key: cord-0889121-mv1wdzq5
authors: Cecconello, M. S.; Mizukoshi, M. T.; Lodwick, W.
title: Interval nonlinear initial-valued problem using constraint intervals: Theory and an application to the Sars-Cov-2 outbreak
date: 2021-08-16
journal: Inf Sci (N Y)
DOI: 10.1016/j.ins.2021.08.045
sha: c78118b7c30da3a20a5bf679c3433955beb8a2ed
doc_id: 889121
cord_uid: mv1wdzq5

This article discusses the theory of constraint interval solutions to interval nonlinear initial value problems and applies the notion of constraint interval solutions to analyze the asymptotic behavior of a susceptible-infected-recovered (SIR) epidemiological nonlinear differential equation model, specifically the covid pandemic, in the presence of interval uncertainty to illustrate the efficacy of this approach. Furthermore, constraint interval solutions are used to estimate the intervals for the parameters by fitting solutions to the Brazilian’s Sars-Cov-2 pandemic official data. Simulations and graphical solutions incorporating constraint interval uncertainties are presented to help in the visualization of the pandemic’s behavior.

Real-valued mathematical analysis of processes and systems that incorporate uncertainty in its theory has a history that dates back to at least Archimedes 1 e-mail:moiseis@gmail.com. 2 e-mail:tuyako@ufg.br. 3 e-mail:Weldon.Lodwick@ucdenver.edu.

( [1] ) where the perimeter of a circle was approximated via outer and inner polygonal perimeters. Quantitative uncertainties or inaccuracies occur due to data 5 measurement errors, lack of complete information, assumption of physical models, variations of the system, and computational errors. Uncertainty is often represented by distributions ( [2] ). However, a simpler representation of uncertainty is the interval where only the bounds of the uncertainty are known, but not the distribution within the bounds. Intervals naturally arise in measurement 10 errors and in bounding roundoff error. This study limits itself to quantitative uncertainties represented by intervals.

The context in which these uncertainties will be studied is the nonlinear differential equation initial value problems. That is, of interest to this study are differential equations with interval uncertainties in the initial condition or/and 15 coefficients. For intervals, there are two types of interval arithmetic analyses that can be considered: (1) Standard arithmetic, the Warmus, Sunaga, Moore arithmetic (WSMA)( [3] ) where every interval is considered as independent; (2)Constraint interval arithmetic (CIA)([4]), where it is possible to consider total or partial independence, or total dependence. This study limits itself 20 to the constraint interval approach.

It will be seen that the interval solutions obtained via CIA, whether dependent or independent, can result in solutions that may not have the flow property. This can happen because to obtain an interval solution requires a global optimization problem, which, in turn, means that a solution can be found in the 25 resultant interval that was not in the deterministic case. Even in this case, the stability of the deterministic solution, which, in terms of an interval solution, is a degenerate fixed point (the left endpoint is equal to the right endpoint), can be analyzed considering the interval solution over compact sets in R n .

There is another constraint interval analysis, a subset of CIA, of interval dif-30 ferential equations using total dependence called single level arithmetic ( [5] , [6] ). However, this line of analysis will not be pursued here. Epidemiological mathematical modeling has been used to assist the decisionmaking processes and many research articles have been published with the aim of analyzing the dynamics of the epidemic in the population ( [7] ,[8], [9] ). In Brazil, 35 which is used in the development of the interval SIR model, there are many states with diverse social-economic situations. This means that the information obtained that is used to determine the parameters of the model (inflection rates, recovered rates, for example) have varied precision. Even the tests given to determine how many people in the population are infected at any period of time 40 are not deterministic. Thus, the application of uncertainty in the SIR nonlinear differential equation model is not only necessary, it is useful and timely. The parameters for the case of the epidemic in Brazil are obtained via a constrained interval regression procedure on Brazil's Ministry of Health Sars-Cov-2 data.

This work is organized as the following: in Section 2 we present the basic 45 concepts on the CI theory. In Section 3 we define the concepts of interval solution of initial value problems and asymptotic behavior of such solutions. In Section 4 analyze the SIR's model interval solution. In Section 5 we estimate the parameters of SIR's model interval solution using the Sars-Cov-2 data of Brazil. 50 

This section describes the main concepts on the constraint interval (CI) theory. IR will denote the set of all closed intervals on R and P the space of linear functions p(γ) = a + bγ over the interval [0, 1] with non-negative slope (b 0).

The main feature of the CI theory consists in identifying a given interval [x] = [x, x] ∈ IR with a polynomial p x ∈ P by means of the mapping I : IR → P defined as

It is easy to check that I([x]) is a function so that each function p(γ) = a + bγ in P defines an unique interval [x] = I −1 (p) = [a, a + b] in IR. Denotes by IR n the set of all closed hypercubes on R n . In this way,

∈ IR n and a vector a = 60 (a 1 , a 2 , · · · , a n ) ∈ R n the notation a ∈ [x] means that there are

This mapping approach allows for the concept of extension of a function f : R n → R to be defined, (see [4] ). Next, P n is the cartersian product of the representation P. That is, P n = P n−1 × P, for n 2.

Definition 1 (CI extension). Consider a function f :

The CI extension of f is the functionf : IR n → IR such thatf ([x]) is computed by the following procedure: a) Mapping to representation space: First, the vector [x] ∈ IR n is embedded into P n by means of the map I n : IR n → P n defined as

, · · · , p n (γ n )) = (x 1 + γ 1 w 1 , x 2 + γ 2 w 2 , · · · , x n + γ n w n ),

Evaluating the function: Next, define g : IR n → F by computing

F is a space of multivariate functions. 3   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 Given a function f : U ⊂ R n → R m , the CI extensionf is defined taking the CI extension on each one of the coordinates 

The approach considered in Definition 1 allows IR to be endowed with the basic arithmetic operations, say, addition, subtraction, multiplication and di-80 vision, by considering the following functions, respectively:

Furthermore, IR n can be also turned into a metric space by considering the Pompieu-Hausdorff distance for compact sets given by:

[x], [y] ∈ IR n ([10]).

Now, consider A and B subsets of R n , the projection π i : R n → R is such that π i (x 1 , x 2 , · · · , x n ) = x i . Thus π i (A) and π i (B) are unions of points of the metric space IR and, since |x i − y i | ||x − y|| for all x ∈ A and y ∈ B, then

Therefore, it follows that: 

||f (t, b) − g(a)|| < ε.

(3) for all t > T. Now, 

||f (t, a) − g(a)||.

By the uniform convergence hypothesis, given ε > 0 there exist T > 0 such that ||f (t, x)−g(x)|| < ε for all t > T and x ∈ U. Therefore, for these conditions, it turns out that

||f (t, a) − g(a)||} ε for all t > T, proving the statement. 

The main goal in this section is to consider constraint interval uncertainties on initial value problems as well as to analyze the impact of these uncertainties in the asymptotic behavior of the solution. To this end, we are going to consider two cases: i ) -uncertainties only on initial conditions; ii ) -uncertainties on both 100 initial conditions and parameters.

Consider an initial value problem

f : U → U, U ⊂ R n where x t : U → U denotes the (deterministic) flow of the initial value problem defined by Eq. (4).

Now, consider an initial condition x 0 subject to generalized uncertainties. The CI extension of the flow x t isx t : IR n → IR n such that for each [x 0 ] ∈ IR n we associate the constraint interval vectorx t ([x 0 ]) ∈ IR n for all t ∈ R. In this way,x t ([x 0 ]) is defined as the constraint interval solution, for the interval initial condition [x 0 ], of Eq. (4).

Although the applicationx t : IR n → IR n is not necessarily a flow over IR n (see Figure 2 ) its asymptotic behavior can be analyzed by looking at the behavior of x t over compact sets of U ⊂ R n . For a dynamical system x t defined over a metric space X we say that S ⊂ X ⊂ R n is an invariant set if for x t (S) = S for all t ∈ R. Two of the most ordinary examples of invariant sets 115 are the equilibrium points and periodic orbits.

Moreover, according to [11] in the case of S being uniformly stable, then for any neighborhood V of S and a compact

The next statements establish a relationship between the flow x t and its CI 125 extensionx t . 5   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 Theorem 2. Consider the flow x t : R → R given by Eq.(4) and its CI extension x t : IR n → IR n . The following statements are true:

The first part of this statement follows directly from the identity

). To prove the second and third parts of the statement it is enough to check that for any neighborhood W

is also a neighborhood of the x when x is seen as a point in IR n . The converse is also true. The neighborhood

Thus, the statements follows directly from the definition of stable and uniformly stable. 

For a given neighborhood V of S it is true that:

Proof. To prove a), given ε > 0 let us consider the ε-neighborhood V of S defined 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 To prove b), suppose S uniformly stable. Thus, S is stable and for a given neighborhood V of there is a neighborhood V ′ of S such that for any neighbor-

∈ W for all t > T and the statement is proved.

Exemple 1. Let us consider the two-dimensional system defined by the equations ( [12] )

Now, let us consider an initial condition [x 0 ] ∈ IR 2 defined by p 1 (γ 1 ) = x 01 + γ 1 w 1 and p 2 (γ 2 ) = x 02 + γ 2 w 2 , w i = x 0i − x 0i , i = 1, 2. Thus, the constraint interval solution is given by:

towards the circumference S of radius k for all x 0 = (0, 0) so that S is an asymptotically stable invariant set. Therefore, Theorem 3 ensures that the set 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 compute the interval solutionx τ ([x 0 ]) at τ = 2π. As we clearly see, it turns out thatx 2π ([x 0 ]) (the blue box on the right) is not equal tox 3π/2 (x π/2 ([x 0 ])) (the yellow box) so thatx t is not a flow over IR 2 . The red boxes are the set x t ([x 0 ]) at t = π/2 (left) and t = 2π (right).

The results previously presented can also be used to analyze the behavior of initial value problems with uncertainties on parameters. In fact, given the equation

where f : U × V → R n , U ⊂ R n , V ⊂ R k is a sufficiently smooth function such that (7) has a solution x t (x 0 ), ∀t ≥ 0. Given (7), we can defined the equation 8   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 

Our main goal in this section is to consider interval uncertainties in the SIR model as well as to analyze the impact of these uncertainties in the asymptotic behavior of the solution. To this end, we are going to consider interval 185 uncertainties for both initial conditions and parameters.

A classic mathematical model to describe the spread of an infectious disease on a population as function of time is the SIR model, proposed by Kermack -McKendrick in 1927 [13] . Despite of its apparent simplicity, the SIR model 

where α and β are both positive parameters [14] .

Since that S(t) + I(t) + R(t) = 1, defining Z(t) = I(t) + R(t) the SIR model can be rewritten as

where the solution x t (I 0 , Z 0 ) of Eq. (10) is defined on the set

The reproductive number ρ = α β which is the ratio, defines the asymptotic behavior dynamics of Eq. (10) and, according to [15] , it is a key parameter determining whether an infectious disease will persist. In fact, looking at the 9   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 equations of Eq. (10), and considering I 0 > 0, it can be seen that Z(t) is a increasing function of time. Yet, for I(t) it turns out that

Thus 

so that the eigenvalues are λ 1 = α(1 − z) − β and λ 2 = 0. We can easily check that z < 1 − 1/ρ implies λ 1 > 0 so that E = (0, z) is unstable. Therefore, we can conclude that the set S is invariant and attracts all solutions of Eq. (10). In 205 fact, since the dynamics of Eq. (10) are bounded on U then for every compact set B ⊂ U −S the ω-limit, ω(B), is a nonempty, compact and invariant subset of U ( [16] ). Since S ⊂ U is the a invariant set on U then we must have ω(B) ⊂ U and, therefore, S attracts compact subsets B of U.

Using the chain rule for derivatives it turns out that

whose the solution at (I 0 , Z 0 ) is given by

Therefore, the equilibrium for Eq. (10) is the solution of I(I 0 , Z 0 , Z) = 0, which can be expressed as

with W being the Lambert W function ( [17] ) and c = Z 0 − I 0 + ρ −1 ln (1 − Z 0 ). To account for uncertainties on the parameters we follow the recipe presented in the previous section and consider the expanded system 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 so that y t (I 0 , Z 0 , α, β) = (x t (I 0 , Z 0 ), α, β) is the solution to Eq. (13) .

Using the results discussed earlier it turns out that y t (I 0 , Z 0 , α, β) converges uniformly to E(I 0 , Z 0 , α, β) = (0, Z e , α, β) and the S = {0} × [1 − 1/ρ, 1] × {α} × {β} is the attracting invariant set for y t .

Based on the asymptotic behavior of x t and y t together with Theorem 1 and Theorem 3 the following is behavior of the CI extensionsx t andŷ t . To this end, let U ⊂ R 2 be the set U = {(I, Z) ∈ R 2 : I, Z > 0, Z 1, Z I} and as before π i is the orthogonal projection onto the ith axis . Proof. The first item follows directly from the Theorem 3.

To prove b), it is enough to see that y t converges uniformly to E for all 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 5. Regression procedure Several policies to mitigate the spread of the disease have been proposed by Brazilian authorities. Moreover, such policies have changed over time as well as with geographical areas and such changes can be an obstacle to fit the model to the data. In this section we are going to use the constraint interval 250 solution of the model here presented in order to predict the time evolution of the Sars-Cov-2 disease in Brazil.

We are mainly interested in the cumulative of cases, or total cases, of Sars-Cov-2 in Brazil. Among many other useful information, the cumulative number Let a k be denote the cumulative of cases at time k, k ∈ {1, 2, . . . , N }, k ranging from 26 February 2020 to 10 July 2021. From the time series data of 260 total cases we can infer the daily new cases just by computing the difference of two consecutive total cases data, that is n k = a k − a k−1 . We can also infer the number of currently infected by computing the difference i k = a k − a k−τ , where τ = β −1 is the infectious period of Sars-Cov-2. Figure 3 . In 265 order to better fit the model to the data, we split the data set in two parts: the first one ranging from 26 February 2020 to 03 November 2020 and the second one ranging from 04 November 2020 to 10 July 2021.

To the regression procedure we are considering the relative values (cumulative cases and currently infected) to the total population, currently estimated at 270 211 million individuals. Furthermore, following the classical approach used in machine learning ( [20] ), we split the first part of the data set (from 26 February 2020 to 03 November 2020) into training and testing sets (85% − 15% of the data, respectively) in order to validate the model. 12   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 

The infectious period is key parameter to any epidemiological model and several studies have been made in order to estimate it to Sars-Cov-2. A study conducted in [21] suggests that infectiousness may start 1 to 3 days prior to the onset of symptoms and declines within seven days. Furthermore, transmission after 7 to 10 days of illness is unlikely [22] .

It is also important to consider the under-report aspect of the data published by Official Brazilian government sources. According to [23] , under-reported infections range from 11 to 30 times the number of confirmed cases and in [24] suggests that reporting rate is at 9.2% (IC95: 8.8% -9.6%) of confirmed cases. We use these estimates to explore some scenarios, comparing the results with 285 the best-fit scenario.

Let a k and i k be the proportions of reported total cases and infectious at time k, respectively. As we have described previously, the behavior of the dynamics relies on the ratio ρ = α/β so that we suppose uncertainty just on α, considering 290 β a deterministic unknown. Therefore, we assume that the initial conditions I 0 and Z 0 , with Z 0 = I 0 , as well as α are unknown intervals.

Let

) be the CI extension of solutions I(t) and Z(t) from Eq. (13) and let us defineĨ t and Z t as

To 

where s > 0 is a constant accounting for under-report of cases and a 0 is trade-off parameter.

For the optimization process we use the Octave's fmisearch algorithm ( [25] ), setting both TolFun and TolX options as 1e-7.

The pandemic behavior, as previously mentioned, is defined by the reproductive number ρ = α/β and the previous procedure allows us to determine the 305 reproductive number for the whole data set. However, due to changes in both policies and behavior of the population this reproductive number can change over time and nevertheless we would like to measure it.

The main idea is to track the value of reproductive number over time consists in finding the CI solution that best fits a subset of the data set. That is, to compute the instantaneous reproduction number R k+1 at a time k + 1, the objective function

is minimized whereĨ t andZ t are as in Eq. (14) and the CI solution is computed considering

Finally, R k+1 is given by R k+1 = α k+1 /β. That is, in the above procedure, for each k in the range of 315 the data set, given the initial condition (I 0 , Z 0 ) = (i k , a k ) the interval [α k+1 ] = [α k+1 , α k+1 )] is found such that the CI solution best fits the data (i k , i k+1 , i k+2 ) and (a k , a k+1 , a k+2 ).

The results of this section are obtained following a machine learning approach on the first part as follows: for each β and s fixed, the objective function (15) is minimized over the training set ( [20] ). Next, we compute the Mean Absolute Percentage Error (MAPE) using

over the testing set in order to find the reporting rate s that best predicts the Table 1 .

We use the estimations for β and the reporting rate s of the first part of the data set to find the parameter α that best fits the model to the data of the 325 second part of the data set. 14 1 2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 reported rate of s = 9.2%.

As well know, parameters α and β are paramount to describe the dynamics of spread and they are usually not known at the beginning of a new infectious disease, as it is the case of the Sars-Cov-2 infection. Estimations as presented in Table 1 could be informative to predict and control the spread of the disease 335 as well as be useful in comparison to other estimation methods.

To illustrate the results, in Figure 4 we present the CI solution that best predicts the data on the test set as indicated in Table 1 . The reproductive numbers estimated from Table 1 are obtained by fitting   15   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 the CI solution to the first part of the data set. Thus, these estimates refer to 340 an overall dynamics and, since social distancing policies as well as the behavior of the population change over time we should expect fluctuations on the reproductive number as time evolves. Since policies of mass testing can also change over time, the reporting rate s can also be affected. Therefore, although the parameters in Table 1 give reasonable predictive values on the test set the CI 345 solutions based on them is not able to predict the second wave of infections.

To the second wave of infections we use previous estimates for β and then the objective function is optimized over the second part of the data set (from 04 November 2020 to 10 July 2021) to find the [α] and s that best fits the model to the data from 04 November 2020 to 10 July 2021. Figure 5 shows the behavior 350 of the CI solution that best fit the data in this scenario. Figure 6 shows the 7-day moving average of the daily reproductive number R t estimated following the recipe described in Subsection 5.4. The mean of daily reproductive number R t by epidemiological week of 2020-2021 is also shown. 16 1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 Finally, we compare our approach to estimate the time-varying R t with the method presented in [26] . Denoting by d t the number of new cases at time t, the authors in [26] define the time-varying reproductive number by R t = d t /( t j=1 w s d t−s ) in which w s is a probability measurement related to infectivity. Considering β = 0.1214 then the infectious at time t are they who get 385 infected up to 8 (1/β) days earlier. The red lines in Figure 6 are the time-varying reproductive number estimated according to this approach (CFFC method) assuming w s = 1/8 for s = 1, 2, . . . , 8. The mean value for the R t estimated by the CFFC method is 1.14 (CI 95%: 1.09 − 1.20) and the data defining the black and red lines on the left in Figure 6 3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 coefficient.

This study presented the theoretical foundations of the constraint interval representation in nonlinear differential equations and least squares problems under interval uncertainties in initial conditions, parameters, and data. Using 395 the constraint representation of interval uncertainty, this study showed how to define and obtain resulting solutions to interval nonlinear differential equations and interval least squares problems. The efficacy and efficiency of the constraint interval approach was demonstrated in the analysis of a current nonlinear differential equations applied problem, the SIR analysis of the current COVID 400 pandemic in Brazil, where the interval parameters and initial conditions were calculated using interval least squares approaches together with machine learning .  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 

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