A reductive analysis of a compartmental model for COVID-19: data assimilation and forecasting for the United Kingdom
AA reductive analysis of a compartmental model for COVID-19: data assimilation andforecasting for the United Kingdom
G. Ananthakrishna , ∗ and Jagadish Kumar , ∗ Materials Research Centre, Indian Institute of Science, Bengaluru 560012, India Department of Physics, Utkal University, Bhubaneswar 751004, India
We introduce a deterministic model that partitions the total population into the susceptible,infected, quarantined, and those traced after exposure, the recovered and the deceased. We hy-pothesize ’accessible population for transmission of the disease’ to be a small fraction of the totalpopulation, for instance when interventions are in force. This hypothesis, together with the struc-ture of the set of coupled nonlinear ordinary differential equations for the populations, allows us todecouple the equations into just two equations. This further reduces to a logistic type of equationfor the total infected population. The equation can be solved analytically and therefore allows for aclear interpretation of the growth and inhibiting factors in terms of the parameters in the full model.The validity of the ’accessible population’ hypothesis and the efficacy of the reduced logistic modelis demonstrated by the ease of fitting the United Kingdom data for the cumulative infected anddaily new infected cases. The model can also be used to forecast further progression of the disease.In an effort to find optimized parameter values compatible with the United Kingdom coronavirusdata, we first determine the relative importance of the various transition rates participating in theoriginal model. Using this we show that the original model equations provide a very good fit withthe United Kingdom data for the cumulative number of infections and the daily new cases. Thefact that the model calculated daily new cases exhibits a turning point, suggests the beginning of aslow-down in the spread of infections. However, since the rate of slowing down beyond the turningpoint is small, the cumulative number of infections is likely to saturate to about 3 . × aroundlate July, provided the lock-down conditions continue to prevail. Noting that the fit obtained fromthe reduced logistic equation is comparable to that with the full model equations, the underlyingcauses for the limited forecasting ability of the reduced logistic equation are elucidated. The modeland the procedure adopted here are expected to be useful in fitting the data for other countries andin forecasting the progression of the disease. I. INTRODUCTION
The highly contagious SARS-CoV-2 has infected morethan six million people worldwide since its first detectionin China on December 31 [1]. The novel coronavirus isthe fourth wave in the class of coronaviruses. In less thantwo months, the virus has spread all over the world, pos-ing serious threats to healthcare systems and economies.The alarming speed of transmission, the virulence of thedisease, and the unprecedented high proportion of fatal-ities even in countries with high healthcare indices haveraised questions about what kind of interventions are ap- ∗ Corresponding [email protected]@utkaluniversity.ac.in propriate for a given setting. The wide variability ininfected numbers and fatalities in different countries andsettings has also brought into sharp focus a debate aboutthe underlying causes of the variability. In the absenceof any treatment for the disease and non-availability ofvaccines in the near future, policy makers have resortedto standard epidemiological interventions, such as socialdistancing, isolation, contact tracing, and quarantining,and more recently a complete lock-down.At a basic level, the purpose of all non-pharmacologicalinterventions is to control disease transmission by limit-ing the proportion of the population exposed to the virusas much as possible. Furthermore, inherent in the pro-cess of implementation of these interventions are delaysat each stage. The delay time-scales are specific to theparticular intervention. a r X i v : . [ q - b i o . P E ] J un The importance of mathematical models describing thespreading dynamics of infectious diseases has been rec-ognized since early days [2]. In particular, the fact thattimely models that include realistic features have oftenbeen helpful in decision making on healthcare issues iswell recognized [2–4]. In the short period since the emer-gence of the coronavirus, there have been several math-ematical models [5–17], to name a few. Some of thesemodels attempt to evaluate the contribution from differ-ent transmission routes, such as contact tracing and isola-tion [10, 11, 18], travel restrictions [16, 17] social distanc-ing [19, 20], lock-down measures [16, 21, 22], and a com-bination of these interventions [12, 23–26]. These modelsbroadly fall into three categories, deterministic, stochas-tic, and simulations. Several new mathematical tech-niques used in different disciplines have been employed togain insights, which would not be possible with the tradi-tional approaches in the field. These include the humanmobility model [16], differential evolution[19], heuristicoptimization technique [19], stochastic agent-based dis-crete time simulation [27], supply chain risk simulations[28], etc.One class of epidemiological models attempts to de-scribe the transmission dynamics by partitioning thepopulation into smaller subsets based on the disease sta-tus such as the susceptible, exposed, infected, quaran-tined, recovered, etc [5–9]. Most models of this kind ig-nore age-dependent infection and fatality rates, and theheterogeneous spatial distribution of the population. Ina sense, these models describe the evolution of the meanresponse of each type of population. Despite these limi-tations, these models have the ability to include severalrealistic features.In the compartment type of models, the disease statusof individuals changes with the development of the dis-ease, i.e., transitions occur between two compartmentseither due to interaction of the infected with the suscep-tible or due to interventional actions. These models in-clude delay time-scales inherent in the dynamics of trans-mission, for instance, the period spent in quarantine andthe time required for tracing individuals exposed to theinfected. These models have the ability to include severalrealistic features, such as the response of the populationto interventional measures. However, generally, inclusionof more and more realistic features requires a larger num- ber of partitions. Then, the number of differential equa-tions increases and so does the number of parameters,making calibration of the parameters difficult [5–9, 13].Motivated by the complexity of such models, we havedevised a compartment-based model having the suscep-tible, infected, quarantined, traced, recovered, and thedeceased populations. The susceptible and the infectedform the core populations in the sense that it is throughthese two populations that inward/outward transitionsoccur with other populations. However, even this sim-ple model contains several parameters, leaving numericalsolutions as the only option for further analysis. Consid-ering the fact that the model equations form a coupled setof nonlinear differential equations, we adopt mathemat-ical methods drawn from nonlinear dynamical systemsmost appropriate for analyzing such equations [29]. Wehypothesize ’accessible population for infection’ to be asmall fraction of the total population. The validity of thishypothesis can be appreciated by noting that the purposeof interventions is to minimize the exposure of the popu-lation to virus transmission, thereby limiting the spreadof infection. We further assume that the order of magni-tude of the accessible population is similar to that of theinfected population . This assumption is made more quan-titative. This, together with the structure of the modelequations, allows us to decouple them into two equations.These two equations further reduce to a logistic type ofequation for the total infected population with well de-fined parameters namely, the ’testing rate’ and ’contacttransmission rate’ parameters [30, 31]. The equation canbe solved analytically, thereby allowing for a clear inter-pretation of the parameters controlling the growth andinhibiting factors. The validity of the ’accessible popula-tion’ hypothesis and the efficacy of the reduced logisticmodel is demonstrated by the ease of fitting the cumu-lative number of infections and daily new cases for theUnited Kingdom (UK). The procedure further allows usto forecast the progression of the disease. Using this in-formation and calibrating the relative importance of thevarious transition rates (equivalently the associated pa-rameters), we optimize the parameter values specific tothe UK. This procedure allows us to delineate the vari-ous time scales contributing to the various regions in thetime development of the disease. Using this, we numeri-cally solve the full model equations. The calculated totalinfected population fits very well with the available datafor the UK [32]. (UK does not publish data on the re-covered and the active populations.) The model exhibitsa turning point in the active infected population aroundMay 15. However, since the rate of slowing down be-yond the turning point is poor, the projected end timeof the epidemic would be around late July with the pre-dicted saturation level of the total number of infections ∼ . × assuming lock-down conditions continue. II. THE MODEL
The total population N is partitioned into the suscep-tible S , active infected I , quarantined Q , those traced T after being exposed to the infected, recovered R , and thedeceased D . The respective populations are denoted by N s , N i , N q , N tr , N r and N d .Testing is one of the standard protocols used for iden-tifying the infected. If α s is the rate of testing per dayper million and p s is the probability of testing positive,then α s p s N s is the transition rate from S to I . Infectedindividuals coming into contact with the susceptible classcan transmit the virus. If β i is the transmission rate percontact, p i is the probability of transmission of the dis-ease and F ( d i ) is a distance-dependent interaction, then, p i β i F ( d i ) N i N s is the transition rate from S to I . Con-sidering the fact that one of the primary routes of trans-mission is through airborne aerosols generated by the in-fected, larger separation is known to reduce the risk oftransmission [19, 20, 33]. This distance-dependence of F ( d i ) is generally expressed as F ( d i ) ∝ /d or 1 /d i .However, in the present context where we will be dealingwith a lock-down situation for most part of the progres-sion of the disease, we set F ( d i ) = 1.During testing, some individuals would always exhibitmild or ambiguous symptoms. These are identified aspre-symptomatic. If the probability of finding the pre-symptomatic is p q , then, α s p q N s transition out of S to Q . Subsequently, when tested again, say after a quaran-tine duration [5, 34, 35], some of them may either testpositive with a probability q or negative with a proba-bility (1 − q ). If positive, the transition out of Q (to I )is q λ q N q . Here, 1 /λ q is the quarantine duration, usuallyof the order of the incubation period [34, 35]. Similarly, if tested negative, the transition rate out of Q into S is(1 − q ) λ q N q . The total loss rate to ˙ N q is λ q N q .Tracing those exposed to the infected, and testing tofind if they are infected, are important steps in control-ling the spread of the disease. Inherent in tracing suchindividuals are delays in tracing. Such delays cause in-creased transmission of the disease. If p t is the prob-ability of tracing such individuals, then, α t p t N s is thetransition rate from S to T . Subsequently, individualstesting positive will move to the infected compartment I with a probability q and the rest with a probability(1 − q ) move to S . The total transition out of T is equalto λ t N tr , where 1 /λ t is the time taken to trace the indi-viduals. (There is also another possibility, namely, someindividuals may show mild symptoms. Then, there wouldbe a transition into Q . For the sake of simplicity, we haveignored this route.) Finally, the outward transitions from I are the recovery and death rates respectively, γ r N i and κ d N i .Collecting these terms, we have the following set ofcoupled nonlinear ordinary differential equations˙ N s = − ( α s p s + α s p q + α t p t ) N s − p i β i N i N s + (1 − q ) λ q N q + (1 − q ) λ t N tr , (1)˙ N i = α s p s N s + p i β i N i N s + q λ q N q + q λ t N tr − ( γ r + κ d ) N i , (2)˙ N q = α s p q N s − λ q N q , (3)˙ N tr = α t p t N s − λ t N tr , (4)˙ N r = γ r N i , (5)˙ N d = κ d N i . (6)(Here, we have suppressed F ( d i ) factor since it has beenset equal to unity. Definitions of the notations are givenin Table I). Note that the total infected population isgiven by N t = N i + N r + N d .To begin with, we highlight a few features of the modelequations. Our model, much as other compartment-typemodels, has several parameters. However, several of theseare directly measurable and therefore can be obtainedfrom the literature. A few others are related to test-ing protocols and again can be obtained from the liter-ature or from relevant open sources [32]. For instance, α s p s , α s p q and α t p t are directly related to testing ratesand therefore, these are known for a given situation. Afew other rate parameters such as λ q , λ t , γ r and κ d , areinversely related to measurable time-scales, such as theduration of quarantine τ q , time required for tracing τ t ,time for recovery starting from illness τ r and the timefrom illness to death τ d , respectively [35, 36]. TABLE I. Definitions of the notations for our calculation.
Notation Definition N a (0) Accessible population α s Testing rate per day per million p s Probability of testing positive p i Probability of transmission of the disease β i ∼ /N a (0) Transmission rate per contact p q Probability of finding thepre-symptomatic α t Tracing rate per day per million p t Probability of tracing λ q Rate of quarantined individualstesting positive λ t Rate of tracing individuals exposedinfected q Probability of quarantinedindividualstesting positive q Probability of traced individualstesting positive γ r Recovery rate of infected individuals κ d Death rate of infected individualsThe present model includes two delay loops defined byEqs. (3) and (4). These delays are natural to the im-plementation of the protocols. For instance, once quar-antined, subsequent tests are conducted after quarantineduration to identify if quarantined individuals test posi-tive or negative. Similarly, delays in tracing individualsare common. A more transparent way to describe thesedelay loops is through the integral representation of Eqs.(3) and (4), which forms the definitions of the two popu-lations N q and N tr , respectively. For instance, N q ( t ) = α s p s (cid:90) t dt (cid:48) N s ( t (cid:48) ) K ( t − t (cid:48) ) . (7)When the kernel K ( t ) is modeled using an exponentialform with a single time scale 1 /λ q , i.e., K ( t ) = e − λ q t , one can easily verify that differentiating Eq. (7) (using theLeibniz rule) leads to Eqs. (3). The convoluted natureof the integral physically implies that those quarantinedearlier will leave the quarantine sooner than those quar-antined later.Equations (1-6) constitute a set of coupled nonlineardifferential equations. A standard procedure for furtheranalysis of such equations is through numerical integra-tion. While this is a necessary step, here we adopt areductive analysis of the model equations by exploitingthe fact that there are two main populations, namely, thesusceptible (Eqs. 1) and the infected (Eq. 2). Further-more, Eqs. (5,6) are essentially decoupled from the rest(transitions to R and D are from I ). These two featuressuggest that Eqs. (1,2) can be decoupled from the rest ofthe equations. We refer to the decoupled equations as thereduced model equations. Since the two equations can befurther reduced to a logistic-type equation (referred toas the reduced logistic equation), it can be analyticallysolved. As we shall see, analysis of this equation providesinsights that prove to be useful for the analysis of the fullmodel Eqs. (1-6). (We shall often refer to Eqs. (1-6) as”full model equations” to avoid confusion.) III. CONCEPT OF ACCESSIBLEPOPULATION: THE REDUCED MODEL
We now introduce the concept of ’accessible popula-tion for transmission of the disease’. To appreciate thisconcept, consider the spreading dynamics of a contagiousdisease in the absence of any interventions. Then, in prin-ciple, the entire population is exposed to the disease, andit may spread to the entire population (barring the pos-sibility of the population acquiring herd immunity). Inthis case, the entire population is the accessible popula-tion. However, since no Government would like to seethe entire population infected, interventional measuresare enforced precisely to mitigate the risk of transmis-sion and limit the population exposed to the disease toa minimum. In this case, the accessible population isexpected to be a small fraction of the total population.These two limiting cases can be accommodated by hy-pothesizing the accessible population to be a fraction ofthe total population, where the fraction is determined bywhether interventional measures are imposed or not.Consider dropping all terms except α s p s N s and p i β i N i N s in Eqs. (1,2). Then, these two equations getdecoupled from the rest of the equations. Further, be-cause all other inward/ outward transitions are removed,the character of the compartment I changes from theactive infected to the cumulative infected I t with N t de-noting the corresponding population. Then, we have˙ N s = − α s p s N s − p i β i N t N s , (8)˙ N t = α s p s N s + p i β i N t N s . (9)Noting that ddt ( N s + N t ) = 0 , (10)we have N t + N s = constant. Without loss of generality,we set N t + N s = N s (0), the total population. Then, weget a single equation governing the cumulative infectedpopulation, given by˙ N t = c + bN t − aN i , (11) a = p i β i , (12) b = p i β i N s (0) − α s p s , (13) c = α s p s N s (0) . (14)Equation (11) has the well known form of logistic equa-tion, extensively studied in the context of population dy-namics [37], with a notable difference, namely, the pa-rameters a, b and c have a well defined interpretation asdiscussed above. We refer to Eq. (11) as the reducedlogistic equation. (For brevity we often refer to α s p s and p i β i as testing and contact transmission rates, respec-tively.)We begin with a few observations on the relative mag-nitudes of the model parameters in the absence and thepresence of interventions. Consider a situation whenthere are no constraints. Then, one should expect thatthe testing rate ( α s p s ) to be low (compared to the lock-down period) due to absence of any guidelines from pol-icy makers. Similarly, since infected individuals carry onwith their routine activity, the number of contact trans-missions is high and hence, the contact transmission rate( p i β i ) is expected to be high (compared to when interven-tions are in place). Then, the total accessible populationdenoted by N a (0) is the entire population of the region or the country, i.e., N a (0) = N s (0). In contrast, wheninterventions are in place, testing rates are high to en-sure identification of the infected, therefore, α s p s is high.In this situation, since the mobility of individuals is re-stricted, the number of contacts is severely limited, i.e., p i β i will be small. Therefore, the accessible population N a (0) is expected to be small compared to the total pop-ulation N s (0). These two limiting cases can be writtenas N a (0) ≈ F N s (0). These qualitative statements aboutthe accessible population will be made quantitative bycarrying out a detailed analysis of Eq. (11).Consider the initial growth of Eq. (11) by droppingthe quadratic term. Then, we have ddt N t = c + bN t . (15)The solution is given by N t = cb (cid:0) e bt − (cid:1) + N t (0) e bt , (16)where N t (0) is the initial number of infections. As can beseen, the growth rate given by b ≈ p i β i N s (0) depends on N s (0), the total population. Therefore, the growth ratecan be high. In addition, the pre-factor for the exponen-tial growth term (in Eq. 16) depends not only on N t (0)but also on c/b = α s p s /p i β i . Thus, the initial growthdepends on relative magnitudes of N t (0) and α s p s /p i β i .It is straightforward to obtain the solution of Eq. (11).(See Appendix for details.) Here, it is adequate to con-sider the solution in terms of the parameters a, b, and c ,given by N t = (cid:0) ba N i (0) + ca (cid:1) e bt + ca + acb ( N t (0) + acb (cid:1) e bt − N t (0) + ba . (17)We now examine two limiting cases. For short times, N t tends to ( N t (0) + cb ) e bt (since the denominator is dom-inated by b/a = N s (0)), consistent with the short timesolution given by Eq. (16). For long times however, N t tends to b/a = N s (0), the total population.The self-limiting nature of Eq. (17), a characteristicfeature of logistic equations, is evident from the fact that N t tends to N s (0). In other words, the entire popula-tion becomes accessible for transmission of the disease.Clearly, the situation can only represent the growth ofinfection in the absence of any kind of interventions.On the other hand, the effect of all interventions is tolimit the contact transmission rate, thereby limiting theproportion of the exposed population to the disease toa small fraction. It is this that we call the accessiblepopulation. In other words, the accessible population N a (0) is of the same order as the infected population.This can be written as N t ∼ N a (0) ≈ F N s (0), where F is a small fraction. It must be noted here that the valueof the fraction F depends on the nature of interventionsin force.However, within the scope of the reduced logisticmodel, the evolution of N t is independent of the values ofthe parameters α s p s and p i β i during the absence or pres-ence of interventions. As a consequence, the asymptoticvalue of the cumulative infected population is always N t = N s (0), the entire population. Therefore, demon-strating the accessible population is a small fraction ofthe total population is outside the scope of Eq. (11)and the full model Eqs. (1- 6). An independent way ofdemonstrating N a (0) ≈ F N s (0) is desirable. A. Quantitative estimate of the accessiblepopulation
Since the factor F is not well determined, there is anecessity to get a better estimate of this parameter orthe accessible population N a (0). Assuming that the ac-cessible population is of the order of N t , we assume that N a (0) ≈ F N s (0). This is equivalent to using N a (0) inplace of N s (0) in Eq. (17). Then we numerically evaluatethe dependence of N t on the parameters on N a (0) , α s p s ,and p i β i . Given the fact that the disease evolves, weexpect that the accessible population N a (0) also evolveswith time and in the early stages of evolution, N a (0) willbe small, even in the absence of interventions.Consider the dependence of N t on N a (0), keeping α s p s and p i β i fixed. We find that even for relatively large val-ues N a (0), N t grows exponentially; for intermediate val-ues, a near saturation value is reached in a relatively shortduration of 10-15 days; and for small values, the satu-ration value is not reached even after 100 days. Thesefeatures are illustrated in Fig. 1(a) in plots (i-iii) for N a (0) = 8 × , . × and N a (0) = 1 . × re-spectively, keeping p i β i = 3 . × − and α s p s =1 . × − . We have also examined the influence of p i β i , FIG. 1. (Color online) (a) Plots of N t for decreasing values of N a (0) as a function of time : (i) N a (0) = 8 × , (ii) 2 . × ,and (iii) 1 . × respectively, keeping p i β i = 3 . × − and α s p s = 1 . × − fixed. (ii) shows decreasing N a (0)by a factor of 5.51 leads to slow increase in N t . (iv) Plotof N t for p i β i = 4 . × − , keeping N a (0) = 2 . × and α s p s = 1 . × − . Smaller values of p i β i take longertime for N t to grow as is clear (see (ii) and (iv)). (v) Plotof N t for α s p s = 1 . × − , keeping N a (0) = 2 . × and p i β i = 3 . × − fixed. Increase in α s p s leads toa faster initial growth seen in (v) and (iii). Also shown is thecumulative number of infections ( • ) for the UK. (b) Figureshows the two-phase evolution of the disease. The inset showsthe good fit using Eq. (17) with the cumulative infected casesfor the UK ( • ) prior to March 23, 2020. Parameter used are N a (0) = 3 . × , N t (0) = 13 , α s p s = 1 . × − , p i β i =6 . × − . Post lock-down period: The four curves (i-iv)correspond to the four iterations of N a (0) values. (i) N a (0) =5 . × , α s p s = 0 .
0, (ii) N a (0) = 4 . × , α s p s = 1 . × − ,(iii) N a (0) = 3 . × , and (iv) N a (0) = 2 . × , with α s p s =4 . × − for latter two. The initial value of N t (0) = 5687on March 23, 2020. (c) Plots of daily new cases for the UKand the calculated daily new cases using the reduced logisticmodel predicted. keeping N a (0) = 2 . × and α s p s = 1 . × − . Smallervalues of p i β i , N i take longer time for the infection ( N t )to grow. This feature can be seen from the curves (iv)and (iii) for p i β i = 4 . × − and 3 . × − respectively. We have also examined the growth depen-dence of N t on α s p s , keeping the other two parametersfixed. The dependence of N t on this parameter is sim-ilar to that on p i β i . The curve (v) taken together with(ii) shows that increasing α s p s also leads to faster initialgrowth of N t . In the same plot, we have also plotted thetotal number of infected cases ( • ) for the UK.A careful scrutiny of the total coronavirus cases ( • )in the UK shows that it is similar- both in magnitudeand shape -to the plot of N t corresponding to N a (0) =0 . × marked (ii) shown in Fig. (1a). This similaritysuggests two important points. First, noting that theUK is under lock-down, one expects that the accessiblepopulation is a small fraction of the total population,and therefore we see that the order of magnitude of theaccessible population N a (0) used is comparable to that ofthe infected population N t shown in curve (ii). The figurealso shows that as much as all populations dynamicallyevolve during the development of the pandemic, N a (0)also keeps evolving with time. Second, the similarity inshape of the UK data ( • ) with the sigmoidal shape of thelogistic solution raises a question whether the similarityis accidental. If not, can this be used to fit the UK data? B. Data Assimilation and fitting
However, considering the complex dynamics of thehighly contagious virus and the fact that logistic equationcan at best represent simple situations, any attempt tofit the data appears ambitious. Even so, it is tempting toexamine if Eq. (17) could be used to fit the coronavirusdata for some country/region. To do this, we first notethat the reduced model equation contains just three pa-rameters and the dependence of N t on these parametershas already been examined [see Fig. 1(a)].In most countries, the development of the disease fallsinto two phases, namely, the initial period when Govern-mental constraints are absent, referred to as phase oneand the period beyond the lock-down date, called phasetwo. In the case of the UK, the first case was reported on January 31, 2020. Subsequently, the lock-down wasimposed on March 23. Thus, we need to fit the data forthe period January 31 to March 23 and then the rest.Consider the period between January 31 and March23, 2020. Briefly, the fitting procedure adopted here isto equate the initial growth rate of infections obtainedfrom the coronavirus data with the model growth rategiven by Eq. (16) (or Eq. 17). Using the fact that theaccessible population is of the order of the total numberof infections, we use a trial value of N a (0) (assumed tobe a few times larger than the infected population) to fixthe parameter β i . Then, the correct value of N a (0) thatprovides the best fit for the entire data is found iterativelyby decreasing N a (0) so as to fit an increasing number ofdata points. The procedure is illustrated below.Here, we use the analytical solution given by Eq. (17)(or solving Eqs. 8-9) with parameters and initial condi-tions appropriate for the unconstrained growth. Recallthat the testing rate parameter α s p s is low during the ini-tial period and the contact transmission rate parameter p i β i would be high. The values of these two parametersin the lock-down period are just the opposite.Consider the first phase where virus transmission isunconstrained. A careful perusal of the UK data showsthat a smooth increase in the infected numbers startson Feb. 26, 2020, when the number infected stood at N t = 13. The local growth rate obtained from the dataover 8 days was found to be 0.25849/day. Equating thiswith the model growth rate given by p i β i N a (0) (in Eq.16), with a trial value of N a (0) = 4 . × fixes a valueof β i = 6 . × − . The solution of Eq. (17) (or Eqs.8-9) obtained using the initial condition N t = 13 keeping α s p s = 0, passes through several more data points than 8.In the next iterations, we reduce N a (0), keeping in mindthat the solution should pass through a larger numberof data points. In addition, since the initial growth rate(Eq. 16) depends also on c/b = α s p s /p i β i , a proper valueof α s p s is required for a good fit. We find that just oneiteration of reducing N a (0) to N a (0) = 3 . × with α s p s = 1 × − fits the data well for the period fromFeb. 27 to March 23, 2020, as shown in the inset of Fig.1(b).Fitting the data for the second phase follows the sameiterative procedure except that the number of iterationsis greater for the second phase due to the large numberof data points. The number of infections as on March 23stood at N t = 5687. This number matches with the pre-dicted value of N t as on March 23, 2020, obtained fromEq. (17) for the first phase. (See the inset in Fig. 1(b)).Since the effect of lock-down is expected to take sometime to manifest, we have taken the local slope over 17points from the lock-down day is 0.13/day. This slopeis equated with model growth rate using a trial value of N a (0) = 5 . × ( α s p s = 0) to obtain β i = 2 . × − .Using the initial condition N t = 5687 in Eq. (17) (orsolving Eqs. 8-9), we find that the solution (i) (with α s p s = 0) passes through a few more than 17 points.In the next iterations, we reduce N a (0) = 4 . × andcompute the solution taking into account the contribu-tion from α s p s = 1 . × − . The solution (ii) passesthrough several more data points. Two further itera-tions for successively smaller values of N a (0) = 3 . × and N a (0) = 2 . × are used to obtain the solutionmarked (iii) and (iv), respectively. ( α s p s = 4 . × − is used in both cases.) This is shown in Fig. 1(b).As is clear from the Fig. 1(b), solutions (ii) and (iii)are seen to pass through successively larger number ofpoints. Surprisingly, the solution curve labeled (iv) with N a (0) = 2 . × fits the entire data fairly well. Notethe increasing trend of the values of α s p s for successiveiterations. This feature is consistent with the steadily in-creasing testing rates routinely used for proper enforce-ment of lock-down. This feature is easily incorporatedby parameterizing α s p s with time.Now consider the estimation of the end time of the epi-demic, usually defined as the time when no new infectedcases are reported. Noting the close fit of the model pre-dicted cumulative infected population N t with the UKdata seen in Fig. 1(a), dN t dt gives the model calculatednew infected cases. This can be compared with the UKdata for the daily new cases. This is shown in Fig. 1(c).As can be seen from the figure the general profile of themodel predicted daily new infected cases matches wellwith the published data for the UK. The estimated endtime of the epidemic to be last week of June.Apart from providing a close fit for the entire data, themethod appears to have predictive power, as is clear fromthe curve (iv), which shows that the rate of slowing of thetotal number of infections is decreasing. The predictedsaturation value is ∼ . × . A near-saturation value is likely to be seen by the second week of June. Theseresults suggest that the reduced logistic model can beused for obtaining fit for the COVID-19 data for othercountries as well. The close fit in itself is attributableto the fact that the dominant contribution to the growthof the total infected population N t comes from the twodirect transitions. On the other hand, Eq. (17) doesnot include outward transitions (the recovered and thedead), and also the inward quarantine and tracing tran-sitions. Therefore, the estimated saturation value andthe projected future development should be taken withsome reservations. This will be clear once the full modelequations are analyzed and a fit with COVID-19 data forthe UK is accomplished. Despite these limitations, be-cause the reduced logistic equation retains basic growthcontributions to the cumulative infected N t , the fit withthe data appears reasonable.There are attempts to use logistic equations to getinsights into the dynamics of COVID-19 transmission[30, 31]. For instance, a five-parameter hierarchical logis-tic model has been used to fit the observed data to projectthe cumulative number of cases for several countries [31].The parameters entering in the model are determined bythe fitting procedure. IV. THE FULL MODEL
One of the challenges of compartmental models isthe difficulty associated in making accurate predictions,mainly attributable to the uncertainties in obtainingproper estimates of the parameters [13–15, 35, 36]. Forthe same reason, forecasting is even more challenging.Often, several factors may also contribute to the sameparameter, making it difficult for proper interpretation.In our model, however, several parameters in Eqs. (1-6) are related to measurable quantities. For instance,the parameters α s p s , α s p q and α t p t respectively rep-resent rates of testing positive, rates identified as pre-symptomatic, and tracing rate of those exposed to theinfected. Similarly, parameters λ q , λ t , γ r and κ d are in-versely related to quarantine duration τ q = 1 /λ q andtime required for tracing τ t = 1 /λ t , time from illnessto recovery τ r = 1 /γ r , and time from illness to death τ d = 1 /κ d . Though these quantities are country/region- FIG. 2. (Color online) Calibration of parameters for iden-tifying the relative importance of transitions contributing to N i by varying one parameter, keeping all others parametersfixed at reference values listed in Table II. The dotted curveis the reference plot for the active infected population N i cor-responding to the values in the Table II. (i) Plot of N i fora 50% increase in p i β i showing a substantial increase in thepeak height (38%) and a shift by 25 days. (ii) Similar ef-fects are seen when α s p s is increased by a factor 8. (iii) Theplot shows a substantial decrease in the peak height with amarginal shift in its position when α s p q is increased by a fac-tor of 2. (iv) The plot shows a decrease in the peak heightas κ d is increased by 60%. (v) Similar effect is seen when therecovery rate γ r is increased by 60%. specific, their values have been estimated in the literature[5, 6, 8, 35, 36, 38–42]. Some values are also available inthe public domain [32, 43]. One parameter that is hardto estimate is the contact transmission rate β i , which isalready estimated in the context of the reduced logisticequation. TABLE II. Post lock-down period: Select set of parametervalues serving as a reference set. q = q = 0 .
08 and p i = 0 . α s p s α s p q α t p t p i β i . × − . × − . × − . × − λ q λ t γ r κ d /
14 1 / /
42 1 / A. Calibration of relative strengths of theparameters: insights into disease evolution
However, the dynamical evolution of a nonlinear cou-pled set of equations such as Eqs. (1- 6) is necessarilycomplex. Therefore, in the absence of appropriate valuesrelevant for the country/region , a systematic method offinding optimized values of parameters that fit the dataunder consideration requires calibration of all parametersin the model. This will also help us to delineate the differ-ent time-scales participating in Eqs. (1- 6). For instance,several of these parameters represent the growth or de-cay rates of these populations. Our model contains eighttime-scales which are inversely related to the transitionrates. These are β i , α s p s , α s p q , α t p t , λ q , λ t , γ r and κ d .These time scales control how these populations evolvewith time. From the structure of Eqs. (1-6), it is clearthat N i , N q and N tr exhibit peaks as a function of time(days) whereas N r and N d grow monotonically. How-ever, at what point of time do the peaks appear in thesepopulations with the progression of the disease cannot beeasily determined since these are coupled nonlinear dif-ferential equations where the evolution of any populationdepends on the evolution of all other populations [29].More importantly, if one is interested in fitting the modelpredicted growth of populations, which convey the dis-ease status (such as the total infected, active, recoveredand dead), estimating the relative proportion of the pop-ulations as the disease evolves is necessary. Further, thetotal infected population commonly used to convey thedisease status in daily briefings has contributions from allpopulations. Therefore, delineating and determining atwhat points of time each of these populations contributeto the total populations would provide required insightinto further analysis.Following the recently developed method in the area ofplasticity [44–46], we investigate the influence of the pa-rameters to identify the relative importance of the tran-sition rates. Since it is a multi-parameter space, we varyeach parameter, keeping all other parameters fixed at thereference set of values listed in Table II. The results are il-lustrated using plots of the active infected population N i .The dotted curve shown in Fig. 2 is the reference curvecorresponding to the reference set of parameters given inTable II. As in the case of the reduced logistic model, the0growth of N i depends sensitively on the contact trans-mission parameter p i β i . (Note that in our model, this isthe only parameter that directly contributes to growthof infections.) A 50% increase in the parameter inducesa substantial increase (38%) in the peak height with theposition shifting towards earlier time (by 25 days) as isclear from the curve (i). Noting that the position of thepeak, i.e., the turning point of N i is indicative of slowingdown of the rate of infection, increasing peak height of N i suggests slowing down of infections occurs at higher val-ues of infections, where as a shift towards shorter timesimplies that slowing down occurs earlier. A similar effectbut of lesser magnitude (20%) is seen when testing rate α s p s is increased by a factor 8 as is clear from (ii).In contrast, an increase in the quarantining rate α s p q by a factor 2 decreases the peak height comparable inmagnitude to that induced by β i with a shift in the peakposition away from the origin by 20 days (see iii). A sim-ilar effect, but of lesser magnitude, is seen when α t p t isincreased (not shown). The reduction in the peak heightof N i is understandable because the total inward transi-tion is α s p s + α s p q + α t p t , and therefore increasing oneof these changes the relative weights. Physically, the de-crease in the peak height (commonly referred to as ’flat-tening the curve’ ) accompanied by a shift in the peakposition for longer times implies that as more individualsare quarantined, the disease control is facilitated.We have also investigated the dependence of the re-covery ( γ r ) and death rate ( κ d ) parameters on N i . Anincrease in γ r by 60% decreases the peak height substan-tially as is clear from (iv). Similarly, increase in the deathrate κ d leads to a decrease in the peak heights of N i asis clear from (v). Clearly, increased recovery rate, a de-sirable feature, also leads to a decrease in the numberof active infections. This feature is also intuitively obvi-ous. On the other hand, increased death rate, though notdesirable, also leads to decrease in N i . We have also in-vestigated the influence of other parameters and find that N i is relatively insensitive to these parameters. Notingthat any change in the parameter values relative to thosecorresponding to the reference curve changes the peak po-sition and height, we conclude that the parameters listedin Table II are close to the optimized values. The above analysis on the relative importance of thetransition rates (equivalently the corresponding param- eters), and the accompanying discussions, can now beused to delineate the contributions from different timescales participating in the transmission dynamics of thevirus. As discussed above, the direct transitions from S to I , namely, p i β i , to a lesser extent α s p s , control theinitial growth of N i . The same also to the initial growthof the total infected population N t . Similarly, the de-layed routes, namely, the quarantine S and tracing T contribute to the mid region of the evolution of N i andto therefore to the total population N t also. Recall thatthe turning point of N i is controlled by the balance be-tween all inward transitions (from S to I , and Q and T to I ) and outward transitions ( I to R and D compart-ments). Then, the peak position of N i can be identifiedwith the inflection point of N t . Therefore, the time devel-opment beyond the point of inflection of N t is controlledby a balance of all inward and outward transitions. Theinsights from the above analysis identify three distinctstages in the evolutionary period of the disease, namely,the initial growth period, the mid developmental periodand the final approach to saturation. As we shall see,this identification will be helpful in obtaining a good fitto the UK data. B. Data assimilation and forecast
Having demonstrated that the two direct transitionrates p i β i and α s p s are the dominant contributions tothe growth of N i and having assessed the relative impor-tance of other transitions, we now consider the solutionof the full model Eqs. (1- 6) with a view to obtainingthe best possible fit with the COVID-19 United King-dom data. Attempt will also be made to forecast thefuture progression of the disease.Recall that the spread of coronavirus in the UK fallsinto two phases of development. During the first phaseprior to March 23, 2020, there were no constraints andthe disease transmission was free. After the lock-downdate, the transmission is restricted. Therefore, the modelparameters and the initial conditions relevant for the twophases are different. As in the reduced model, we assumethat the dynamics of the disease transmission is limitedby the accessible population N a (0) and not by the totalpopulation N s (0), i.e., N i ∼ N a (0) ≈ F N s (0). Note that1 FIG. 3. (a) Inset: Plot of the total infected population (con-tinuous curve), along with the UK data ( • ) from February27, 2020 till March 31. Also shown is the active infected N i (dotted curve). Post Lock-down period: Plots of the totalinfected population (curve marked i) along with the corre-sponding data for the UK ( • ) from March 23, 2020. Curvemarked (ii) shows the active infected population. (b) Plotsof the reported daily new cases for the UK along with themodel predicted daily new cases. (c) Plots of the infected (i),quarantined (ii), traced (iii), recovered (iv) and deceased (v),starting from March 23, 2020. The values of the parametersused are given in Table II. both N i and therefore N a (0) depend on whether thereare any interventions or not, and also on the type ofinterventions.Consider the period between January 31 and March23 corresponding to the initial phase. Here, we note thatprior to the lock-down date, there would be no quarantin-ing and tracing procedures and therefore, it is adequateto solve Eqs. (1,2,5,6), More specifically, we ignore thedelayed inward transitions into I . Furthermore, in thefirst few days of the development of the disease, we mayassume that the total number of the infected cases N t isequal to the active infections N i . Finally, since, we planto fit the model solution with the UK data [32], publiclyavailable coronavirus data for the total number infected,active infected, recovered and the dead are useful in fur-ther optimizing the parameters. Unfortunately however,only the total numbers of the infected and the dead aremade available in the UK.Now we are in a position to solve the relevant equationsfor the first phase. As discussed in Section III B, we useFebruary 27, 2020 as the starting day for the first phaseevolution of Eqs. (1,2,5,6). The local growth rate of0.25849/day (on the starting day) over 8 days obtainedfrom the log-linear plot of the cumulative infected casesfor the UK is equated with the model growth rate givenby p i β i N a (0) to fix β i = 6 . × − by using theinitial value of N a (0) = 4 . × . Further, using theinitial conditions for N t (0) = N i (0) = 13 , N r (0) = 0 and N d (0) = 0, we solve Eqs. (1,2,5,6) from February 27 toMarch 23 by choosing a value for α s p s that gives the bestfit to the data for the period. (Here, α s p s = 3 . × − and the values of other relevant parameters are thoselisted in Table II.) The model-predicted total infectedpopulation N t (continuous curve) along with the datapoints ( • ) is shown in the inset of Fig. 3(a). Clearly, thematch is seen to be very good. Also shown is a plot ofactive infections N i (dotted curve). Equations (1,2,5,6)also provide the values of N i , N r and N d on March 23,2020. These are N i = 5407 , N r = 400 , N d = 285.Now we consider the solution of Eqs. (1- 6) with aview to obtaining the best fit for the UK data for the pe-riod starting from March 23, 2020. Here again, we firstfind the growth rate from the data and equate it withthe model growth rate. Here, we note that the effect oflock-down is expected to manifest after some time. For2this reason, we use a 17-point slope in the log-linear plot.Equating 0.13/day with p i β i N a (0) and using N a (0) =3 . × we get β i = 3 . × − . The initial valuesused for evolving Eqs. (1- 6) are N i (0) = 5407, N q (0) =0 , N tr (0) = 0 , N r (0) = 0 , N d (0) = 0. (The reason forusing zero initial conditions for N q (0) , N tr (0) , N r (0) and N d (0) is that the values of these populations would notbe recorded during the first phase. However, using thevalues obtained from the first phase for N r and N d makeslittle difference. Note that N i (0) = 5407 is smallerthan the total number of infected cases. Again, using N i (0) = 5687 does not alter the results.) The parame-ters values used are those listed in Table II. Figure 3(a)shows plots of the calculated total infected population N t and the total infected cases in the UK ( • ). Clearly, thefit is very good. The asymptotic saturation value of thetotal number of infections N t turns out to be 3 . × .Also shown is the active infected N i labeled (ii). Further,the plot of the model predicted active infected popula-tion N i (ii) exhibits a turning point (peak) around May15. Subsequent decrease in N i is seen to be slow. Atthis rate of slowing-down, the model predicts that a nearsaturation value of 3 . × .Now consider the estimation of the end time of theepidemic. A natural candidate for estimating this is thedaily new infections. Considering the close fit of themodel predicted cumulative infected population N t withthe UK data (see Fig. 3a), the model calculated dailynew infections dN t dt can be compared with the UK datafor the daily new cases. This is shown in Fig. 3(b). Ascan be seen the general profile of the model predicteddaily new infected cases matches well with the publisheddata. Recall that the end time of the epidemic is definedas the time at which no new cases are reported. Thisdefinition is clearly impractical for forecasting since thestate of saturation is always reached asymptotically. Forthis reason, we use an arbitrarily small value, say 300new cases as the end date of the epidemic. Then, theestimated end time is late July.To the best of our knowledge, we are not aware of anymodel that fits the COVID-19 data for any country ashas been done here, particularly over such long periodswith the ability to forecast the future progression of thedisease. However, there have been some efforts to fit datafor the initial periods [8, 21, 22, 24, 26, 38, 47]. In view of the good fit for the UK data, the model can be usedto fit the COVID-19 data for other countries and also toforecast the progression of the disease. V. SUMMARY, DISCUSSION ANDCONCLUSIONS
Recent literature has focused on abstracting the effectof various types of interventions through epidemiologicalmodels to make projections of how the disease progressesunder different conditions. Recall that one limitationparticularly applicable to the deterministic compartmen-tal models is the difficulty in getting proper estimates ofthe parameters, particularly when the number of com-partments is large. In this respect, simpler models withfewer compartments have an advantage. However, sev-eral factors may contribute to a single parameter. Thisis also a model-dependent feature. Therefore the abilityof such parameters to represent the mitigating efficacy ofinterventions appears limited (see below). Furthermore,the number of parameters in such models is not neces-sarily small, making numerical solutions often the onlychoice. Therefore, any method - whether mathematicalor conceptual - which simplifies analysis and easy inter-pretation is welcome.Motivated by this, we hypothesize accessible popula-tion for transmission of the disease that can be the totalor a small fraction of the total population depending onwhether the transmission dynamics evolves in the ab-sence or presence of interventions. Indeed, the effect oflock-down is evident in all counties where the disease hasbeen controlled or nearly eliminated. At the mathemati-cal level, we introduce a decoupling scheme to aid mathe-matical analysis that also helps easy interpretation. Themodel equations have been devised in such a way thatthe susceptible and active infected populations form themain populations. The decoupling is affected by drop-ping all inward and outward transitions excepting thedirect transitions ( p i β i N a (0) and α s p s N a (0)). Because,all outward transitions from I are ignored under this de-coupling, the active infected population N i takes the roleof the cumulative infected population N t . The simplicityof the reduced logistic equation (Eq. 11) allows easy iden-tification of the growth and inhibiting factors in terms ofthe dominant growth factors (direct inwards transitions3or parameters). Surprisingly, this simple equation pro-vides a good fit to the reported cumulative number ofinfections for the United Kingdom, as is clear in Fig.1(b).The full model Eqs. (1- 6) contain several param-eters whose range has been estimated in a number ofstudies[8, 21, 22, 24, 38, 40–42, 47]. However, when itcomes to explaining or capturing the growth character-istics for a specific country, optimized parameters suit-able for the situation are required. Following [44–46], wehave determined the relative importance of the varioustransition rates (equivalently the associated parameters)subject to the constraint that the parameter values pro-vide the best fit for the given data. In this work, we havemade use of publicly available data on the total infectedand daily new infected cases for the United Kingdom [32].Figure 3(a) shows the fit obtained for the period tillMarch 23, 2020 (shown in the inset) and for the periodbeyond. Clearly, the fit is seen to be very good for boththe period till the lock-down date and the period there-after. Comparing Fig. 3(a) with Fig. 1(b) for the re-duced logistic map, we see that while the fits in bothcases are comparable, the projections of the future pro-gressions are significantly different. The saturation valuepredicted by the full model (shown in Fig. 3(a) is close to3 . × , whereas that predicted by the reduced logisticequation in Fig. 1(b) is ∼ . × . Conventionally, theend time of epidemic is defined as the day on which nonew infections are reported. However, since the approachto the end point is generally slow, a better variable topredict the end point of the epidemic is by comparingthe model computed daily new case with the reporteddata on the daily new cases for the UK. Then, the endtime of the epidemic predicted by the full model turnsout be late July (see Fig. 3b). In contrast, the end timefor the epidemic predicted by the reduced logistic modelis late June. Clearly, the results obtained from the fullmodel emphasize the limitations of the reduced model.A natural question is: what are the underlying causes?The fact that the reduced logistic model provides agood fit also means that the major contributing factorsfor the growth of infection are included in Eq. (2). Tosee this, consider Eqs. (1-6). The growth of N i ( t ) ormore appropriately the daily new infected cases dN i dt hastwo types of inward transitions, namely, direct and de- layed. However, the dominant direct transition from S to I given by p i β i controls the growth rate of N i . The otherdirect transition α s p s into I also contributes to a lesserextent. Now, consider the delayed inward transitions to I coming from Q and T . These transitions are smallerin magnitude and contribute to sub-exponential growthof N i in time. More importantly, the turning point in N i or dN i dt is due to a competition between the growth fac-tors (all inward transitions) and the outward transitions(recovery and fatality terms). Further, since the timeevolution beyond the turning point of dN i dt is controlledby a balance between all inward and all outward tran-sitions, the approach towards the state of no infectionsor the saturation value of N t is slow in our case. Thesefeatures are clear from Fig. 3(a,b). Note that the fit, tillJune 1, is just two weeks beyond the turn point of dN i dt and it has a long way to evolve to the end point of theepidemic.These arguments clarify two features of the data fitobtained using the reduced logistic equation. Becausethe total number of infected cases N t has the dominantgrowth contributions, the good fit is not surprising. Onthe other hand, growth dynamics beyond the turningpoint in the daily new infected cases (i.e., dN i dt ) is con-trolled by a balance between growth factors (all inwardtransitions) and inhibiting factors (the rate of recoveryand dead). However, these competing time scales areabsent in the logistic equation. This clarifies why theprojected saturation value of N t and the end time of theepidemic is not well captured.A few comments above model are in order. As statedin the introduction, the standard compartmental modelsignore both age dependence and the heterogeneous spa-tial distribution of the population, and therefore cannotabstract the aspects that depend fundamentally on thesetwo features. For the same reason, the populations inthese models represent only the mean response of eachof these populations. From this point of view, the goodfit obtained by the full model, to lesser extent by the re-duced logistic model, may come as a surprise. However,the success story of the mean field approaches has beenestablished in physics literature [29, 44–46] and the limi-tations of the approach has also been well established. Tothis extent, the success of the present model can be at-tributed to the way the model equations are structured4and the underlying nonlinear dynamical methods usedfor analysis.Here, we mention that the parameter values used forfitting have been obtained using an optimization proce-dure subject to the constraint that the optimized val-ues should fit the data for the cumulative infected casesfor the UK. Although, each parameter is varied withinthe range of values estimated in the literature (and opensources), since the optimization has been carried out sub-ject to only one constraint, the values may not be unique.From this point of view, more number of constraints suchas the data for the active infected, recovered, quarantinesetc., would be helpful in removing the non-uniqueness ofthe optimized values, at least partially.As stated in the text, the contact transmission rate pa-rameter β i is one of the crucial parameter in the modelbecause this parameter largely controls the time devel-opment of the disease. In our approach, this parameterhas been estimated by using the local initial slope in thelog-linear plot of the UK data. The slope itself, however,depends on whether the disease development occurs inthe absence or presence of interventions, which in turndepends on the accessible population N a (0). Since themodel growth rate depends on p i β i N a (0), the value of β i depends inversely on N a (0) corresponding to the ab-sence or presence of interventions. However, the valuereported by Tang et al. [6] is two orders smaller thanthat estimated in our paper. However, in both cases, thevalue of β i is inversely proportional to the relevant pop-ulation used for modeling. In this context, we mentionthat there is only independent estimate both under freeevolution [5]. Such an independent estimates are desir-able.One point that needs some discussion is about the val-ues of the recovery γ r and the death κ d rates obtainedfrom the optimization procedure used that fit the UKdata very well (see Table II). These rates are inverselyrelated to the time duration between detection of illnesstill recovery and death respectively. While these valuesare within the published range of values [40–42], they ap-pear to be on the lower side of the mean. However, inprinciple, the parameter values depend on the structureof model equations. In our model, the recovery and fatal-ity are outward transitions from a single compartment,namely, I to R and D . This feature is clearly because our model equations were structured to have only twocore populations. However, the recoveries can occur ifother kinds of compartments (populations) are included[5]. Thus, the best fit for lower values may be the resultof simplicity of the model.In conclusion, the simple compartmental model notonly provides a good fit to the United Kingdom coron-avirus data but also makes concrete long term predictionsfor the future. We believe that these results have beenmade possible due to the reductive approach adoptedhere. Appendix
Recall the equation governing the cumulative infectedpopulation N s ( t ) from Eqs. (8-9) is given by˙ N t = c + bN t − aN t , (A.1) a = p i β i , (A.2) b = p i β i N s (0) − α s p s , (A.3) c = α s p s N s (0) . (A.4)Equation (A.1) has the well known form of the logisticequation extensively studied in the context of populationdynamics. However, the parameters a, b, and c have awell defined interpretation.Now consider the solution of Eq. (A.1). Let α , = b ±√ b +4 ac a be the roots of the quadratic equation. Then,in terms of a, b and c , the two roots can be written as α ∼ ba = N s (0) and α ∼ − ac/b <
0, which is smallcompared to b . Then the solution is given by N t = Aα e a ( α − α ) t − α Ae a ( α − α ) t − Aα e bt − α Ae bt − . (A.5)The constant A is given by A = N t (0) − α N t (0) − α . (A.6)Then, we have N t = (cid:0) ba N i t
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