Dengue is an infectious disease that is caused by the dengue virus (DEN virus or DENV). This virus is transmitted through the bite of an infected female Aedes aegypti or Aedes albopictus [1, 2]. It is estimated that ~50% of the world population is at risk of dengue every year [3]. Dengue has been the subject of main concern in many tropical and subtropical countries, including Indonesia. Since the first case of dengue in Indonesia was reported in Jakarta and Surabaya way back in 1968, the incidence of dengue in Indonesia continues to spread to this day [4]. Based on a new report from the Ministry of Health Indonesia, from early 2022 until 20 February 2022, the cumulative number of dengue cases was recorded as 13,776 cases. Meanwhile, the number of deaths due to dengue was recorded as 145 cases [5].

There are four different serotypes of DENV, namely, DENV-1, DENV-2, DENV-3, and DENV-4 [6, 7]. In many cases, the primary infection of dengue is often asymptomatic. In contrast, the secondary infection with different serotypes from the primary infection may develop more severe symptoms, such as bone pain and headache, up to more occasionally fatal symptoms [8, 9]. Individuals who have already recovered from the primary infection will maintain a lifelong immunity to the first DENV that had caused the primary infection, but only temporal to the other three serotypes [10]. When the short-term immunity to other serotypes wanes, the recovered individuals may acquire a secondary infection that can be even more severe than the primary infection. This phenomenon is called the ADE process [11, 12].

There is no specific treatment to cure dengue-infected individuals of the disease. The main action plan to treat dengue-infected individuals is rendered feasible by giving them supportive care, or if the case is severe, then the patient requires hospitalization which becomes an obligation to be done. Recently, several candidates for dengue vaccines have been in the process of development [13, 14]. An affordable and effective dengue vaccine will give importance to the control of dengue spread around the world. The main control program adopted by many governments worldwide to control the spread of dengue is the vector control program and steps are taken to reduce the probability of a successful infection through a mosquito repellent. Another option to prevent the spread of dengue (and other diseases) is by developing community awareness on the danger of the disease [15–18]. Community participation in eliminating or at least suppressing the spread of dengue can be done through several activities, such as through media campaigns to disseminate knowledge about how to prevent acquiring infection from mosquitoes from individual levels up to community levels. The author in [15] implies that the risk of dengue may be increased when there is a lack of community awareness due to misunderstanding between the community and the government. Therefore, maintaining community awareness by reducing the ignorance of dengue is essential to guarantee intervention success in controlling the spread of dengue.

Mathematical models have been used widely by researchers to understand how vector-borne diseases spread among the population [19–23]. For the dengue transmission model, many authors have used mathematical modeling to guide public health strategies to control the spread of dengue. The mathematical modeling process is very challenging due to the complexity of the dengue transmission mechanism. A more complex model may bring in a more realistic modeling, but finding the analytical results and conclusion often entails difficulty. Hence, the researcher needs to develop a realistic but simple model with realistic assumptions. The use of real incidence data is also needed to calibrate the performance of the model. There are many approaches that can be used to construct the dengue transmission model, such as with ordinary differential equations [24, 25], partial differential equations [26, 27], fractional-order differential equations [28, 29], stochastic differential equations [30–32], and other approaches.

Many mathematical models for dengue transmission use a deterministic approach. Although the transmission process of dengue involves a vector animal (Aedes mosquito) as the prime spreader, some authors use a host-to-host modeling approach [30, 33, 34]. This approach does not involve the dynamics of mosquitoes in their model since it can be argued that the mosquitoes' life expectancy is very short compared to the human life expectancy. Hence, the dynamic of mosquitoes is much faster compared to that of the human. The authors in [35] find that the only essential dynamics are coming from the human population, and mosquito dynamics only slightly perturb them. The other approach is adopted by considering the dynamic of mosquitoes [24, 25]. With this approach, the mosquito population is explicitly involved in the model. With the involvement of mosquito dynamics, such implementation of vector control can be modeled into the equation. When the vector control is involved in the model, an optimal control approach can be used to understand the short-term impact of the intervention and determine the most effective strategy [36–38]. Modeling dengue transmission is not only for the macro scale (population scale). Some of the authors also construct the model to understand the dynamic within the host [39, 40]. This modeling is conducted to understand the interaction between the free virus with susceptible targeted cells. Some interesting factors are involved in this modeling approach, such as the infectivity of the virus and immune response.

From the aspect of the impact of community awareness on the dengue transmission model, there are some models which have been introduced by authors. The authors in [41] introduced a mathematical model of dengue where the effect of media awareness was included. Mathematical analysis on the equilibrium points and the basic reproduction number was included in it. The author in [42] introduced a multistrain dengue model that combined mosquito control programs and human awareness. They found that the control of a large number of mosquitoes and human awareness was required to control dengue effectively. The author introduced an optimal control problem of dengue with human awareness and vector control in [43]. The authors used Pontryagin's maximum principle to characterize the necessary conditions for the optimal control problem. The author in [44] introduced a modified host–vector model by considering low- and high-risk susceptible populations. The author analyzed global stability on all equilibrium points. The author in [45] introduced an optimal control model of dengue transmission. The author developed the model by considering five control variables: information spread, bed nets, treatment, screening, and insecticide. The impact of a media campaign that can reduce the rate of infection was developed by the author in [24]. The author conducted a cost-effectiveness analysis to understand the most cost-effective strategy that can be employed to control dengue transmission. Recently, the authors in [25] combined vaccination, vector control, and media campaign in their model where the seasonality was accommodated. All the mentioned references consider the same assumption that (1) aware individuals have a smaller probability of being infected due to their awareness and (2) more prominent infected individuals will reduce the infection rate more. The second assumption is reasonable when we wish to model the spread of dengue among a population where awareness of dengue could increase the participation of the population in the dengue control programs.

To calibrate the proposed dengue model, many authors use incidence data to estimate their parameters. The idea behind this development is to find the best-fit parameters, such that the model simulation output can fit the time series of the data. Please see the following references for the use of incidence data in their dengue research: Aguiar and Stollenwerk [30], ten Bosch et al. [31], Aguiar et al. [34], and Aldila et al. [46]. With this parameter estimation, the researcher can make a short time prediction on their model. Some interventions can be included in their model and the possible outcomes predicted in the near future.

Motivated by the above discussion, no authors had discussed the impact of community ignorance on the spread of dengue. In this circumstance, more infected individuals will increase the probability of infection in the human and mosquito populations. In some countries where dengue fever continues to emerge throughout the year, the level of public ignorance of the spread of dengue fever is no longer as high as for several newly discovered diseases and it is quite a concern, such as Zika in 2018 or COVID-19 in the late 2019. Hence, it is important to consider the community's ignorance of our proposed model. Based on this background, here in this article, we introduce our SIR-UV mathematical model to describe the spread of dengue under the impact of community ignorance. The Quasi-Steady State Approximation method was used to simplify the model. We used the weekly incidence data of dengue from Jakarta during the period from January 2008 to December 2021 to estimate the parameter values in our model. We used the fast Fourier transform to extract the most significant frequency from our data. With this dominant frequency, we fit our model output with the data by assuming the infection rate as a sinusoidal function that depends on time. Some mathematical and numerical analyses were conducted to understand the qualitative behavior of our model and how it was related to the basic reproduction number. Furthermore, we also analyzed how community ignorance can trigger the appearance of a backward bifurcation, which can cause dengue to exist, even though the basic reproduction number is already <1. The layout of this article is as follows: In Section 2, we construct our model. In the same section, we perform our data assimilation to find out the best-fit parameters of our model. The model analysis is given in Section 3, which is followed by some sensitivity analyses and numerical experiments in Section 4. The concluding remarks are given in Section 5.

In this section, we conduct several numerical experiments to understand the behavior of our model with respect to the level of community ignorance (α). The first simulation will be the bifurcation diagram, followed by numerical simulation on the dynamic of the model with respect to time.

As previously mentioned in Theorem 7, a backward bifurcation occurs when α > α^*, where α presents the ignorance level of the community. Larger α means more ignorance in the community about the spread of dengue. To present the situation, we conduct numerical experiments to show a possible type of bifurcation that could appear from our model. At first, we set up all coefficients on the polynomial (12) as a function of R0. By solving R0 with respect to β_h, we have βh=(γ+μh)κvR0MN, and substituting it in (12), yields:

where

Next, we substitute the parameter values as given in Section 2.3 which gives us

Using these parameter values, we have the value of α^* as 6.729 × 10⁻⁷. Therefore, we choose α = 5 × 10⁻⁸ < α^* to find the forward bifurcation as shown in Figure 3A and α = 5 × 10⁻⁶ > α^* to find the backward bifurcation as shown in Figure 3B.

Backward bifurcation phenomena imply that a condition R0<1 will not be enough to guarantee the disappearance of dengue from the community. We can see that for some parameter value when R0<1, we can have multiple stable equilibrium points, i.e., the dengue-free equilibrium and the endemic equilibrium point (please see Figure 3B). When the ignorance level of the community is small enough (at least smaller than α^*, a condition R0<1 is enough to guarantee the disappearance of dengue from the community (please see Figure 3A). To illustrate the bistability phenomenon when a backward bifurcation appears, we show the phase portrait of system (5) with some initial conditions. The results are given in Figure 4, where we have a stable node dengue-free equilibrium and a stable-spiral endemic equilibrium point. It can be seen that different initial conditions may lead to a different final state condition. To see further impact of α on the bifurcation phenomena of our model, we conduct numerical experiments as shown in Figure 5. These numerical experiments confirm our previous results on the impact of α on the appearance of backward bifurcation phenomena at R0=1. Smaller the level of ignorance of the community, higher the chance to have a free endemic equilibrium when R0<1.

The public health implication of backward bifurcation is that it is not enough to only reduce the basic reproduction number to eliminate dengue. Another factor, which in our case is the community ignorance level of dengue, should also be considered for further intervention in the field. Therefore, it is necessary to find the impact of model parameters on the critical level of community ignorance. To determine this, we calculate the normalized sensitivity of α^* with respect to γ, μ_h, δ, and κ_v. Using the formula given by [55], we have:

From a previous analysis, we know that a backward bifurcation will appear when α > α^* and a forward bifurcation if α < α^*. We can see from the expression of Γα*, we have that Γα*γ>0, which indicates that increasing γ will increase α^*. Hence, a larger recovery rate will increase the chance of non-appearance of backward bifurcation phenomena at R0=1, since we have a larger interval of α ∈ [0, α^*]. On the other hand, we can see that Γα*μh,Γα*δ, and Γα*κv are negative, which indicates that increasing natural death rate of human (μ_h), waning immunity (δ), and mosquito dynamic parameters (κ_v) will reduce α^*. Hence, different with the effect of recovery rate, increasing μ_h, δ, and κ_v will increase the chance of appearance of backward bifurcation phenomena, since the interval of α ∈ [0, α^*] is getting smaller. Therefore, we can conclude that longer the temporal immunity of human (smaller δ⁻¹) will increase the chance of finding the only possible condition that dengue disappears when R0<1. Furthermore, we also find that when κ_v increases (larger life expectation of mosquitoes or a smaller infection rate in mosquitoes) will increase the possible existence of dengue-endemic situation in the field, even though R0 is already <1.

Next, we carry out numerical simulation in Figure 6 using MatLab to understand the impact of the human level of ignorance on the spread of dengue. We use the same parameter values that we used to produce Figure 2. We can see that less ignorance of the community (smaller α) to the dynamics of infected individuals will reduce the number of infected individuals. The impact will be more significant as time increases.

A mathematical model was presented and studied in this article to assess the impact of the level of human ignorance on the spread of dengue. At the beginning of the study, we introduced our SIR-UV model. Using the QSSA approach, we simplified the model to an IR-model. With this approach, we converted our host–vector dengue model to a host-to-host dengue model. A host-to-host dengue model is a common approach adopted by several researchers to reduce the complexity of their model, by considering the fact that the dynamic of mosquitoes is very fast compared to that of human dynamics [30, 33, 34]. Two types of equilibria emerged from the model, namely the dengue-free equilibrium and the endemic equilibrium point. The basic reproduction number, denoted by R0, was calculated. We found that the dengue-free equilibrium point was always locally asymptotically stable when R0<1. The center manifold theory was used to establish the stability of the endemic equilibrium point, and it showed that the existence of backward bifurcation appears when the level of community ignorance increases. In this situation, we conclude that ensuring the size of the basic reproduction number to be <1 does not always guarantee the disappearance of dengue. Several authors have shown the appearance of a backward bifurcation in the dengue transmission model in their models [56–59]. Their analysis showed that some crucial aspects were not included in the calculation of the basic reproduction number. This aspect may trigger the backward bifurcation phenomena, making the dengue control program more difficult to achieve. In our model, we show that, even though the level of community ignorance does not appear in the basic reproduction number, it does trigger the backward bifurcation. More ignorant the population about dengue, the more difficult it is for dengue to be controlled since the condition of basic reproduction number <1 no longer guarantees the disappearance of dengue.

To test our model, we fit our model output with dengue incidence data in Jakarta, Indonesia. Our preliminary analysis of the time series dengue data reveals the existence of periodicity of dengue incidence data in Jakarta from 2008 to 2021. Three dominant frequencies of the data related to a periodicity of 53, 74, and 147 weeks. These results indicate that dengue cases in Jakarta always recur every year. A numerical experiment on the bifurcation diagram has shown that reducing community ignorance can significantly change the endemic situation. The chance of the existence of dengue-endemic equilibrium when the basic reproduction number is <1 can be avoided when the community ignorance is relatively small. To reduce community ignorance, a media campaign to increase people's awareness of dengue could be an alternative intervention. On the other hand, we find that we can increase the chance of the non-existence of backward bifurcation by increasing the recovery rate of a human, prolonging the temporal immunity, or reducing the life expectancy of a mosquito. Our non-autonomous simulation was conducted by assuming the infection parameter as a sinusoidal function with three dominant frequencies. It has been shown that reducing community ignorance of dengue could suppress the incidence of dengue in Jakarta. Although the outbreak still appears, the outbreak can be reduced significantly. The longer period of intervention of media campaigns to reduce community ignorance will give a more significant reduction in dengue outbreaks.

The data analyzed in this study is subject to the following licenses/restrictions: It can be available due to personal request to the corresponding author. Requests to access these datasets should be directed to aldiladipo@sci.ui.ac.id.

DA, CA, and RR contributed to the concept and design of the study and wrote the original version of the article. CA organized the data used in the article. DA wrote the final version of the manuscript. All authors contributed to the article and approved the submitted version.