On the dynamics of the Coronavirus epidemic and the unreported cases: the Chilean case
OOn the dynamics of the Coronavirus epidemic and theunreported cases: the Chilean case
Andr´es Navas & Gast´on Vergara-Hermosilla Introduction
One of the main problems faced in the mathematical modeling of the coronavirus epi-demic has been the lack of quality data. In particular, it is estimated that a large numberof cases have been unreported, especially those of asymptomatic patients. This is mostlydue to the strong demand of tests required by a relatively complete report of the infectedcases. Although the countries/regions that have managed to control the epidemic have beenprecisely those that have been able to develop a great capacity of testing, this has not beenachieved in most of the situations.In general, the number of unreported patients has been estimated by extrapolating datafrom the reported cases assuming that both numbers vary proportionally. However, thisview can be criticized (al least) in two directions:– It does not consider the dynamical role of the unreported cases in the evolution of theepidemic and, by extension, in the number of reported cases;– it does not allow a possible variation of the proportion between the numbers of reportedand unreported cases when the conditions of social distancing remain unchanged.In this work, we address these points incorporating the unreported cases into the mod-eling. In a first introductory section, we discuss from a mathematical perspective whathappens when the testing capacity is very low. Using a very simple argument we show that,in this context, the dynamics of the disease is actually governed by the growth of the numberof unreported cases, and that of the reported patients becomes much smaller (in a progres-sive way) than that of the total cases. Although the discussion is presented in a context ofextreme (hence “ideal”) conditions, it certainly illustrates how essential is to consider therole of the unreported cases to analyze the global evolution of the epidemic.Several mathematical models have been proposed to deal with the unreported cases andtheir role in the progression of the epidemic. In the second section of this work we address oneof them, with the acronym SIRU, recently proposed by Zhihua Liu, Pierre Magal, Ousmane Supported by MICITEC Chile. Supported by the European Union’s Horizon 2020 research and innovation programme under the MarieSklodowska-Curie grant agreement No 765579. a r X i v : . [ q - b i o . P E ] J un eydi and Glen Webb [10, 11, 12]. After a brief presentation of the model and an explanationof why it is more adjusted to the current epidemic, we proceed to establish a series ofstructural results. Although this does not completely close the study of the qualitativeproperties of the underlying differential equations, we can already glimpse an analogy withthose of the classical SIR model. However, we stress an important difference: in the SIRUmodel, the curves that appear do not necessarily have a single peak. Specifically, we exhibita simple method to detect parameters that give rise to curves with at least two peaks.The model of Liu, Magal, Seydi and Webb has already been used to describe the evolutionof the epidemic in various countries (China, South Korea, the United Kingdom, Italy, Franceand Spain). In the last section of this work, we implement this modeling to the global Chileanscenario using the official COVID-19 data provided by the Chilean government [13]. However,unlike [10, 11, 12], our implementation is novel in that it uses a variable transmission ratefor the disease, which is more pertinent according to the local epidemiological evolution.This work concludes with a section of general conclusions in which some future lines ofresearch are also described. We denote by U the number of positive cases that one would expect to detect within acertain unit of time in some region using the maximal capacity of testing that is available. Suppose that the epidemic has evolved to a point where the number of detected cases in thisunit time is systematically very close to this threshold U . The reported contagion curve isthen taking a plateau shape, but a question immediately arises: has the contagion processentered into an stationary phase -with a reproduction rate R equal to 1 or slightly higher-,or is a significant number of positive cases being indetected?It is impossible to answer a priori to this question. However, it is very unlikely thatthe process enters a stationary regime exactly when the threshold is reached. We arguebelow that, if the entry into an stationary regime has not occurred, then not only a fixed“proportion” of the number of cases is being undetected, but this is the case of a much largeramount; more precisely, the latter follows an exponential rate growth. We note that this quantity shouldn’t be equal to the total number of tests, but to a fraction of this.Indeed, it is expected (and international statistics confirm this) that, for each positive test, a certain fixedamount of tests will have negative outputs. This percentage depends on numerous factors, in particular, onthe bias to preferably test symptomatic patients. .2 A mathematical argument To simplify the discussion, our unit time will be equal to the total period of the disease. Foran instant i , we denote by c i the number of reported positive cases, and by C i that of totalcases. We assume that, while c i remains below the threshold, it assumes values very closeto it in the future evolution. We write this as follows: U − v ≤ c i ≤ U, where v is relatively small compared to U .We will also assume that we are not in a stationary regime (that is, the apparent sta-tionarity is actually a consequence of a default of testing). Therefore, the value of C i issubstantially greater than the threshold and, consequently, than c i . At the instant i , thenumber of cases that are not being detected is C i − c i . These individuals will have a highersocial activity than those detected as infected (since the latter will be quarantined). There-fore, the average number of new infected individuals by these undetected agents will be avalue ˆ R strictly larger than R .Thus, on average, the detected infected individuals of time i generate R new infectionsat time i + 1, while those undetected at time i (whose number is C i − c i ) each generate ˆ R new infections. Therefore, the following inequality holds: C i +1 ≥ R c i + ˆ R [ C i − c i ] . Since
R ≥
1, this implies C i +1 ≥ c i + ˆ R [ C i − c i ] , hence C i +1 − c i +1 ≥ c i − c i +1 + ˆ R [ C i − c i ] . Since c i ≥ U − v and c i +1 ≤ U , the above implies C i +1 − c i +1 ≥ ˆ R [ C i − c i ] − v. By simple recurrence, for an initial time i and all n ≥
1, this gives C i + n − c i + n ≥ ˆ R [ C i + n − − c i + n − ] − v ≥ ˆ R [ ˆ R [ C i + n − − c i + n − ] − v ] − v = ˆ R [ C i + n − − c i + n − ] − v [1 + ˆ R ]... ≥ ˆ R n [ C i − c i ] − v [1 + ˆ R + ˆ R + . . . + ˆ R n − ] = ˆ R n [ C i − c i ] − v (cid:34) ˆ R n − R − (cid:35) ≥ ˆ R n (cid:20) C i − c i − v ˆ R − (cid:21) .
3n other words, C i + n ≥ c i + n + ˆ R n (cid:20) C i − c i − v ˆ R − (cid:21) . It is natural to expect that the term v/ ( ˆ R −
1) is (very) small with respect to C i − c i .Indeed, on the one hand, ˆ R does not approximate indefinitely to 1 (since the undetectedinfected individuals do not significantly reduce their social activity); on the other hand, ouranalysis deals with times for which C i is very (although, a priori , not exponentially) greaterthan c i . (A more robust argument consists in choosing not only a single initial time i , but toimplement the previous inequality along a sequence of consecutive times and finally averagealong these inequalities.)Assuming all of the above, the conclusion is clear: the number of infected people C i + n ismuch higher than the number c i + n of reported infected individuals. Indeed, the differencebetween the two is bounded from below by an exponential of ratio ˆ R , while that the evolutionof the detected cases is governed by the rate R , which is strictly less than ˆ R . So, the curvethat we see (that of the values c i + n ) is not only very far from the real one (that of C i + n ),but the difference between them has an exponential growth that we are not perceiving. The “toy argument” above shows something evident: if we are not able to follow the evolutionof an epidemic through appropriate mass testing, then we lose track of the infectious curve.In more sophisticated terms, what it reveals is that as long we are not aware that thereproduction rate R is strictly smaller than 1, the dynamical system of the epidemic movesin a regime of either slight exponential growth or, at least, of instability . In such a regime,small variations of the initial conditions can lead to exponential explosion. Now, to a largeextent, these initial conditions are provided by official data. However, if these move aroundthe maximum of what the system can detect, then we can hardly know how accurate they areand, therefore, whether we really are in a stationary situation or whether we have advancedto an exponential explosion of cases that we are not perceiving. For this reason, for eachinstance in which the number of positive detected cases becomes nearly constant (that is,when the curve of reported cases begins to acquire a plateau shape), it seems reasonable toapply a substantive increase in the number of tests (both in quantity and spectrum). If thisleads to a significant increase in the number of positive cases, then most likely this wouldmean that, actually, the regime was not stationary, but was simply exceeding a thresholdabove which a significant amount of infections cannot be reported. We will return to thispoint in the general conclusions of this work.4 About the model of Liu, Magal, Seydi and Webb
The key argument in the preceding section is that unreported patients have a greater dy-namic role than those that are reported (since the former do not enter into quarantine), andtherefore they contribute more importantly to the epidemic. However, in the traditionalSIR model, both types of patients are part of the same compartment. In [10], Liu, Magal,Seydi and Webb solve this problem by separating them into two compartments. Denotingrespectively by
S, I, R and U the susceptible individuals, infected individuals who do not yethave symptoms (and are at incubation stage), reported infected individuals, and unreported(either asymptomatic or low symptomatic) infected individuals, they consider the followingdiagram flux: Figure 1: Diagram fluxThe differential equations attached to the diagram above and that govern the dynamicsof the epidemic are the following: S (cid:48) ( t ) = − τ S ( t )[ I ( t ) + U ( t )] I (cid:48) ( t ) = τ S ( t )[ I ( t ) + U ( t )] − νI ( t ) R (cid:48) ( t ) = ν I ( t ) − ηR ( t ) U (cid:48) ( t ) = ν I ( t ) − ηU ( t ) , (1)5here t ≥ t corresponds to time, with t being the starting date for the study (as in [10,11, 12], in the implementation, we will consider the time t corresponding to the beginningof the epidemic). Although this system of differential equations makes perfect sense whenprescribing any initial condition, in epidemiological modeling one is naturally lead to usedata of the following type: S ( t ) = S > , I ( t ) = I > , R ( t ) ≥ U ( t ) = U ≥ . The parameters used in the model are described in the Table below. In particular, note that ν = ν + ν . In addition, all the parameters that are considered τ, ν, ν , ν , η are positive.Symbol Interpretation t Initial time. S Number of individuals susceptible to the disease at time t . I Number of infected individuals (in incubation period) withoutsymptoms at time t . R Number of reported infected individuals at time t . U Number of unreported infected individuals at time t . τ Transmission rate of the disease.1 /ν Average time during which the infectious asymptomatic individu-als remain in incubation. f Fraction of asymptomatic infected individuals that become re-ported infected. ν = f ν Rate at which asymptomatic infected cases become reported. ν = (1 − f ) ν Rate at which asymptomatic infected become unreported infectedindividuals (asymptomatic or mildly symptomatic).1 /η Average time during which an infected individual presents symp-toms.Table 1: Parameters and initial conditions of the model.Note that in the first of the equations of (1), namely S (cid:48) ( t ) = − τ S ( t ) [ I ( t ) + U ( t )] , the role of I and U in the spread of the infection is the same. This is the essential pointof the model: it gives the same dynamic role to those who are infected and do not have6ymptoms as to those who are not reported, because the latter do not go into quarantineand, in fact, have a similar social activity. Certainly, the model can be refined in manydirections; for example, one could give different though still dynamical roles to I and U by attaching to them different positive parameters τ (cid:54) = τ (this could be justified by that U includes individuals with low symptomaticity who can practice self-care). However, it isalready worth to visualize some of the main properties of this model and to implement it inspecific situations following this general format. We next consider the system of equations (1) in more detail. We will assume an epidemicsituation, which is summarized by the condition τ S − ν > . (2)We will also consider the initial conditions of Liu, Magal, Seydi and Webb: S > , I > , R = 0 and U = ν I η + χ (3)for a certain χ >
0. (The precise value of the parameter χ will be given in the next section;here we just retain the fact that it is positive.) For simplicity, our starting time will be t := 0. Theorem 2.1.
In an epidemic situation (2) and starting with the initial conditions (3), thefollowing three properties are fulfilled: (i)
For all time t > , the values of S ( t ) , U ( t ) , I ( t ) and R ( t ) exist, are positive and strictlysmaller than P := S + I + U (the total population); (ii) S ( t ) converges to a certain positive limit value S ∞ as t → ∞ , while I ( t ) , R ( t ) , U ( t ) converge to ; (iii) R ( t ) /U ( t ) converges to ν /ν from below.Proof. We first note that, due to (2) and (3), we have S (cid:48) (0) = − τ S I < ,I (cid:48) (0) = ( τ S − ν ) I + τ S U > , (cid:48) (0) = ν I > ,U (cid:48) (0) = ν I − ηU = ν I (cid:18) − ηη + χ (cid:19) > . Therefore, there exists ε > S ( t ) , I ( t ) , R ( t ) and U ( t ) (are defined in [0 , ε ) and)are strictly positive. The arguments that follow are inspired by an observation contained inthe classical book of Vladimir Arnold [1].(i) Suppose S vanishes, and let T >
C > I ( t ) + U ( t ) ≤ C for all t ∈ [0 , T ]. Then S (cid:48) ( t ) ≥ − τ CS ( t ), and hence, for t ∈ [0 , T ), we have S (cid:48) ( t ) S ( t ) ≥ − τ C. Integrating between 0 and s < T , this giveslog( S ( s )) − log( S ) ≥ − τ Cs, and then, S ( s ) ≥ S e − τCs . Letting s go to T , this contradicts the assumption S ( T ) = 0.Suppose now that U vanishes, and let T > I hasnot vanished until this time, then U (cid:48) ( t ) = ν I ( t ) − ηU ( t ) ≥ − ηU ( t ) for all t ∈ [0 , T ]. Byintegration, this gives U ( T ) ≥ U e − ηT , which contradicts our assumption. Hence, I should have vanished in [0 , T ].The same argument above shows that if R vanishes at a time T >
0, then I must havevanished at some time in [0 , T ].Finally, suppose that I vanishes, and let T be the first moment this occurs. Then, U ( t ) ≥ t ∈ [0 , T ], and therefore I (cid:48) ( t ) = τ S ( t ) [ I ( t ) + U ( t )] − νI ( t ) ≥ − νI ( t ) . However, by integration, this again gives a contradiction, namely I ( T ) ≥ I e − νT .We next show that neither I nor U explode (that is, none of them tends to infinity alongan increasing sequence of times tending to a finite time T ). Indeed, if anyone does it in8ime T then, from the above, P ≥ S ( t ) ≥ , T ), and U, I are positive on this interval.Therefore, on [0 , T ), ( I + U ) (cid:48) = τ P [ I + U ] − ν I − ηU ≤ τ P [ I + U ] , which implies by integration that ( I + U )( t ) ≤ ( I + U ) e τP t for t ∈ [0 , T ). Letting t go to T , this contradicts the explosion.To see that R does not explode, we proceed again by contradiction: if this occurs attime T , then from R = ν I − ηR ≤ ν I we deduce R ( t ) ≤ R e ν C for all t ∈ [0 , T ) and C := max t ∈ [0 ,T ] I ( t ).Finally, S does not explode because it is decreasing and positive.In conclusion, S, I, R, U are all positive and do not explode. To prove that they arebounded from above by P , we introduce the equation of the deletted (removed) individualsfrom the system: D (cid:48) ( t ) = η [ R ( t ) + U ( t )] , D (0) = 0 . We have a constant population P = S ( t ) + I ( t ) + R ( t ) + U ( t ) + D ( t ), and since D (cid:48) >
0, wehave that D ( t ) > t >
0. Now, since
S, I, R, U are positive for t >
0, we concludethat each of them must be strictly smaller than P .(ii) First we show that S converges towards a positive limit. Let S ∞ be the limit of S (whichexists because S is decreasing). Denoting c := min { η, ν } , we have( I + U ) (cid:48) = τ S [ I + U ] − ν I − ηU ≤ ( τ S − c ) ( I + U ) . Since S (cid:48) = − τ S [ I + U ], this implies( I + U ) (cid:48) ≤ ( τ S − c ) S (cid:48) − τ S = − S (cid:48) + cτ S (cid:48) S .
Integrating between 0 and t >
0, this gives( I + U + S )( t ) − ( I + U + S ) ≤ cτ log (cid:18) S ( t ) S (cid:19) . Since I + U + S = P , this implies − P ≤ cτ log (cid:18) S ( t ) S (cid:19) . Thus, for all t >
0, we have S ( t ) ≥ S e − τc P , and therefore, S ∞ ≥ S e − τc P > . I and U converge to 0 (the convergence of R to 0will be then a consequence of the convergence of R/U to ν /ν proved in (iii) below). To dothis, we first note that, since S (cid:48) = − τ S [ I + U ] , it follows that there is a sequence of times t n → ∞ such that ( I + U )( t n ) →
0. Otherwise,there would exist c > I + U )( t ) ≥ c for all t >
0, which implies S (cid:48) /S ≤ − τ c ,and therefore S ( t ) ≤ S e − cτt . However, for t large enough, this contradicts the inequality S ( t ) ≥ S ∞ > ε >
0, we may fix T such that ( I + U )( T ) ≤ ε/ S ( t ) ≤ S ∞ + ε/ t ≥ T . We claim that ( I + U )( t ) ≤ ε for all t ≥ T . (Since ε > I + U towards 0.) To prove this, note that, since( S + I + U ) (cid:48) ( t ) = − ν I − ηU <
0, we have ( S + I + U )( t ) ≤ ( S + I + U )( T ) for all t ≥ T ,hence ( I + U )( t ) ≤ ( I + U )( T ) + [ S ( T ) − S ( t )] ≤ ε ε ε, as we wanted to show.(iii) To prove that R/U converges to ν /ν , we first remark that (cid:18) RU (cid:19) (cid:48) = R (cid:48) U − RU (cid:48) U = ( ν I − ηR ) U − R ( ν I − ηU ) U = ν IU (cid:18) ν ν − RU (cid:19) . (4)Therefore, RU > ν ν = ⇒ (cid:18) RU (cid:19) (cid:48) ( t ) < ,RU < ν ν = ⇒ (cid:18) RU (cid:19) (cid:48) ( t ) > . In other words, if
R/U is smaller (resp. greater) than ν /ν at a point t , then it is increasing(resp. decreasing) around this point.We first prove that R/U cannot be equal to ν /ν at any point. To do this, we note that R /U = 0 (cid:54) = ν /ν . We define ϕ := ν ν − RU . Equality (4) then becomes ϕ (cid:48) = − ν IU ϕ. If T were the first instant at which R/U = ν /ν , then T would be the first zero of ϕ . By(i), there exists C > ν I/U ≤ C on [0 , T ]. Since R = 0, one has ϕ ( t ) > t . Choosing such a t smaller than T we obtain, on [ t, T ) ,ϕ (cid:48) ϕ ≥ − C. By integration, this gives ϕ ( T ) ≥ ϕ ( t ) e t − T , which contradicts the fact that ϕ ( T ) = 0.To prove the convergence of R/U towards ν /ν , which is equivalent to that of ϕ towards0, we will use the fact (proven below) that I/U is bounded from below by a positive constant c . Assuming this, we have ϕ (cid:48) ϕ ≤ − ν c, and hence, ϕ ( T ) ≤ ϕ ( t ) e ( t − T ) ν c , which converges to 0 as T → ∞ .To conclude, we must show that I/U does not approach zero. For this, we begin bynoting that (cid:18) IU (cid:19) (cid:48) = I (cid:48) U − IU (cid:48) U = ( τ S [ I + U ] − νI ) U − I ( ν I − ηU ) U = τ S − ν (cid:18) IU (cid:19) + IU ( τ S + η − ν ) . Since S ≥ S ∞ , if I/U is very small, then the derivative (
I/U ) (cid:48) becomes positive, and therefore I/U grows. As a consequence,
I/U cannot arbitrarily approximate 0.There are several remarks to the proof above.
Remark 2.2.
The proof above applies to more general initial conditions than (3): it onlyrequires that the values S > , I > , R ≥ U ≥ I (cid:48) ( t ), R (cid:48) ( t ) and U (cid:48) ( t ) are all positive. However, note that, at this level of generality, in statement (iii) abovethe convergence of R/U towards ν /ν can occur from above, and even the quotient R/U can remain constant and equal to ν /ν throughout the whole evolution. Remark 2.3.
In the classical SIR model, the fact that the population of susceptibles con-verges towards a positive limit is often presented as a consequence of the so-called final sizerelation [5]. In our context, S (cid:48) not only depends on I , but also on U . For this reason, it ishard to expect such a simple relation, and this partly justifies the use of robust estimates inthe preceding proof. 11 emark 2.4. The convergence of R and U towards 0 must hold at a speed lower than e − ηt .Indeed, from the last two equations of the system one obtains R (cid:48) + ηRν = I = U (cid:48) + ηUν , which can be rewritten in the form ν ( Re ηt ) (cid:48) = ν ( U e ηt ) (cid:48) . By integration, this gives ν Re ηt = ν U e ηt + C, where C := − ν U . Therefore, RU = ν ν + Cν U e ηt . Since
R/U converges to ν /ν , the product U e ηt must tend to infinity, thus showing ourclaim for U . The claim for R immediately follows from this. Remark 2.5.
Another observation is related to the proportion
R/U . Since this numbergrows throughout the epidemic, the dynamic variability that we mentioned at the beginningof this work holds. Moreover, the convergence of
R/U towards ν /ν tells us what shouldbe the values of the parameters in the implementation. We will return to this point for aspecific case in the next section. Remark 2.6.
Finally, we must mention that one of the basic results on the SIR model hasnot been incorporated into the theorem above, namely, that of the uniqueness of the peakof the curve of infected cases. In fact, this uniqueness is no longer valid for the SIRU model.Below we draw an example of a double peak for the reported and unreported cases curves.Note that though this occurs under initial conditions different from (3), it holds in a contextin which the theorem is still valid, according to Remark 2.2.Figure 2: A double peak for the curves.12xamples of this type can be easily obtained. For the curve R , one starts by imposingthe conditions R (cid:48) ( t p ) = 0 and R (cid:48)(cid:48) ( t p ) > t p > S, I, R and U atthat time. Then the SIRU equations are implemented in both directions of time around t p .This moment will hence correspond to a local minimum of the curve, necessarily locatedbetween two peaks.The same argument allows to build examples in which the curve of infected cases hastwo peaks. Naturally, this is due to the presence of U in the derivative of I , which is anabsent element of the SIR model. In fact, in this one, the condition I (cid:48) = 0 necessarily implies I (cid:48)(cid:48) <
0, as a straightforward computation shows. This occurs only at the peak of the curve,which corresponds to the time when herd immunity is achieved.
Our goal now is to compare the data available on COVID-19 in Chile until May 14 (2020)[13] with what is predicted by the SIRU model, so that we can estimate the evolution ofthe number of unreported cases throughout the period. We point out that a first estimateof unreported cases in (some regions of) Chile using the data of March 2020 and the SIRUmodel was carried out by M´onica Candia and Gast´on Vergara-Hermosilla in [6].To begin with, let us point out that in outbreaks of influenza disease, the parameters τ, ν, ν , ν , η , as well as the initial values S , I and U , are generally unknown. However,it is possible to identify them from specific time data of reported symptomatic cases. The cumulative number of infectious cases reported at a time t , denoted by CR ( t ), is given by CR ( t ) := ν (cid:90) tt I ( s ) ds. This is data that is openly available. Likewise, the cumulative number of unreported casesat a time t is CU ( t ) := ν (cid:90) tt I ( s ) ds. We note that, up to constants, these values coincide respectively with R ( t ) + η (cid:90) tt R ( s ) ds and U ( t ) + η (cid:90) tt U ( s ) ds. CR ( t ) has an (almost) exponential form: CR ( t ) = χ exp( χ t ) − χ . For simplicity, we will assume that χ = 1. The values of χ and χ will then be adjusted tothe accumulated data of cases reported in the early phase of the epidemic using a classicalleast squares method (after passing to logarithmic coordinates). According to the above(see [10] for details), for numerical simulations, the initial time for the beginning of theexponential growth phase is fitted at t := − χ · log ( χ ) . Again, for simplicity, we will identify the initial value S to that of the total populationof Chile (since there is no prior immunity against the virus). Once the values of ν , η and f are set, the conditions at the beginning of the disease are naturally fitted as I := χ f ( ν + ν ) = χ ν , U := (cid:18) (1 − f ) ( ν + ν ) η + χ (cid:19) I = ν I η + χ , R := 0 . We recall that 1 /ν corresponds to the average time during which patients are asymp-tomatic infectious. As in [10, 11, 12], we will let this parameter be equal to 7 (days), and thesame value will be used for the average time in which patients are reported or unreportedinfectious: τ = η = 1 / . Note that although the incubation period has been reported as being slightly smaller, thelag in the delivery of the results of the PCR tests in Chile justifies this choice.
To implement the SIRU system we still need to establish a good value for the parameter f (the fraction of symptomatic cases that are reported). Once this is fitted, we will have thevalues of ν = f ν y ν = (1 − f ) ν, Specifically, to adjust χ , χ we consider the next 28 days since the detection of the first case in Chile(Talca). τ .Before continuing, it is worth pointing out that in [10, 11, 12] there is no major discussionon the criterion used to establish the value of f in the different scenarios. Actually, a valueissued by the medical counterpart is assumed as valid. (For example, f = 0 . Since Baeza-Yates’ argument is simpler and is not included in an academic publication, weborrow it below in a language closer to that of the SIRU model. As we will see, it yields amethod to adjusting the value of f that can be used in almost all contexts.Since we know that R/U converges towards ν /ν , we assume for this calculation that R/U is simply equal to ν /ν . In addition, we will argue in discrete units of time (in days).Then we have R ( n ) = f [ R ( n ) + U ( n )]If d is the average time of illness to death and M ( n ) the number of deaths in day n , thenthe case fatality rate L corresponds to L = M ( n )[ R ( n − d ) + U ( n − d )] . Moreover, the reported case fatality rate is L R = M ( n ) R ( n − d ) = M ( n )[ R ( n − d ) + U ( n − d )] · [ R ( n − d ) + U ( n − d )] R ( n − d ) = Lf .
This gives f = LL R . In the Chilean context, deaths in the period studied occurred within 9 . d = 9, the computation of L R is made feasible from thedata available in [13].Finally, regarding the natural case fatality ratio L of the disease, it is deduced frominternational studies that, after adjusting it to the age distribution of the Chilean population,it should vary between 0 .
2% and 1%, with a very high tendency to be around 0 . f varying between 0 . .
5, with a high tendency to beclose to . ∼ . They estimate this number between 60% and 70% higher than the one reported for the period studied. .3 Variations in the transmission rate Given the heterogeneity of the safeguard measures taken by the Chilean government, insteadof directly applying the SIRU model, it became more pertinent to us to consider a variabletransmission rate τ (as a function of time). To analyze the variation of the values of τ ( t ),we observe how the percentage of the Chilean population subjected to confinement has beenchanging, which is illustrated below:Figure 3: Variation of the percentages of the population in quarantine in Chile correspondingto the first 49 days from March 13, 2020.We hence propose a function τ ( t ) of the form τ ( t ) = τ if t ∈ I = [ N , N ] ,τ ( t ) = τ exp( − µ ( t − N )) if t ∈ I = ( N , N ] , ... ... τ r ( t ) = τ r − ( t ) exp( − µ r − ( t − N r − )) if t ∈ I r = ( N r − , N r ] , where the I i ’s correspond to successive time intervals. For the Chilean context, accordingto the graph of quarantines illustrated above (Figure 3), the chosen time intervals are thosedescribed in Table 2 below: 16nterval Time frame N i I from March 3 to 22 March 22 I from March 23 to April 1 April 1 I from April 2 to 11 April 11 I from April 12 to 21 April 21 I from April 22 to 31 April 31 I from May 1 to 10 May 10 I from May 11 to 14 May 14Table 2: Time intervals used in our numerical simulations.Following [10, 11, 12], the value of the transmission rate in I is fitted to τ := (cid:18) χ + νS (cid:19) (cid:18) η + χ ν + η + χ (cid:19) . Then, the parameters µ i are chosen in such a way that the reported cumulated cases in thenumerical simulation align with the data of the reported cumulative number of infectionsat time t according to [13]. This is implemented with various values of f , following theguidelines of the preceding subsection (specifically, we work with f = 0 . f = 0 . f = 0 . f = 0 . f = 0 . f = 0 . µ · − · − · − µ · − · − · − µ · − · − · − µ · − · − · − µ · − · − · − µ · − · − · − Table 3: Parameters µ i corresponding to the different values of f considered in our numericalsimulations.Using the data of the cumulated reported cases available in [13], we can finally proceed tothe simulations, which are shown below. They illustrate numerical estimates for the curvesof CR ( t ) , CU ( t ) , R ( t ) and U ( t ). 178igure 4: Plots of the numerical approximations of the functions CR ( t ) , CU ( t ) , R ( t ) and U ( t ) obtained from the numerical solutions of the model (1) applied to the Chilean contextbased on the data of reported cumulated cases up to 14 May 2020 [13]. The plots (A), (B)and (C) were obtained by considering f = 0 . f = 0 . f = 0 . η = 1 / ν = 1 / Discussion and future work
The epidemic outbreak of the new human coronavirus COVID-19 was first detected inWuhan, China, in late 2019. In Chile, the first case was reported on March 3, 2020, in Talca(Maule region). Since then, modeling the epidemic in the country has faced the problem ofthe low availability of disaggregated data [3].The first part of this work was inspired by the situation experienced in Chile duringthe month of April, when the number of reported cases stabilized around 450 per day. Ac-cording to most specialists, this was not accurately representing the genuine epidemiologicalevolution. Although it is difficult to argue that our reasoning from the first section fully19pplies to this situation (in particular, because the number of daily tests during the periodwas variable), the explosive increase of cases that subsequently occurred should call us toreflection on the point. Nevetheless, it is clear that this increase was also due in part tothe relaxation of the protection measures, which is reflected not only in the absolute (andexponential) increase in the number of cases, but also in the proportion of positive caseswith respect to tests (the latter despite of the significant increase in the number of tests [8]).The first part of the work naturally led us to model the dynamics of the epidemic incor-porating a compartment for unreported cases so that we could deal with their active role inthe evolution. To do this, we used the SIRU model recently introduced/implemented by Liu,Magal, Seydi and Webb in [10, 11, 12]. Since the qualitative theory of the underlying dif-ferential equations has not yet been treated, we developed some essenttial points of it in thesecond section, thus rigorously establishing fundamental structure theorems. However, wepointed out an important difference with the SIR model, namely, the possibility of multiplepeaks for the curves of infected, reported and unreported patients. For the future, it wouldbe desirable to complete the qualitative description of the solutions of the equations withrespect to the parameters, with an emphasis on the phenomenon of multiplicity of peaks.For example, until now we do not know whether there can be more than two peaks and/orwhether there is any restriction on the relative position between them. Without any doubt,this discussion is relevant for the implementation of disease containment policies, since itis directly related to the recognition of the moment when the curves begin to definitivelydescend.In the last section of the work, we implemented an extension of the model to the caseof Chile. To do this, we considered a variable transmission rate, which is more appropriateaccording to the local reality. Our rate is coupled to the official statistics provided by thegovernment in [13], information that also allowed us to make the parametric afjustement.The last ingredient to launch the simulations was to fit the value of the fraction f of infectedindividuals that were reported during the period of study. For this we used the work ofBaeza-Yates [4] and that of Castillo and Past´en [7]. Naturally, our modeling is consistentwith their estimates. In particular, in Section 2.3 of [7], the authors estimate the number ofreal cases for April 28 as 43095, which is remarkably close to the number of total cases thatcan be inferred from the plot (B) in Figure 4 obtained by considering f = 0 .
3. We point outthat the prediction of Castillo and Past´en is recovered with complete accuracy through themethod used in this work for the value f = 0 . . . This corresponds to day 57 from the first case detected in Talca. a posteriori , their complementarity puts usin a good position to use them for modeling the future evolution of the epidemic. However,working in this direction requires great caution. In particular, it would be necessary toconsider a variable parameter f , since both the quantity and the criteria of the tests havebeen modified during late May. Despite this, we strongly believe that the basis for pursuingthe implementation of the SIRU model are fullfilled, and it would be very useful to advancein a more compartmentalized and georeferenced implementation of it. Acknowledgments.
We would like to thank Mar´ıa Paz Bertoglia, M´onica Candia, AliciaDickenstein, Yamileth Granizo, Rafael Labarca, Mario Ponce and Marius Tucsnak for theirreading, their kind remarks and suggestions of bibliography.
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