). We modeled

-pulse input; its value can be positive or negative, in which a positive value represents a control effect and a negative value corresponds to a increase in mosquito natality due to, for example, favorable environmental conditions:

In the present study, we adapted the mathematical model (1) and estimated its parameters considering scenarios with and without pulse-type inputs. We identified diverse dynamics among the three studied cases: we realized that case A (no pulse-type input) provides some estimated parameters anchored to estimation limit ranges (see Table 2), e.g., α, θm, θh, and μe. Those estimated values suggest that the estimation algorithm still might find a better minimum outside the estimation intervals, i.e., in a region with a lesser biological sense; moreover, the model does not provide endemic behavior, as shown in Figure 1a. Conversely, both cases B and C did not present anchored parameters, as presented in case A (see Table 2), and they did display the endemic response (see Figure 1b,c).

The model behavior in the three scenarios makes sense from a biological perspective because Bello authorities performed fumigation to mitigate the spread of dengue. Thus, if we did not consider a pulse input in the model, the fumigation effect would be included as an implicit component of the estimated value in μm and other parameters. We highlight that, in case A (um=0), we could not reach an endemic behavior, regardless of the number of estimations we performed. We hypothesize that the lack in endemic behavior is caused by the real data behavior that the model itself could not reproduce without pulse inputs, where dengue cases increased and decreased rapidly during January 2010–July 2011 outbreak because of the anthropogenic or environmental disturbances. For the case of B (um=um1), we incorporated the effect of the exogenous action, allowing the model to reproduce an endemic scenario after the first outbreak.

For the case of C, the model had several available pulses (um=um1+um2+um3+um4). This means that the model can fit the rapid changes in real data caused by anthropogenic and environmental disturbances such as fumigation or climatic variations. Hence, we hypothesize the pulses act as excited signals and theoretically lead to a better estimation [53] that could reproduce trends in epidemiological data overcoming limitations of the non-pulse scenario (case A). Even more, future research could study the changes in the model equilibrium caused by the presence and absence of pulse inputs, which is a theoretical point of view to identify mathematical characteristics that we did not address in this applied study.After we obtained nominal values for cases A, B, and C, we estimated CSB (see Table 2, columns 5, 7, and 9) as described in Section 2.5. Figure 4 shows the CSB and their relation with the biological and initial estimation intervals, which allows us to identify that some parameters are outside of biological enclosures (the respective CSB are also outside of biological enclosures). We performed both UAs and SAs to assess the relevance of the contours we obtained from the CSB; we can see graphically these analyses in Figure 2. In the first example, for an uncertainty level of 30%, the UA shows that the CSB intervals define a band that includes the nominal curve and real data for each case without non-representative curves. This case implies that CSB intervals are suitable for describing the uncertainty related to several processes, such as misreporting in real data and numerical errors during parameter and interval estimations, among others. Moreover, the bands we obtained for case A, following the same behavior of the nominal curve, never showed the endemic behavior. Conversely, the UA of cases B and C provides both the outbreak and endemic behavior, which shows an effect of pulse-type inputs on both parameter estimation and interval estimation.As the next step for assessing the CSB method, we performed an SA focusing on case C to quantify how much each parameter contributes to the uncertainty in the model by varying parameter values over CSB intervals. The results are presented in Figure 3a,b, which correspond to the scalar and vectorial SA, respectively. The vectorial analysis is time-dependent, so the STi indices are calculated over the set of model output for each time step in the simulation; thus, we can determine the relative contribution of the parameters at each time. From vectorial STi, we noticed that the CSB intervals define a region where most of the parameters play an important role, at least from intervals where one could expect this behavior, i.e., the initial conditions are relevant at the beginning of the simulation but irrelevant when the model tends to equilibrium.Note that mathematical models are not perfect representations of a phenomenon; thus, all parameters are unlikely to achieve high relevance in SA using the CSB approach. However, we should not remove from the model all non-sensitive parameters. Instead, we could fix them at any value for a future work in which we want to identify those dynamics. Using the vectorial SA, we can identify that the pulse-related parameters (umj, t0cj and Δtcj) are relevant for the time intervals where they are defined, disturbing the ranking of relevance and taking the model out of a stationary tendency (see Figure 2). Additionally, we found that parameters related to aquatic states are relevant only at the start of the outbreak because they do not contribute to the stationary dynamics of the model. We hypothesize that early dynamics of the aquatic states cause a pulse effect over the mosquito population at the beginning of the outbreak. The early pulse effect could generate an increase in the vector population and, therefore, an increase in infections that led to the first outbreak. Thus, for future analyses, we suggest performing estimations with an initial pulse-type input.The scalar STi indices that are summarized in Figure 3a were obtained using the Xiao method, whose basis relies on distance components decomposition, i.e., it is a method that allows us to synthesize all the information from the vectorial SA into a single scalar SA index. Hence, Figure 3b constitutes a rapid insight into the relevance of each parameter throughout the entire simulation. The goal of the CSB approach is to achieve intervals such that all the parameters have nearly the same contribution to the model output; however, this was not possible in our study. Thus, readdressing the discussion in the previous paragraph, we attribute the non-uniformity of the SA for CSB to the following causes: (i) the model is an approximation of reality and does not contemplate all spread components and variables; (ii) the CSB is an approximation and only considers a sub-contour box and does not consider the confidence contour itself; and (iii) even with a significant amount of computational time, there are numerical and computational limitations to approximate the CSB.After performing the SA to validate the CSB intervals, we implemented another SA approach, based on the modified Saltelli method for MSE outputs, to assess parameter influence. This approach allowed us to identify the parameters with the highest potential for changing the nominal output behavior when all the parameters have the same uncertainty level. We chose a unique uncertainty level such that it also defined a band that enclosed the real data. Thus, we determined parameters related to mosquito mortality (α and μm), infection interactions (βh and βm), and human recovery (γh) has the greatest influence in model outputs.We followed these non-classical methods to identify the most important parameters in the model dynamic (UA and SA), which represent, to some extent, a practical identifiability analysis, as proposed by Lizarralde-Bejarano et al. [40]. Therefore, alongside a classical analysis, as presented in the Appendix A, we can reinforce the importance of mortality, recovery, and transition rates in the dengue model behavior. Furthermore, t0ci parameters had a considerable effect on the model output in specific time intervals. It is remarkable that the estimated amplitude for the third pulse input (Am3) always led to a negative total mortality rate for the model (Am3+μm), then it did act as a positive external flow of susceptible, exposed and infected mosquitoes. Regarding such behavior, we hypothesize the rapid increase of dengue cases at the start of the last outbreak to be also related with migration of exposed and infected mosquitoes from neighboring areas, which makes sense, in this case, since Bello is part of a cluster of highly conurbated municipalities. These are significant results since they allow us to propose effective control strategies and suggest that using pulse-type inputs, which change mosquito mortality, could be an appropriate strategy to model extrinsic perturbations.After performing the UA and SA to validate CSB and estimated nominal parameters, we implemented the pulse-type inputs into the model to determine more information about dengue spread dynamics in three simulation scenarios. In Figure 6, we identify a direct effect of the extrinsic conditions on the mosquito population. Indeed, the pulse-type inputs generate knock-down effects on the vector population, where the negative and positive pulses act as outbreak starters or finishers because of the increase or decrease in the mosquito populations, respectively.Figure 7a,b show that the amplitude of chemical control input is a determining factor that reduces the outbreak, but the high intensity of insecticide spraying does not produce any effect after a certain value. Figure 7c,d show that late chemical spraying increases the dengue incidence. The simulations provide information about the consequences of performing control in the wrong moment. Figure 7c shows that, if the Health Surveillance Office had applied the chemical control before the first outbreak that occurred in Bello (following the endemic channel criteria proposed by Lizarralde-Bejarano et al. [52]), the dengue outbreak would not have been as serious. We show the same result in Figure 7d for the second outbreak.In Figure 8, we compared the effect between chemical and vaccination control strategies modeled as pulse inputs, where a vaccination campaign is performed as a long-term strategy in a specific time interval instead of considering it as a constant value over time. As shown in literature, the vaccination is a better option than chemical control in long-term implementation [54] because the disease incidence could disappear from the population if we inoculate enough individuals.

Finally, we implemented pulse-type inputs to estimate and simulate extrinsic conditions, e.g., a decrease in mortality rates because of chemical control. This method could simulate other control strategies such as vaccination, cleaning up breeding sites, Wolbachia introduction into the mosquito population, or biological control. The definition and implementation of these inputs could vary according to the environmental dynamics or by the researcher’s criteria, e.g., instead of a pulse input, we could use a ramp, stairs, or sine curves, among others. Thus, future research could introduce and change these input signals to study the effect of the disease dynamics and the associated uncertainty.