Exact closed-form solution of a modified SIR model
EExact closed-form solution of a modified SIR model
Angel Ballesteros , Alfonso Blasco , Ivan Gutierrez-Sagredo , Departamento de F´ısica, Universidad de Burgos, 09001 Burgos, Spain Departamento de Matem´aticas y Computaci´on, Universidad de Burgos, 09001 Burgos, Spaine-mail: [email protected], [email protected], [email protected]
Abstract
The exact analytical solution in closed form of a modified SIR system is presented. This is, to the bestof our knowledge, the first closed-form solution for a three-dimensional deterministic compartmental model ofepidemics. In this dynamical system the populations S ( t ) and R ( t ) of susceptible and recovered individuals arefound to be generalized logistic functions, while infective ones I ( t ) are given by a generalized logistic functiontimes an exponential, all of them with the same characteristic time. The nonlinear dynamics of this modifiedSIR system is analyzed and the exact computation of some epidemiologically relevant quantities is performed.The main differences between this modified SIR model and original SIR one are presented and explained interms of the zeroes of their respective conserved quantities. We recall that both models have been recently usedin order to describe the essentials of the dynamics of the COVID-19 pandemic. Keywords: epidemics, compartmental models, dynamical systems, Casimir functions, exact solution, SIR
The dynamics of an epidemic outbreak is intrinsically nonlinear, and this nonlinearity is obviously reflected in anydynamical system amenable to model epidemics. The simplest among these models are the so-called deterministiccompartmental ones, and a large number of them have been recently considered in relation to the COVID-19pandemic (see, for example, [1, 2, 3] and references therein).While exact solutions have been found for some two-dimensional compartmental systems, like for instance theSIS (susceptible-infective-susceptible) model [4, 5, 6], this is by no means the case for three-dimensional ones.We stress that for an infectious disease that produces some kind of immunity, like COVID-19, two-dimensionalmodels do not appropriately describe the dynamics since a third compartment is needed in order to account forthe recovered individuals that do not become immediately susceptible.Among three-dimensional models, the well-known SIR (susceptible-infective-recovered) model˙ S = − β S I, ˙ I = β S I − α I, ˙ R = αI, (1)proposed by Kermack and McKendrick [7] is probably the best known one. Despite its apparent simplicity, it hasbeen succesfully used to predict relevant features of the dynamics of a number of epidemics, including the actualCOVID-19 pandemic [8, 9]. Nevertheless, the system (1) does not admit an explicit exact analytic solution inclosed form [10] (see [11, 12, 13, 14] for different approaches to the mathematical properties of the SIR solutions).Moreover, this lack of an explicit closed-form solution is not a unique feature of the SIR model, but seemed to be acommon feature of three-dimensional (and higher dimensional) compartmental epidemiological models. Therefore,the study of their dynamics typically relies on techniques from dynamical systems theory and numerical studies.Although these techniques allow a deep understanding of their associated nonlinear dynamics, the simplicity andaccurateness provided by exact simple solutions would be indeed helpful both from the mathematical and theepidemiological perspectives.In this letter we present the exact analytical solution in closed form of the modified SIR system˙ S = − β S IS + I , ˙ I = β S IS + I − α I, ˙ R = α I, (2)1 a r X i v : . [ q - b i o . P E ] J u l here α, β ∈ R + . This system has been proposed [15, 16, 17] as a more realistic model than (1) when the recoveredindividuals are removed from the population (not only due to death, but also to quarantine or other reasons). Thegeneral solution of this modified SIR system is S ( t ) = S (cid:18) S + I S + I e t/τ (cid:19) βτ , I ( t ) = I (cid:18) S + I S + I e t/τ (cid:19) βτ e t/τ , R ( t ) = 1 − ( S + I ) βτ ( S + I e t/τ ) βτ − , (3)where τ = ( β − α ) − .This is, to the best of our knowledge, the first exact solution of a three-dimensional compartmental epidemiolog-ical model in closed form. We also analyze the model and show that, in the typical range of the model parameters,the dynamics of (2) is actually quite close to the one of the SIR model (1).In the next Section we derive this exact solution by making use of the fact that any epidemiological three-dimensional model has a conserved quantity, which in turn is straightforwardly derived from the the more generalresult (recently proved in [12]) stating that any three-dimensional compartmental epidemiological model is a gen-eralized Hamiltonian system. Moreover, the conserved quantity turns out to be just the Casimir of the Poissonalgebra of the underlying Hamiltonian structure. In Section 3 we present the analysis of the modified SIR sys-tem (2) both from a dynamical systems approach and from a Poisson–algebraic point of view, and we show thatthe exact solution (3) is helpful in order to obtain some relevant epidemiological quantities in a simple and exactform. Finally, the main differences between the SIR and modified SIR systems are analysed in Section 4, where weshow that these differences can be understood in terms of the the zeroes of their respective conserved quantities,which are again the Casimir functions for both models that are obtained through the formalism presented in [12]. In order to find the exact solution of (2) we make use of the following recent result (see [12] for details).
Proposition 1. [12] Every epidemiological compartmental model with constant population is a generalized Hamil-tonian system, with Hamiltonian function H given by the total population. For the system (2) the generalized Hamiltonian structure is thus explicitly provided by the Hamiltonian function H = S + I + R, (4)together with the associated Poisson structure, which is found to be given by the fundamental brackets { S, I } = 0 , { S, R } = − β S IS + I , { I, R } = β S IS + I − α I , (5)and leads to the system (2) through Hamilton’s equations˙ S = { S, H} , ˙ I = { I, H} , ˙ R = { R, H} . (6)Since every three-dimensional Poisson structure has a Casimir function C , i.e. a function C : U ⊆ R → R suchthat { S, C} = { I, C} = { R, C} = 0, then C is a conserved quantity for any generalized Hamiltonian system (6)defined on such a Poisson manifold. Therefore: Corollary 1.
Every three-dimensional epidemiological compartmental model with constant population has a con-served quantity, which is functionally independent of the Hamiltonian function.
Note that in case that H (4) is functionally dependent of C , the dynamics (6) would be trivial. For the specificPoisson algebra (5) the Casimir function is found to be C = S − αβ ( S + I ) . (7)We can use this Casimir function to restrict the dynamics of (2) to the symplectic leaf defined by the value of C given by the initial conditions S (0) = S , I (0) = I , R (0) = R , namely C = S − αβ ( S + I ) . (8)2his can be also used in order to reduce the system (2) to a nonlinear ODE, since from (7) and (8) we obtain thephase space equation I ( S ) = ( S + I ) (cid:18) SS (cid:19) αβ − S, (9)which can be inserted within (2) in order to get the following nonlinear ODE for the variable S :˙ S = − βS (cid:32) − S α/β S + I S − α/β (cid:33) . (10)This ODE suggests the change of variable A ( t ) = S ( t ) − α/β , (11)thus obtaining ˙ A = − ( β − α ) A (cid:32) − S α/β S + I A (cid:33) . (12)If we now set B ( t ) = S α/β S + I A ( t ) , (13)we obtain ˙ B = − ( β − α ) B (1 − B ) . (14)The general solution to this ODE is a logistic function with characteristic time τ = ( β − α ) − , i.e. B ( t ) = 11 + e ( β − α ) t + d = 11 + e t/τ + d . (15)The integration constant d is fixed by the initial condition B (0) = S S + I , thus obtaining e d = I S . Therefore wecan write B ( t ) = 11 + I S e t/τ . (16)Now, inverting the change of variables (13) we get A ( t ) = ( S + I ) S /βτ S + I e t/τ , (17)and finally, from (11), we obtain S ( t ) = S (cid:18) S + I S + I e t/τ (cid:19) βτ . (18)From the phase space equation (9) we directly get I ( S ) = ( S + I ) (cid:18) SS (cid:19) α/β − S, (19)and inserting (18) we are able to obtain I ( t ) without any further integration. Finally, we have that I ( t ) = I (cid:18) S + I S + I e t/τ (cid:19) βτ e t/τ . (20)Note that I ( t ) is related to S ( t ) by I ( t ) = I S S ( t ) e t/τ , (21)and from the conservation of the total population, we find R ( t ) = 1 − S ( t ) − I ( t ) = 1 − ( S + I ) βτ ( S + I e t/τ ) βτ − . (22)3ummarizing, equations (18) shows that the susceptible population follows a generalized logistic function,or Richards’ curve, with characteristic time τ and the relevant constants set to satisfy that S (0) = S andlim t →∞ S ( t ) = 0. Moreover, the dynamics of the infective population given by (20) is essentially this samefunction multiplied by an exponential with the same characteristic time. This is indeed a very natural dynamicsfor infective processes and, as we will see in the sequel, this dynamics strongly resembles the one described by thefamous SIR model (1), provided that the range of values for the parameters α and β is similar to the one found inactual epidemics. Remark 1
It is worth stressing that the method here presented is indeed applicable to any three-dimensionalcompartmental model, provided we are able to find the Casimir function of the associated Poisson structure.Nevertheless, the distinctive feature of the system (2) is that the resulting ODE admits a closed-form solution. Werecall that in [12] such Casimir function approach was used in order to find the solution for some epidemiologicalmodels in terms of an inverse function.
In this Section we briefly analyze the main features of the modified SIR system (2). Without any loss of generalitywe can assume that R = 0, so S + I = 1, and the solution of (2) reads S ( t ) = S (cid:0) S + I e t/τ (cid:1) βτ , I ( t ) = I e t/τ (cid:0) S + I e t/τ (cid:1) βτ , R ( t ) = 1 − (cid:0) S + I e t/τ (cid:1) βτ − . (23)As we have previously stated, the behavior of S ( t ) is that of a generalized logistic function while the evolutionof the infective population I ( t ) is given by a generalized logistic function times an exponential. This means thatsince βτ >
1, the logistic term dominates for large times and therefore lim t →∞ I ( t ) = 0. However during the firststage of the outbreak the exponential term is the dominating one, and thus the model presents the characteristicinfection peak (for appropriate values of the parameters α and β ). The behavior of the functions S ( t ) and I ( t ) fordifferent values of α and β is shown in Figure 1.A fundamental question to be answered by any epidemiological model is whether, for given values of theparameters, there will be an outbreak. For the modified SIR system we see, simply by evaluating the secondequation from (2) at t = 0, ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 I ( t ) = I (cid:18) βS S + I − α (cid:19) , (24)and the outbreak will exist if and only if βS > α ( S + I ), or equivalently βα − > I S , (25)which in the case S + I = 1 means that S > αβ . (26)Obviously, this same result can be obtained by checking the condition for which I ( t ) has a maximum. Moreover,the analytic solution allows us to exactly determine the time at which the infection peak t peak is reached, and weobtain t peak = τ log (cid:18) S I ( βτ − (cid:19) = τ log (cid:18) S I (cid:18) βα − (cid:19)(cid:19) , (27)which is positive if and only if (25) holds. The fraction of infected population at the infection peak reads I ( t peak ) = (cid:18) βτ − S (cid:19) βτ − (cid:18) S + I βτ (cid:19) βτ − = S (cid:18) βα − (cid:19) (cid:18) αβ (cid:18) I S (cid:19)(cid:19) β/ ( β − α ) . (28)Another interesting insight is gained by computing the area below the infective curve I ( t ). In order to do that,we do not even need to perform the integration of I ( t ), since from the third equation in (2) we get Area ( I ) = (cid:90) ∞ I ( t ) dt = 1 α (cid:90) ∞ ˙ R ( t ) dt = lim t →∞ R ( t ) − R (0) = S + I α = 1 α . (29)4his result is specially interesting from a parameter estimation point of view, since it allows to obtain a valuefor α directly from the data. Afterwards, assuming that S and I are known, β can be obtained, for instance,from (27). Thus, the exact solution in closed form greatly simplifies the fitting with actual data. Moreover, aswe will see below, since the dynamics of the SIR and modified SIR systems are quite close (for a realistic rangeof the parameters), this procedure for the determination of the parameters of the modified model provides a goodapproximation for the parameters of the SIR one.A related interesting quantity from the epidemiological point of view is the removal rate, defined by ˙ R ( t ) = αI .While for the SIR model it can only be approximated by a closed-form expression in certain limits (see [5]), in themodified SIR system it can obviously computed exactly. Therefore, the behaviour of the removal rate (divided by α ) for the modified SIR system can be directly extracted from Figure 1.For any epidemic outbreak, it is also enlightening to analyze the intersection of the susceptible S ( t ), infective I ( t ) and recovered R ( t ) functions. The closed-form solution of the modified SIR model allows us to get some exactresults in this respect, which we write down in the following Proposition 2.
For the modified SIR system given by (2) , with β > α and initial conditions S (0) = S , I (0) = I , R (0) = 0 such that S > I > and S > α/β , any two of the curves S ( t ) , I ( t ) and R ( t ) always intersect exactlyonce, regardless of the exact values of the initial conditions and parameters of the system.Furthermore:i) The curves S ( t ) and I ( t ) intersect before the infection peak if β > α , exactly at the infection peak if β = 2 α and after the infection peak if β < α .ii) The three curves S ( t ) , I ( t ) and R ( t ) intersect in a common point if and only if βα < log 3log 3 − log 2 and S = (cid:0) (cid:1) β/α .iii) The three curves S ( t ) , I ( t ) and R ( t ) intersect exactly at the infection peak if and only if β = 2 α and S = .Proof. The solution of (2) when R (0) = 0 is given by (23). In particular, the unique time at which the curves S ( t )and I ( t ) intersect can be explicitly computed from (21), and it reads t SI = τ log (cid:18) S I (cid:19) . (30)Since we are assuming S > I , this time is always positive.From (23) we can also compute the times t SR and t IR such that R ( t SR ) = S ( t SR ) and R ( t IR ) = I ( t IR ). It iseasy to check that these times are given by the common expression t ∗ = τ log (cid:18) X − S I (cid:19) , (31)where X is a solution of the equation X βτ − X − S = 0 (32)in the case of t SR , while X is a solution of the equation X βτ − X + S = 0 (33)in the case of t IR . An elementary computation shows that equation (32) has only one solution, and therefore t SR is unique. Equation (33) has 2 solutions, the first one living in (0 ,
1) and the second one within (1 , ∞ ). The firstof these solutions results in a negative or complex time, so t IR is defined by the unique solution of (33) in (1 , ∞ ).Therefore, we have proved that t SI , t SR and t IR are unique, which means that the curves S ( t ), I ( t ) and R ( t )intersect exactly once.Now, statement i ) derives straightforwardly from a comparison between (30) and (27).To prove ii ) we first note that t ∗ = t SI if and only if X = 2 S , which is a solution of both (32) and (33) if andonly if S = 13 (cid:18) (cid:19) β/α . (34)5iven that S <
1, then (cid:0) (cid:1) β/α <
1, and therefore βα < log 3log(3 / .Statement iii ) is a consequence of i ) and ii ), since in order that the intersection coincides with the infectionpeak we need that β = 2 α , and substituting this into the condition for triple intersection (34) we get S = .Equivalently, note that if β = 2 α , then by (28), S ( t peak ) = I ( t peak ) = S , so R ( t peak ) = 1 − S , and by imposingthat they coincide we get S = . Remark 2
The condition S > α/β in the previous Proposition is just the condition for the existence of aninfectious peak (26) but we are not using it explicitly in the proof. Therefore, all the previous results which do notinvolve the infection peak are true if this condition is removed. Remark 3
The fact that the functions S ( t ), I ( t ) and R ( t ) always intersect is a striking difference with respectto the original SIR system of Kermack and McKendrick (1). Finally, it is worth performing a more detailed analysis of the dynamics of the modified SIR system (2) whencompared to the original SIR model (1). Although the exact closed-form solution here obtained in the modifiedcase is valid for any values of α and β , for the sake of brevity from now on we only consider the case β > α (recallfrom (25) that this is the regime in which an actual outbreak does exist).Qualitatively, the most significant difference between both dynamical systems is the stability of their fixedpoints. From equations (1) and (2) we see that I = 0 is a line of non-isolated fixed points for both models. Thestability of any of these fixed points p = ( S,
0) is defined simply by the trace of the Jacobian evaluated at thispoint, J ( p ). For the SIR system this trace is Tr J ( p ) = βS − α, (35)while for the modified SIR system it reads Tr J ( p ) = β − α. (36)Therefore, for the modified SIR system all the points belonging to the line I = 0 (with the exception of ( S, I ) =(0 , β > α ). Meanwhile, for theSIR system points such that S > α/β are unstable, while points with
S < α/β are stable. This can be clearlyappreciated in Figures 2 and 3, where the corresponding flows are presented for both systems. Coloured curvescorrespond to the phase space equation I ( S ) for each model. Trajectories of the system starting at any point ofthe appropriate curve will follow this curve (in the direction of the flow) in order to reach the relevant fixed point.The differences regarding the fixed point structure of these two systems can also be analyzed algebraically. Forthe SIR system (1), it is well-known that the phase space equation is I ( S ) = αβ log S − S + C , (37)where C is a constant (this is the equation of the curves in Figure 2, for different values of C ). In fact, C = S + I − αβ log S (38)is the Casimir function for the associated Poisson structure (see [12] for details). It is easy to prove that theequation I ( S ) = 0 always have a solution S ∈ (0 , α/β ). However, the phase space equation I ( S ) = 0 for themodified SIR system, where I ( S ) is given by (9), always has S = 0 as a solution (see Figure 3). This equation isdirectly obtained from the Casimir function (7), and it is interesting to compare this Casimir function (7) with theexponential of (38). 6 % % % % % %
10 20 30 40 50 t0.10.20.30.40.5 %
10 20 30 40 50 t0.10.20.30.40.5 % Figure 1: S ( t ) (black) and I ( t ) (colored) functions for theSIR system (dashed) and the modified SIR system (solid). β = 1. Left: α = 0 .
2. Right: α = 0 .
6. Blue: S = 0 . I = 0 .
9. Orange: S = α/β , I = 1 − α/β . Cyan: S = 0 . I = 0 .
1. Magenta: S = 0 . I = 0 . Figure 2:
Phase space for the SIR system (1). β = 1. Left: α = 0 .
2. Right: α = 0 .
6. Blue line: S = 0 . I = 0 . S = α/β , I = 1 − α/β . Cyan line: S = 0 . I = 0 .
1. Magenta line: S = 0 . I = 0 . Red: Stablepoints. Green: Unstable points.
Figure 3:
Phase space for the modified SIR system (2). β = 1. Left: α = 0 .
2. Right: α = 0 .
6. Blue line: S = 0 . I = 0 .
9. Orange line: S = α/β , I = 1 − α/β . Cyan line: S = 0 . I = 0 .
1. Magenta line: S = 0 . I = 0 . Remark 4
The previous discussion shows that the different qualitative behaviour of the systems (1) and (2) canbe algebraically understood through the differences between the Casimir functions (7) and (38), and in particular,the different structure and location of their zeroes within the phase space.From the epidemiological point of view, the existence of stable fixed points (different from (0 , I = 0axis explains the well-known fact that in the original SIR model of Kermack and McKendrick the whole populationis not infected during the evolution of the infection. While these results can be obtained from a dynamical systemsapproach, it is interesting to note their direct connection with the algebraic and geometric structure of the Poissonmanifold underlying their description of epidemiological models as generalized Hamiltonian systems. Remark 5
For the modified SIR system, the closed-form analytical solution (3) contains all the previous infor-mation. Solutions with I = 0 are constant functions, andlim t →∞ S ( t ) = 0 , lim t →∞ I ( t ) = 0 , lim t →∞ R ( t ) = 1 , (39)for any initial conditions such that I (cid:54) = 0. This shows that the only stable fixed point is ( S, I ) = (0 , S ( t ) and I ( t ) contained in the phase space orbits from Figures 2 and3 are depicted. In the left column β = 1 and α = 0 . β = 1 and α = 0 .
6. Each plotcontains four different curves: coloured ones correspond to I ( t ) while black ones correspond to S ( t ), and solid onescorrespond to the modified SIR system while dashed ones correspond to the original SIR system. The first row7hows the dynamics for initial conditions such that the outbreak rapidly extinguishes. The second row shows thelimiting case given by S = α/β (note that this value is the same for both models since S + I = 1). The third rowshows the typical behavior for values of the parameters and initial conditions for which there is an actual outbreak,and therefore I ( t ) has a maximum. The fourth row shows a situation such that at the beginning of the outbreaka fraction of the total population is immunized.These plots show all possible qualitatively different dynamics for the SIR and modified SIR systems. Essentially,as far as the ratio α/β grows, stronger differences between both models arise. In the left column we have α/β = 1 / α/β = 3 / S ( t ). The most striking difference between both systems can be appreciated in thepicture located at the last row, second column, which corresponds to a small perturbation of the case when initiallyhalf of the population is immunized and the other half is susceptible to the infection. In this case, the SIR systempredicts no outbreak (it is a stable fixed point), while the modified SIR system does predict it. Obviously, this isdue to the fact that in the modified SIR system we are assuming that the recovered population has been removed(death, quarantine, etc) and therefore does not interact (thus not contributing to the so-called ‘herd immunity’,which is of course not attainable in this model). All these considerations can also be deduced from the phase spacerepresentation in Figures 2 and 3.However, it is important to stress that the epidemiologically most relevant scenario, at least for a new epidemiclike the COVID-19 one, in which there is no immunized individuals at the beginning (or they are very few ones),is given by the third row (cyan). So, we can conclude that in this scenario, specially when the ratio α/β is smaller,the SIR and modified SIR systems present similar features, with the modified SIR model always predicting a largerinfection peak that the SIR. Acknowledgements
This work has been partially supported by Ministerio de Ciencia e Innovaci´on (Spain) under grants MTM2016-79639-P (AEI/FEDER, UE) and PID2019-106802GB-I00/AEI/10.13039/501100011033, and by Junta de Castillay Le´on (Spain) under grants BU229P18 and BU091G19.
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