Resume results
1. Parameter definitions and values
Table 1. State and parameter definitions with their reference intervals, where the 
symbol represents the fixed parameter values, i.e., those parameters that were not estimated.
symbol represents the fixed parameter values, i.e., those parameters that were not estimated. 2. Sentitivity analisys
Figure 1. Sensitivity analysis for the general transmission CHIMERA model using the J as output reference. We plot the STi as a bar chart for both localities using the estimation intervals (see Table 1); also, note that for initial conditions as 
and D we took Colombian and Hubei populations as the lower and upper interval values, respectively. We highlight that the three identification parameters (
and 
), have a high importance inside the variance output.
and D we took Colombian and Hubei populations as the lower and upper interval values, respectively. We highlight that the three identification parameters ( and ), have a high importance inside the variance output. 
Figure 2. Vectorial Sensitivity indices for the general model, implementing theestimation intervals presented in Table 1. The vectorial chart represents the parameter effects over the model output in each time step.
3. Identifiability analysis
Figure 3. Identifiability analysis for the parameter estimation; from 9000 and 4000 performed estimations, we filtered the best 114 and 178 ones for Hubei and Colombia, respectively. Both box plots represent the values estimated for each parameter normalized from 0 to 1 using the minimum and maximum estimation intervals; where we can see the tendency values for each estimated parameter. For Hubei, the most identifiable parameters are 
and 
, conversely, the less identifiable ones are 
, 
, and 
. For Colombia, the most identifiable parameters are 
, 
, and ; conversely, the fewer identifiability parameters are a set related to recovery and death of the High symptomatic population as 
, 
, 
, and 
.
and , conversely, the less identifiable ones are , , and . For Colombia, the most identifiable parameters are , , and ; conversely, the fewer identifiability parameters are a set related to recovery and death of the High symptomatic population as , , , and .4. Model fitting
Figure 4. Fitting results for Hubei and Colombia.
5. Uncertainty analisys
Figure 5. Uncertainty analyses for Hubei and Colombia, where the red dotted lines represent the real data for both localities and the blue lines corresponded to 1000
parameter families with Montecarlo simulations, using the interval parameters in Table 1.
Discusion
Our main result was creating and validating a mathematical model developed from the proposed CHIMERA structure using as a case of study the COVID-19 transmission. It was possible by considering parameter estimation performed for two localities with different socio-cultural behavior but sharing the same data-type (infected, recovered, and death cases): Hubei and Colombia. The model fits the epidemic dynamic presented in each study zone, as can see in Figure 4; these results represent by itself suggest a progress on the methodology for modeling and fitting transmission models with real data. We were able to perform a suitable fit after the exponential growth of the disease, which is an issue raised in previous Erlang models as mentioned in [1]. Even more, we were able to estimate the distribution function related to the diffusion systems, as Champredon and Dushoff [2] suggested for future work in the area.
Through an SA, we identified that the model is not over-parametrized and the most representative parameters are those related to social behavior and control policies, whose combined, describes most of the output variance (see Figure 1). At the sight of the CHIMERA model structure, the probability of entry quarantine is the most-relevant parameter and manages the number of people in free circulation altogether with the probability of skip quarantine. This dynamic determines the number of people that could have an infectious contact and, therefore, the magnitude of the disease outbreaks [3].
Parameters related to positive-case identification have a considerable effect on the transmission phenomena and, therefore, on the model outputs; for instance, some studies have found that presymptomatic population generates 44% of the secondary cases \cite{Sjdin2020}. Hence, parameters related to positive-case identification and the sanitary barrier becomes relevant to the reduction of disease propagation according to the model general structure [3, 4]. Besides, we identified that the most relevant infection probabilities are related to presymptomatic and low symptomatic populations, i.e., the human to human contact. The less relevant infection probability is that caused by environmental reservoir infection; nevertheless, it generates a variation in the model output according to the SA, so the virus reservoir contributes to the number of new infections and we can not ignore it [5].
When contrasting the first outbreak dynamic for Hubei and Colombia through its estimated nominal values (see Table 1) we found that the greater value of infection probabilities, the sanitary barrier, and presymptomatic identification in Hubei could indicate that, although its population is more susceptible to develop the disease, they quickly controlled the transmission because of the efficient control policies, as other authors pointed out [3, 5, 6]. Conversely, in Colombia it took more than 150 days for the number of active cases to decrease (see Figure 4) suggesting a reduced capacity to identify presymptomatic carriers, develop an effective sanitary barrier, and make quarantine more strict [7]. Such interpretation coincide with explanation due to nominal parameters in Table 1.
After discussing the results presented for the first outbreaks in Hubei and Colombia, we can not conclude that the parameters or even the natural history of the diseases would be the same over time; instead, we have to note that the pathogen can change its behavior. For example, if the mortality increases, affecting asymptomatic or low symptomatic carriers, it must be necessary to change the basic structure of the diffusion systems appending a flow from low symptomatic to dead compartment. Thus, the researchers must update the structure of the model following the behavior of the disease to achieve an informative model, for instance, modeling the effect of vaccines over the population. Also, it could be necessary, at some point, to introduce input parameters or functions to simulate new outbreaks besides the old ones, following the idea that with more information, we can have a better identification of the model.
All the previous interpretations could apply to the general CHIMERA model and explain its dynamics. We can identify the specific dynamic of a locality through its parameter estimation using real data [8] and, when facing a real-word process as disease transmission, the parameter estimation becomes a multi-objective optimization problem and a complex process [9]. The fundamental challenge in this process is the number of unobservable parameters to estimate and define their corresponding intervals (26 parameters and 2 initial conditions, according to Table 1), followed by the computational effort needed to perform the optimization algorithm [10]. Usually, the simulation of the estimated parameter set could not fit all real data simultaneously. Nevertheless, we identified the opportunity to perform the parameter estimation because of the quantity and quality of the data; thus, not all infections diseases have had this magnitude in free-available data. For this, we successfully followed the identifiability criteria [8, 11]and obtained some parameter set that fits the Hubei and Colombia real data (see Figure 3) beside a validation with the UA for respective parameter intervals (see Figure 5}).
We were able to find a solution that fits real data from three time-series within biological and epidemiological intervals. To achieve this result we focus on four factors that help to turn the parameter estimation into a practical identifiability process [8]: (i) the definition of consistent estimation intervals for the model [10]. (ii) A cost function that supplied some researcher preferences [12]. (iii) A filtering of estimations according to the fit quality and the identification of a suitable set of parameters following the box plot criterion proposed in [11] that the reader can see in Figure 3. (iv) The modeling process which led to the resulting CHIMERA model because trying to describe a complex epidemic with classic 
or 
models is not informative or not even possible [13] given the need to include factors that describe the natural history of the disease and social behaviors. However, it is necessary to perform an in-depth study about the last point to establish where to draw a line between the complexity and goodness of fit for the model that does not affect the biological meaning of the parameters [14, 15].
or models is not informative or not even possible [13] given the need to include factors that describe the natural history of the disease and social behaviors. However, it is necessary to perform an in-depth study about the last point to establish where to draw a line between the complexity and goodness of fit for the model that does not affect the biological meaning of the parameters [14, 15]. Conclusion
The basic-structure we proposed allowed us to built a discrete mathematical model that describes the COVID-19 transmission for two affected very different localities: Hubei and Colombia. We could identify that the strict policies in Hubei helped to finish the first outbreak; conversely, Colombia has not finished it because of the relaxation in its policies. Finally, it will be important that future research investigate a way to estimate parameters during simultaneous outbreaks for different localities.
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