A survey of mathematical models of Dengue fever

In this paper, we compare and contrast five models of Dengue fever. We evaluate each model using different scenarios and identify the strengths and weakness of each of the models.


Dengue Fever
Dengue is a mosquito-borne viral infection that is usually found in tropical and subtropical regions around the world. In recent years, transmission has increased predominantly in urban and semi-urban areas and has become a major public health concern, [1]. This is similar to mass action model in chemistry. The per capita rate of infection and the per capita rate of recovery are assumed to be independent of the length of time the person has spent in each compartment. They are assumed to follow an exponential distribution.
The basic S − I − R model is formulated as: where λ is the force of infection and γ is the mean recovery rate and N is the total population. 2. Frequency-dependent model. However, it has been shown that for most human infections, the number of people each person is in contact with per day is fairly constant across the world, regardless of the population density of the place.
That is why an alternative, known as the "frequency-dependent," formulation of the SIR model is often used to model the transmission of human diseases, where the force of infection is defined as λ = β( I N ). The term I/N is the probability that any random contact that a susceptible person makes will be with someone infectious, which is equivalent to the proportion of the total population that is infectious, [10].

CHAPTER 2 DEROUICH MODEL OF DENGUE FEVER
We first study the model of Dengue fever developed by Derouich et al in [3]. Their model is based on the compartmental diagram shown in Figure 2 For the human population, the model developed by Derouich et al., [3], takes the The parameter values are described in Table 2.1. One of the key features of the model is the fraction, p, that represents a (random) fraction of the human population that can be permanently immunized against the four serotypes that cause Dengue fever.
For the vector population, equations (2.1) and (2.2) can be combined into the single system (2. 3) The main result of Derouich et al in [3] is that system (2.3) has two equilibrium β, M , and R are given by . Analysis of the Jacobian at E 1 and E 2 shows that E 1 is globally asymptoti- To develop a deeper understanding of the model we conduct several simulations.   [3].

Parameter Notation Base Value
Note that all simulations in this thesis were conducted using Wolfram Mathematica, Our first simulation is based on the variation of vaccination levels of a whole population. We numerically demonstrate the change in outbreak behavior using four levels of total population vaccinated in Figure  The second scenario is based on the assumption that for different environment temperatures the activity level of mosquitoes differs [2].
For this model, our final simulation is based on the hypothetical size of mosquito population and its influence on the size of the outbreak in the human population.     Generally, it is thought that warmer weather will cause vectors such as mosquitoes to increase in population size.
The last simulation illustrates the importance of different control measures of mosquito population. In Figure 2.4, we see that a considerable decrease of mosquito population can almost prevent an outbreak of Dengue in the human population.
According to these scenarios it is difficult to identify which parameter affects the severity of an outbreak the most. However, the number of mosquitoes and the vaccination level of the susceptible population appear to be of high importance.
Despite the fact that vaccination campaigns can be easily implemented, they are effective only if just one strain of the virus is present in the environment. Otherwise, the vaccination program is just a waste of resources. So, the best way to decrease the severity of the outbreak is to reduce the actual of mosquitoes.
To obtain the precise result on the existence and stability properties of these equilibrium points it was assumed that Dengue does not produce significant mortality. So, the dimension of the model was reduced by one. Finally, two equilibrium values were considered: For each of the equilibrium points the parameters are defined as σ * 1 = max 0, for E 1 * and σ + 2 = max 0, for E 2 * respectively.
3. Different mosquito activity levels.  The second scenario (Figure 3.2) demonstrates that the mosquito recruitment rate has almost no impact on the outbreak. However, the mosquito recruitment rate can considerably shift the susceptible-infected distribution among vectors.
The third scenario (Figure 3.3) describes the outbreak given different mosquito  area from mosquitoes are also effective in preventing dengue, [10].

CHAPTER 4 SYAFRUDDIN AND NOORANI MODELS OF DENGUE FEVER
The third and fourth models of Dengue fever studied were developed by Syafruddin and Noorani, [7] and [8] respectively. The parameter values they used in both models are the same and are defined in Table 4.1.

The First Syafruddin and Noorani Model
The susceptible-infected recovered (S −I −R) model used by Syafruddin and Noorani in [8] simplifies to (4.1) and δ = µ v . The parameter values are described in Table 4.1. The probability of a susceptible human being infected with Dengue is The main result of the first Syafruddin and Noorani model, [7] is that system (4.1) has two equilibrium points E 1 = (1, 0, 0) and E 2 = (x 0 , y 0 , z 0 ) with the values: Analysis of those equilibrium points for the South Sulawesi outbreak shows that E 1 is globally asymptotically stable point and E 2 is asymptotically stable point, [7].
To illustrate the behavior of this model several simulations were performed. In the first simulation we assumed that the proportions of susceptible and infected population can vary initially (see Figure 4.1).
As shown in Figure 4.1, this scenario illustrates that the more people initially infected, the faster the remaining susceptible population will decrease.    The second scenario describes the situation with different activity levels of mosquitoes.

The Second Syafruddin and Noorani Model
, the susceptible-exposedinfected-recovered (S − E − I − R) model used by Syafruddin and Noorani in [8] simplifies to Refer to In comparison to the previous models, the scaled Syafruddin and Noorani models are convenient because x, u, y, w, and z represent population percents rather than specific numbers. This makes it easier to compare the effects of the virus on the mosquito and human populations.
E 1 and E 2 , equilibrium points for one serotype are locally asymptotically stable when The last equilibrium point E 3 unlike the previous ones, represents the coexistence of two serotypes of viruses. It is locally asymptotically stable if and only if and is defined as the expected number of cases in individuals of type 1 caused by the infected individual of type 1 in a completely susceptible population.
As for previous models, we also explored the behavior of this model based on different biting rates of mosquitoes. As described above, with an increase of atmospheric temperature, mosquitoes become more active and, consequently, the probability to infect an individual increases.
It can be observed that those graphs are different from all previous, which can be explained by the fact that this model describes not only an epidemic outbreak of the disease but the endemic situation of the disease. According to this, after the end of the outbreak the disease will not vanish, but will hide until the required number of   susceptible hosts will not reappear in the environment. After that, a new outbreak will take place in the society.
As we can observe from the top left graphic in Figure 5.1, the each next outbreak is smaller than the previous one. This phenomena is because of the immunity of the group of the population that had a disease during previous outbreaks.
Also, it can be observed that with the increase of the activity of the vectors (mosquitoes), peaks of the outbreaks becomes sharper. However, this does not mean that the number of infected hosts grows.
Unlike some of the previous models, this model does not take deaths into account. However, when compared to the previous models, this model appears to be the most comprehensive yet attempts to capture only the most relevant parameters.
For example, compare the number of values used in system (4.2) to those used in system (5.1).

General conclusions
This thesis reviewed several ODE mathematical models of dengue fever. Five models with different approximations to modeling and different assumptions were considered and for each of them several outbreak scenarios were reviewed.
It was observed that every model is different. The models by Derouich et al. [3] and by Syafruddin et al. [7] are among the simplest one. Both of them are S − I − R ODE models of one strain of the virus.
The model developed by Feng et al. [4] is a more complicated one. This one is also S − I − R ODE model, but of two different strains of dengue.
The most comprehensive model is developed by Nuraini et al. [6]. It not only describes the outbreak with two strains, but also takes into account the separate severe Dengue Hemorrhagic Fever state which is not taken into consideration in any of previous models. In addition, this model describes the endemic behavior of the disease, whereas the other models are modeling only epidemic outbreak.
The other interesting model was developed by Syafruddin et al. [8]. This is the only example of S − E − I − R model considered here, which divide the whole human population into four compartments: susceptible, exposed, infected and recovered (removed).
On the next step in the investigation several hypothetic scenarios for each of the outbreaks were conducted to investigate the behavior of the each model and try to answer the question, "which intervention can be the most efficient in terms of decreasing the number of infected population?" Two different types of interventions are available to reach those goals: vaccination and the direct decrease of the mosquitoes population. Some models show that vaccination can be useful. However, those models assume only one strain of the virus. If there are more strains in the environment vaccination becomes practically useless since currently available vaccines can only protect from one strain, leaving the whole population completely susceptible to others. So, the only feasible working strategy is to decrease the number of mosquitoes.
At the same time another interesting phenomenon was observed. Since the activity of mosquitoes is based on weather condition, mostly on the temperature, global warming, will increase the possibility of being infected and, consequently, the risk of outbreaks.
Finally, there are other problems to consider. One is to develop more models to catch observe important features during the progression of an epidemic. For example, the development of an S − E − I − R model of two different strains will be a step forward in this direction. The ultimate goal is to build a model that will describe the outbreak of four different strains at the same time. However, even a small increase in complexity of the initial model drastically increase the difficulty of its validation.
Moreover, the amount of real data needed for validation also increases and this data is not easy to obtain.

Computational Notes
The graphics and computations in this paper were carried out using Mathematica, [9].
Jim Braselton will send you copies of the notebooks used here if you send a request to him at jbraselton@georgiasouthern.edu.