Source of Information

In the municipality of Cali, the city government issues epidemiological bulletins that monitor the behavior of dengue. Based on these bulletins, the endemic channel is calculated, which represents a graphical representation of the current incidence over historical incidence. It allows for the early detection of abnormally high or low numbers of disease cases (^{Coutin Marie, 2010}). This tool enables us to understand the endemic or epidemic behavior and thereby recognize when the number of actual or expected cases (according to the analysis conducted) exceeds or would exceed normal conditions, potentially surpassing the installed response capacity, and resulting in high social and economic costs, understood as damages and losses (acceptable risk level). Moreover, it helps to identify the need to design mitigation strategies to reduce the cases and/or their impacts.

Therefore, if the infection rate per district is not considered, the threat analysis for each area could be misleading, as absolute figures might distort the real situation-in this case, the risk of infection. To provide a more accurate perspective, we calculated the proportion of infected individuals in each district based on its population, and then determined the percentage represents relative to the total number of infections recorded in that same district over the year (see Table 2 and Fig. 1).

Figure 1 Average annual infection rate by district (2000-2015). 

Upon casually reviewing this table, it is apparent that District 13 has the highest number of total cases with 5,060 cases, while District 3 has one of the lowest with a total of 2,304 cases. This would suggest, initially, that District 13 has a high dengue infection threat, while District 3 has a low threat. However, when the data is converted to proportions by dividing the total number of infected individuals in each district by its total population, it becomes evident that District 3 has a higher percentage of dengue infections, which is not apparent from the absolute numbers.

Therefore, if the proportion of infections per district is not taken into account, the analysis of the threat for each district would be misleading, as there would be a distortion in the data suggesting a particular situation, in this case, the infection. Thus, by using the population projection of Cali in Figures for the year 2006, it is possible to calculate the percentage of the proportion of people infected with dengue per district and per year (See Table 2 and Fig. 1).

This process allows us to identify three dengue threat zones in Cali: high, medium, and low. These zones are effectively established by considering the calculated proportions and grouping together the districts that exhibit similar data patterns. To achieve this, the correlation matrix tool of the SPSS software is utilized. It takes each district and compares its values with all the others, aiming to demonstrate a statistical relationship among the data (See Annex). Therefore, the program would indicate which districts have similar patterns in terms of the proportion of infected individuals.

Risk Theory

Given that the spatial behavior of dengue involves certain environmental, social, and cultural factors that operate differently according to their functional dynamics, and that conducive behaviors to contagion are generated as a result, it is possible to identify threats and vulnerabilities in that context. Therefore, the Risk Theory approach provides an appropriate interpretative framework. Furthermore, from the perspective of Risk Management, it is possible to propose controls that are in line with the social conditions of the population to reduce the likelihood of an epidemic.

Dengue is considered a biological threat, therefore, places where all the conditions for its presence are met will be threatened by the spread of this virus, that is, at risk of epidemics of this disease. Precisely because it is infectious, its spread depends on the mobility of its infectious agents, making this aspect strategic when controls to mitigate this threat are propored.

This threat is tied to the infected mosquito and not its habitat, as not necessarily all habitats favorable for the mosquito (vector) will have infected mosquitoes (Figs. 2 and 3). The convergence of two events or situations is required: first, the habitat that defines favorable conditions for the presence of the mosquito, and second, an infected mosquito carrying the virus. Therefore, a population is considered to be under the threat of dengue if it has the vector that transmits the disease and the virus is circulating, whether the carrier is a human or the carrier is the vector (Fig. 4).

Figure 2 No threat Dengue with vector. 

Figure 3 No threat Dengue without vector. 

Figure 4 Dengue Threat. 

Likewise, vulnerability is linked to two factors traditionally recognized in the literature and validated in the case of dengue: exposure and fragility (^{Blaikie, 1996}) (^{Maskrey, 1993}). The former is understood as the conjunction in time and space of the threat (infected mosquito) and a person who may become infected. The latter refers to the specific condition of susceptibility after being inoculated by the vector to develop the disease. In the case of the population, as mentioned by (^{Niño, 2018}), it is generalized here. As seen in Figure 5 (^{Heydari, 2017}) (^{Ferdousi, 2015}), a population will be fragile regarding a dengue epidemic if its members, in their various combinations, do not have good access to health services to control positive cases of dengue, if they have poor water management that aids the vector in increasing its reproduction rate, if the distribution of the infected population allows the vector a higher probability of becoming infected by biting a human, if the immune system of the population members is deficient, and if their economic activities induce constant population mobility that influences the transmission of the virus to different areas. However, the distribution and concentration of the population are recognized as a preponderant factor of exposure, therefore, in this case, it is an important factor of both vulnerability and threat.

Figure 5 Dengue Vulnerability. 

Therefore, by combining the threat and fragility of a population exposed to the dengue virus, under specific conditions of exposure, population dynamics, and geographical conditions, it will be at risk of infection and likely to suffer a dengue epidemic. This, when materialized, would be a disaster given the high social and labor costs that its consequences would entail, surpassing the Acceptable Risk Level defined by the endemic channel.

Mathematical Model

The mathematical model used was based on the study carried out by Ronald Ross, known as the father of epidemiological models thanks to his work on malaria modeling. In 1950, George McDonald took up the work done by Ross, under similar principles and objectives. In 1956, McDonald published a model as an extension of the model built by Ross, which since then has been known as the Ross McDonald model (^{Anderson R, 1991}) and is still current in works such as (^{Hernández. E. A. G., 2020}) (^{Paredes, 2018}) (^{Sepúlveda-Salcedo, 2015}).

(^{Arias Castro, 2020}) based on Ross Macdonald’s model established a metapopulation model that considered the effect of the mobility of people between zones on the spread of the disease. In this paper, we worked based on this model applied to three zones, which are determined depending on the level of high, medium or low dengue threat. In equations 1, we present how the populations of infected humans and mosquitoes, respectively, in zone are changing over time considering the movement of humans from all zones. The parameters and variables are described in Table 3. This model has six equations.

Equations 1. Equations of the dynamics for humans and mosquitoes in zone i interacting with zones j and k, where i, j, k ∈ {1,2,3} and are different from each other.

As can be seen in Figure 6, there are three zones, in which there are dengue-infected and non-infected mosquito and human populations in each zone. Therefore, for each zone, there is a disease dynamic involving such parameters as the recovery rate in the human population, the mosquito mortality rate, the mosquito bite rate, the probability of a susceptible mosquito being infected by contact with an infected human, and the probability of a susceptible human being infected by an infected mosquito.

Figure 6 Dengue dynamics in isolated areas between them. 

Now, a proportion of the human population in each zone moves to the others with a connecting force between them (Fig. 7), given the daily interactions that occur based on the satisfaction of the population’s daily needs (residence, work, study, access to various goods and services), and a probability that infected humans will move (^{Arias-Castro, 2020}).

Figure 7 Dynamics of dengue with human population movements between zones. 

Thus, the assumptions of the model are as follows:

It is important to be clear about how new infections occur. Humans have two ways of becoming infected through the actual bite produced by an infected mosquito from their area or from the area to which they moved (Fig. 8). Mosquitoes also have two ways of becoming infected by biting an infected person from their area, or one that has arrived in their area.

Figure 8 Dengue dynamics with connected zones and effective populations. 

It should be noted that, with this approach, the human population of each zone changes over time, denoted as effective populations (^{Arias-Castro, 2020}).

Optimal Control

As our objective is to establish strategies for dengue dynamics considering areas with different types of dengue threat, we proceeded to introduce some control variables to the proposed model. Then, the Optimal Control Theory allows us to study the evolution of the time-dependent system controlled, in part, by the decisions of an external agent (^{Lenhart, 2007}).

Our goal is to minimize the number of infected humans in each of the zones and the cost associated with the use of controls. Since there are different ways to control the disease, reducing the number of humans infected with the virus provides the city with a plan to mitigate the risk of the disease and, therefore, of a potential catastrophe (epidemic). Among the most used control measures are lethal ovitraps (^{Morrison, 2008}), spatial repellents (^{Morrison, 2008}), insecticide (^{Morrison, 2008}), larvicide (^{Morrison, 2008}), biological controls like Wolbachia bacteria (^{Cardona-Salgado, 2020}) or species that feed on the aquatic phase of the mosquito’s life evolution (^{Rodríguez Cruz, 2002}) (^{Arias Castro, 2020}), educational campaigns (^{Rodríguez Valdés, 2012}).

For Santiago de Cali, the mayor’s office has established strategies for the control of dengue fever, including information campaigns, education campaigns, control of Aedes aegypti larvae, and control of adult populations of the vector. It is important to note that when epidemic outbreaks occur in the city, the response strategy to reduce the outbreak is the application of insecticide in common areas and in homes where there are reports of cases (^{Cali, 2019}).

For this reason, when the occurrence of a dengue epidemic is established as a disaster, the mayor’s office of Cali assumes, as a control measure, the use of insecticide to minimize the mosquito population, as a response to the epidemic, but not as a preventive measure. In this article, we want to establish control strategies when there is an epidemic outbreak and therefore, we will consider as control the use of insecticides to reduce the mosquito population in each zone. Thus, as shown in Figure 9 (the mosquitoes eliminated by spraying have cross-shaped eyes), a control is established that reduces the number of mosquitoes in each zone. Consequently, the only dynamic that is altered is that of the mosquitoes and is given by

Equations 2. Equation of infected mosquitoes in the zone with control given by the increase in mosquito mortality rate, and are different for each other.

Figure 9 Dengue dynamics with zones connected by human movement and optimal control. 

In addition to implementing controls, we are interested in reducing infected humans with a minimum cost. Consequently, we have an optimal control problem to solve since we have a dynamic to control with a target to minimize. This optimal control problem has solution, based on Pontryagin’s maximum principle, optimal controls are characterized, and numerical simulations are performed with GPOPS II software to obtain the strategies to be implemented.

We want to analyze different control strategies depending on their effectiveness and associated costs. Each strategy may result in a different number of infected individuals. Our objective is to determine which strategy provides the best costeffectiveness-that is, the greatest health benefit (in terms of infections avoided) per unit of cost. To compare the strategies, we use cost-effectiveness ratios, specifically:

and

, (Phillips, 1997),

the ACER index indicates the cost per unit of benefit for each individual strategy, while the ICER index measures the additional cost per additional unit of benefit when moving from one strategy to a more effective (and possibly more expensive) one.

Dengue threat zones

The results of the SPSS correlation process showed that districts 1, 2, 3, 4, 5, 8, and 9 have similar behavior in their data, with an average proportion of infection of 0.23%; this zone was denoted high threat (red) because it contains the highest proportion of cases in the city; Communes 6, 7, 10, 11, 12, 13, 14, 15, 16 and 21, also with a strong correlation in their data, make up the medium threat zone (yellow), with an average proportion of infection of 0.17%, and communes 17, 18, 19 and 22, the low threat zone (green), with an average proportion of infection of 0.14%. It should be clarified that commune 20 showed a strong correlation with most of the communes, except with communes 1, 3, 13, and 22, being communes 1 and 3 closes to it; therefore, by spatial coalescence criterion, this commune was assigned to the low threat zone, since it has communes 18 and 19 as neighbors.

However, neighboring communities show similar patterns in their data. Therefore, when a community presents a high level of contagion, it is very likely that its neighboring communities will also be affected. This spatial correlation indicates that implementing epidemiological controls by treating each commune as an isolated case will not yield the expected results. The clear spatial interconnection must be considered in public health strategies. Based on these patterns, the previously defined three dengue threat zones in the city of Cali are illustrated in Figure 10.

Figure 10 Map of connection between dengue threat zones in Santiago de Cali. 

Once the zones were defined, the average of the proportion data for each of the three zones was calculated using data from all the years, in order to determine which zone had the greatest impact of the disease. However, this calculation is not shown in Figure 10, which only illustrates the previously defined dengue threat zones.

Strength of connectivity between the different threat zones

The strength of the connection of the zones is then established based on the mobility of the zones between the three determined zones. This analysis is based on the origin-destination survey (^{DAPM, 2009}), which seeks to determine how connected the three zones are to each other (see Tables 6, 7, and 8), in order to obtain the spread of the disease in Cali and thus establish dengue control strategies accordingly. Table 8 and Figure 11 show the connection forces between the three zones for the city of Cali.

Figure 11 Strategy 000. Without controls taking into account mobility and without mobility. 

We note that the high zone is the most strongly connected, then there is the medium zone, and finally, the low zone. This could be one of the reasons for the spread of dengue fever in Cali.

Table 6 reports the number of inhabitants in each zone at the time the survey was conducted. Table 7 provides the proportion of the population in each zone that engages in daily commuting and the proportion that does not. Finally, Table 8 indicates the distribution of commuters from each zone to the other zones. For instance, among commuters from the high zone, 80% travel to the medium zone and 20% to the low zone.

Control strategies

With the results of sections 3.1 and 3.2, the parameters of the mathematical model and the initial populations given in Table 10, and assuming that controls with a low (cheap), medium, or high (expensive) effectiveness could be used in each zone, numerical simulations were carried out using the GPOPS II program (^{Lasluisa, 2019}). A total of 64 strategies were obtained as a result of the different combinations in the types of control to be carried out.

The identification of these simulations is done by a three-digit number, the first one represents the type of control applied in the high zone (zone 1), the second, the one applied in the medium zone (zone 2), and the third, the one applied in the low zone (zone 3), where 0 means no control, 1 means a low effectiveness control, 2 means a medium effectiveness control and 3 means a high effectiveness control. For example, simulation 032 represents the situation where no control was applied in the high zone, a high effectiveness control was applied in the medium zone and a medium effectiveness control was applied in the low zone.

It is important to mention that the numerical simulations took into account the parameters estimated by (^{Arias, 2018}) and that the proposed control is to increase vector mortality using insecticides, the cost of which varies according to their effectiveness. Also, keep in mind that the model was carried out for 90 days and that the total number of inhabitants in each area was obtained from “Cali en cifras 2021”. See table 10.

First, we present the uncontrolled dynamics (000), then we will present some of the most relevant strategies in our development.

Case 1 Strategy 000. Without controls taking into account mobility and without mobility

Intuitively, one would expect that in the numerical simulations, the area with the highest incidence of dengue would correspond to the previously described high-threat zone. However, it’s important to clarify that when risk materializes, its greatest impact does not always occur in high-threat zones, as risk involves more than just threat-it also encompasses other variables.

In our simulations, we first analyzed the scenario without any control measures (000) and without considering human mobility (see Fig. 11). As expected, the high-threat zone showed the highest proportion of dengue infections, while the low-threat zone exhibited the lowest proportion of infections in both humans and mosquitoes.

However, when human mobility is incorporated, the results shift: the low-threat zone now presents the highest proportion of infections in both humans and mosquitoes. Although counterintuitive, this outcome can be explained by examining level of outbound movement (see Table 7) is precisely the low-threat zone. People from this area travel to medium- and high-threat zones, becoming infected there rather than in their own community.

Furthermore, the proportion of infections in the high-threat zone decreases compared to the scenario without mobility. This is because, when mobility is considered, this zone receives the largest influx of people from other areas, reducing the mosquito-to-human ratio. As a result, the likelihood of a local resident being bitten by an infected mosquito decreases.

Case 2 Strategy 010. Only low control in zone 2 Comparing the results of strategy 010 (Fig. 12) with those of strategy 000 (Fig. 11), it is observed that:

Figure 12 Strategy 010. Only low control in the medium zone. 

Case 3 Strategy 133. Low control in zone 1 and high control in zones 2 and 3

This simulation (Fig. 13) shows that:

Figure 13 Strategy 133. Low control in zone 1 and high controls in zones 2 and 3. 

Cost Benefit Analysis

As we said before, we have 64 strategies and each of them has an impact on dengue dynamics. Since we are interested in proposing a strategy to be applied, we must establish a criterion that allows us to compare them. Therefore, we must identify the benefit of each strategy and then, using the ACER index (see Table 11), determine the best strategy according to its benefit and cost, where the cost of a strategy is given by the cost of the application of controls and the cost to the health system for the number of infected.

In this case, the final benefit of each strategy is given by the reduction in cumulative incidence due to the applied controls, this reduction is in comparison with the uncontrolled dynamics. Consequently, according to the ACER index, cost per benefit of the strategy, the best strategy to apply in the city is 010 (see Fig. 12), controlling only the medium zone.

In addition, we used the ICER index (see Table 12), which indicates the cost-benefit ratio to compare two strategies. The results of this index are not exclusive, so any of the four strategies shown could be applied in the city depending on the need and resources of the city, considering that those with a higher ICER would be more expensive but with a better benefit. The best strategy according to ICER is the same as that of ACER, 010. In addition, if we wish to move to a strategy with higher benefit, the next strategy is strategy 133 (Fig. 13), where the index corresponds to the cost per unit of additional benefit of switching from one strategy to another.