.Figure 1 shows the elementary processes of the SIQR model. From Eqs. (1) and (2), it is easy to show that the trajectory in the ( x, y ) plane is determined by dydx = q + γβx − . (5)4igure 1: Elementary processes of the SIQR model. x and y obey a reaction x + y → y .Therefore, the trajectory in the ( x, y ) plane is given by y = 1 − x + q + γβ ln x, (6)where the initial condition is set to y = 0 at x = 1 . Figure 2 shows the trajectories (a) forvarious β at q = 0 . and (b) for various q at β = 0 . when γ = 0 . .The peak position ( x ∗ , y ∗ ) of the trajectory can be obtained from Eqs. (5) and (6) by setting dydx = 0 . I find that x ∗ = q + γβ , (7) y ∗ = 1 − q + γβ + q + γβ ln q + γβ . (8)Figure 3 shows the dependence of y ∗ on β and q : (a) three dimensional plot of y ∗ ( β, q ) , (b)the β dependence at q = 0 . and (c) the q dependence at β = 0 . when γ = 0 . . The peakheight becomes lower for smaller β and enhanced q , and the latter is more effective in reducingthe peak. 5 q = 0 0.1 0.2 y x (a) (b)Figure 2: Trajectories in the ( x, y ) plane. (a) for various β when q = 0 . and (b) for various q when β = 0 . , where γ = 0 . . Combining Eqs. (3) and (4) together, I obtain d ( w + z ) dt = ( q + γ ) y ( t ) . (9)Therefore, the set of Eqs. (1), (2) and (9) is identical to the basic equations of the SIR model asstated before, and the exact solution can be written as[13] x ( t ) = x u ( t ) , (10) y ( t ) = 1 − x u ( t ) + q + γβ ln u ( t ) , (11) w ( t ) + z ( t ) = − q + γβ ln u ( t ) , (12)where time t is related to u through an integral t = Z u dξξ [ x βξ − β − ( q + γ ) ln ξ ] . (13)6a)(b) (c)Figure 3: Peak heights of infected at large as a function of β and q . (a) Three dimensional plot,(b) for various q when β = 0 . and (c) for various β when q = 0 . , where γ = 0 . .7 x(t)y(t) w(t)z(t) x ( t ) , y ( t ) , w ( t ) , z ( t ) t Figure 4: The exact time dependence of x ( t ) , y ( t ) , w ( t ) and z ( t ) when β = 0 . , q = 0 . and γ = γ ′ = 0 . . The peak positions of y ( t ) and w ( t ) differ about 10 days for the choice ofparameters.Once y ( t ) is known, then w ( t ) is obtained from Eq. (3), w ( t ) = qe − γ ′ t Z t e γ ′ t ′ y ( t ′ ) dt ′ , (14)and thus z ( t ) is given by z ( t ) = − q + γβ ln u ( t ) − qe − γ ′ t Z t e γ ′ t ′ y ( t ′ ) dt ′ . (15)Figure 4 shows the time dependence of x ( t ) , y ( t ) , w ( t ) and z ( t ) for β = 0 . , q = 0 . and γ = γ ′ = 0 . . It is interesting to note that the number of quarantined patients w ( t ) takes itsmaximum at some time later than the time that the number of infected y ( t ) becomes maximum.The observable in COVID-19 is the daily confirmed new cases ∆ w ( t ) = qy ( t ) , which issimply q -times smaller than y ( t ) as shown in Fig. 5.8igure 5: The time dependence of y ( t ) and ∆ w ( t ) for β = 0 . , q = 0 . and γ = γ ′ = 0 . . Since x ( t ) ≃ x and y ≃ in the early stage of outbreak, I introduce a new integration variable η = 1 − ξ in Eq. (13) t = − Z − u dη (1 − η )[ x β (1 − η ) − β − ( q + γ ) ln(1 − η )] . (16)Taking up to the first order term in − u in the integrand on the right-hand side of Eq. (16) andup to the first order term in y , I obtain u ( t ) = 1 + y βq + γ − β (cid:16) e ( β − q − γ ) t − (cid:17) . (17)Therefore, the short-term solution of the SIQR model is given by x ( t ) = x − y ββ − q − γ (cid:16) e ( β − q − γ ) t − (cid:17) , (18) y ( t ) = y e ( β − q − γ ) t , (19) w ( t ) = y qβ − q − γ + γ ′ (cid:16) e ( β − q − γ ) t − e − γ ′ t (cid:17) , (20)9 ( t ) = y ( q + γ ) β − q − γ (cid:16) e ( β − q − γ ) t − (cid:17) − y qβ − q − γ + γ ′ (cid:16) e ( β − q − γ ) t − e − γ ′ t (cid:17) . (21)These solution can also be obtained by setting x ( t ) = x = 1 in Eqs. (1) ∼ (4) and have beenused in the analysis of the outbreak of COVID-19 [7, 8, 9, 11].The short-term solution indicates that the initial growth rate of the number of infected atlarge is determined by λ = β − q − γ. (22)In order to control the outbreak, a measure must be formulated to make the growth rate λ negative under various restrictions in economic activities and medical care systems. As shown in Fig. 4, the present model like the SIR model shows that the epidemic curve willconverge to an equilibrium state after several months, passing a maximum number of infectedwhich could be ∼ % of population depending on the value of parameters. This means thatif we wait the natural epidemic equilibrium, the number of causalities will become unaccept-ably large. In order to reduce the number of causalities, all governments in the world have beenstruggling against COVID-19 with various measures. Lockdown and social distancing are mea-sures to reduce the effective transmission coefficient β , but it has a severe damage on economicactivities. For COVID-19, quarantine measure including self-isolation has been employed inmany countries to increase the quarantine rate q .Since γ in the SIQR model cannot be altered, β and q are the essential parameters whichcan be modified by policy. I parameterize β as β = (1 − a ) β where a represents the strengthof lockdown measure; a = 0 corresponds to no measure on social distancing and a = 1 denotescomplete lockdown. I consider a measure M = M ( a, q ) which is a function of a and q . The10ransmission rate β can be determined from the growth rate of the epidemic at the earliest stagewhen no measures are imposed.I consider a cost function C ( a, q ) of a measure characterized by a and q which must be anincreasing function of a and q . The problem is to move M in a desired direction, making thecost as small as possible. Namely, for a given M , an optimum set ( a, q ) minimizing the costis obtained which in turn determines the optimum trajectory in the ( a, q ) space. I introduce aLagrange multiplier µ and consider a function F ( µ, a, q ) defined by F ( µ, a, q ) = C ( a, q ) − µ [ M − M ( a, q )] . (23)It is well known that the optimum value of C ( a, q ) is given by the solution to ∂ F ∂a = 0 , (24) ∂ F ∂q = 0 , (25) ∂ F ∂µ = 0 . (26)In the following discussion, I consider a model cost function in an arbitrary unit[14] C ( a, q ) = a + k qβ ! . (27)Here, k is a parameter characterizing relative importance of measures. When k = 1 , bothmeasures cost equally. When k > , cost for medical treatment is larger than the economiccost, and when k < , economic cost due to social distancing is larger than the medical cost. As shown in Fig. 3, the epidemic peak and hence the number of quarantined strongly dependon parameter a and q . Taking Eq. (8) as a measure, I set M ( a, q ) = 1 − q + γ (1 − a ) β + q + γ (1 − a ) β ln q + γ (1 − a ) β . (28)11t is straightforward to find that the optimum trajectory ( a ∗ , q ∗ ) satisfies a ∗ (1 − a ∗ ) β = kq ∗ ( q ∗ + γ ) . (29)Figure 6 shows the optimum strategy for k = 2 , , . . In this subsection, I discuss an optimum measure to stamp out the epidemic at the beginning ofthe outbreak. The growth rate of the number of infected at large is given by Eq. (22) in the earlystage of the outbreak, and I set M ( a, q ) = (1 − a ) β − q − γ. (30)The aim of a measure is to bring the state from the region M ( a, q ) > at ( a = 0 , q = 0) tosome region M ( a, q ) < .It is straightforward to find that the optimum trajectory ( a ∗ , q ∗ ) satisfies a ∗ = k q ∗ β . (31)Figure 7 shows the optimum trajectory on the ( a, q ) plane: (a) k = 2 , (b) k = 1 and (c) k = 0 . . In this paper, I presented exact properties of the SIQR model relevant to COVID-19 in the entiretime span and in the early stage of the outbreak. In particular, I investigated dependence of theoutbreak on parameters, transmission coefficient and quarantine rate, which can be controlledby measures. It is important to note that the peak of the number of quarantined patients w ( t ) appears about 10 days later than the time that the number of infected y ( t ) becomes maximum,12a)(b)(c)Figure 6: The solid thick curve shows the trajectory of the optimum strategy to reduce theepidemic peak for γ = 0 . . Straight lines are constant y ∗ lines: y ∗ = 0 . , . , . , . , . , . from left to right, and oval curves represent constant cost curves: C = 0 . , . , . , . , from left to right. (a) k = 2 , (b) k = 1 and (c) k = 0 . .13a)(b)(c)Figure 7: The increasing thick straight line is the trajectory of the optimum strategy to stampout the epidemic for γ = 0 . . Decreasing straight lines shows the constant growth rate linesfrom 0.7 (the lowest line)to -0.4 (the highest line), and the dashed line shows λ = 0 . Ovalcurves represent constant cost curves: C = 0 . , . , . , . , from left to right. (a) k = 2 ,(b) k = 1 and (c) k = 0 . . 14or the choice of parameters used in Fig. 4. Within this model, the number of infected at large y ( t ) can be estimated from the daily confirmed new cases ∆ w ( t ) , once the quarantine rate q isobtained from the infection trajectory. I also discussed a theoretical framework for optimizingmeasures to control the outbreak on the basis of the expected utility theory, when the social costis given by a function of the social distancing policy and the quarantine measure. The optimumstrategy depends on the aim; reducing the epidemic peak or accelerating the stamping out theoutbreak. It should be emphasized that a simple lockdown ( a = 1 ) is not the optimum strategy.Given the cost function and the aim of policy in each country, it will be possible to formulatethe optimum policy specific to the country for controlling the outbreak on the basis of the presenttheoretical framework.The SIQR model does not consider any memory effects of the epidemic like the incubationperiod and the infectious period functions. 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