Dengue is a virus belonging to the Flavivirus genus. The Flavivirus genus includes mostly mosquito-borne viruses such as the West Nile virus and the yellow fever virus. The dengue virus exists in four different serotypes. A serotype is a distinct variation within a species of viruses that may present a different configuration or slightly different kind of antigen. All serotypes of the dengue virus can cause the full spectrum of disease symptoms [1] .

The World Health Organization estimates that nearly 50 million infections occur annually in over 100 countries [2] . As there are no specific anti-viral treatments for dengue infection, supportive care is the usual treatment. This may include bed rest, antipyretics and analgesics. A small subset of infections result in dengue hemorrhagic fever which can be fatal.

The incubation period of the virus in an infected host ranges from 5 to 10 days [3] . At the end of the incubation period, viral particles enter the bloodstream and cause the onset of symptomatic fever. Viremia, the presence of virus in the blood stream, occurs roughly two days before the onset of symptoms and lasts 5 to 6 days [4] . Viremia tends to peak at the time of or shortly after the onset of illness. The clearance of virus is performed by the immune system.

There have been many mathematical studies of dengue infection. Of those, relatively few [5] [6] [7] [8] [9] are concerned with within-host dynamics. In these, it is assumed that the production of target cells is constant. This assumption is adequate in healthy individuals but the production of monocytes can vary, especially during infection [10] . In fact, the data in [10] show that monocyte levels are actually elevated during dengue infection, which is rather counter-intuitive. In general, the production is controlled by the Macrophage Colony Stimulating Factor (M-CSF). We account for this additional aspect in our model and show that this modification allows for better agreement with the data.

The remainder of the paper is organized as follows: in Section 2 we formulate the homogeneous viral infection model. Section 3 is the analysis of the model’s equilibria. Section 4 contains the parameter sensitivity analysis and comparisons with previous models. In Section 5 we make some concluding remarks.

Within this section, we formulate a model of population growth of the dengue virus within the human body based on the model in [9] . The model starts with the beginning of the detectable viremia period. It is assumed that one serotype of dengue virus circulates within the infected host and that the virus infects the monocyte cell population of the host.

In [9] , the authors studied the following model:

where, and represent the density of susceptible monocytes, infected monocytes, free virus particles and immune cells in blood at time, respectively. The production of susceptible monocytes is assumed to be a constant and they also have a constant death rate. This model always shows that monocyte population significantly decreases during infection, which is not always the case [10] . In order to obtain a model that more closely resembles the data in [10] , we have chosen to model the production of monocytes dynamically. First, we account for the fact that the primary catalyst for monocyte production is a cytokine called the Macrophage Colony Stimulating Factor (M-CSF). Other components of the blood are also controlled in a similar way. For example, the production of erythrocytes is controlled by the hormone erythropoietin. In several previous works, including [11] [12] [13] , it was assumed that the rate of production was proportional to the hormone concentration. We will make the same assumption here and will require that in the absence of infection, the rate of production is indeed the constant. This changes the first equation in (1) to

where is the normal concentration of M-CSF. We are now required to model the dynamics of the M-CSF production. First, we will model how the body regulates its control under normal conditions, i.e., no infection. In this case, the production’s purpose is to maintain a normal monocyte count [14] , which we will call. We want a function that increases when and decreases when To achieve this, we

have chosen the function

It is also known that M-CSF production increases as a result of susceptible cells being infected [15] [16] . Therefore, we will assume that the rate of increased production is proportional to the rate of infection,. Thus we will have the term, where is the constant of proportionality. Finally, M-CSF has a natural decay rate which we will call This results in the equation

The infection of susceptible monocytes depends on the successful invasion rate of virus into susceptible cells per unit time. The infection period of infected monocytes

is assumed constant as. Upon infected cell death, free virus particles are released into the blood. The free virus particles are assumed to be cleared at a rate of.

It is assumed that the immune cells are produced at a constant rate and they have a lifespan of. Additionally, we assume immune cell production is stimulated by the

current level of infection at a constant rate, as well as from contacts with infected cell at constant rate. Lastly, we assume that immune cells will eliminate the infected monocytes at a constant rate.

With these assumptions, we formulate the model for with-in host dengue viral infection with immune response and variable monocyte production rate, as the following.

We were able to find the values for normal susceptible counts and the normal M-CSF concentration, and in the literature. The same is true for the decay rate of M-CSF, [17] . The value of was chosen so that in the absence of infection, and form part of the disease-free equilibrium. This results in

The value of was determined by a statistical analysis based on data

found in [10] , which we will discuss in detail in Section 4. Since we can express and in terms of the other parameters, we decided to do so and work with the following version of the model:

All model parameters are assumed to be positive.

We will focus on the disease-free equilibrium, where, , and and the death equilibrium.

The Jacobian of the model is expressed below

Substituting the disease-free equilibrium into the Jacobian matrix results in

And the tedious calculation of gives the characteristic polynomial:

which, conveniently, is a product of three linear polynomials and a quadratic. Finding the roots of these four polynomials gives us the expressions for the eigenvalues given below.

where

This allows us to formulate the following theorem:

Theorem 1. If then the disease-free equilibrium, , is locally asymptotically stable.

Proof. Recall that all parameter values are positive. Upon inspection, we can clearly see that all aside from are either negative or will have real parts that are negative. In order to ensure that has a negative real part we must require

which leads to the result. W

By substituting, the model reaches the equilibrium

We refer to this as the “death” equilibrium even though the immune

cells are still present. We call this the “death” equilibrium since individuals do not function without monocytes. The resulting Jacobian in this case is:

Here the resulting characteristic polynomial is:

In this case also we can get expressions for eigenvalues, which are given below:

Since for all sets of positive parameters, we see that this equilibrium is always unstable. We have also seen an interior persistence equilibrium in numerical simulations where.

In this section we provide numerical simulations of different theoretical treatment techniques. We were able to find all parameters in the model in literature [9] [17] except the parameters and. The parameter was calculated as mentioned in Section 2. To calculate the parameter, we used the data provided in [10] , which gave monocyte counts in individuals infected with dengue. It was found that the individuals had a mean monocyte count of with a standard deviation of 370. We used MATLAB’S “randn” function to generate 200 random data sets (each with 15 points) from a normal distribution with that same mean and standard deviation. The 15 points were used as data values measured every 12 hours during a one week period. For each data set, we used MATLAB’S “fminsearch” function to find the value of that minimized the function

where. After these 200 values of were found, we calculated their mean. We repeated this process many times and consistently got values in the lower-fifties, and we finally settled on the value.

The rest of the parameter values are given in Table 1. We should also mention that this parameter set and all of the modified ones that follow result in.

In this paper we have presented a new model for within-host dengue infection. The new approach does not assume that the monocyte production is constant throughout infection and includes a fifth equation that models the production of the primary stimulant for monocyte production, Macrophage Colony Stimulating Factor (M-CSF). By modeling the production of monocyte counts dynamically, our model has produced qualitative behavior not seen in previous models. Namely, that monocyte counts are elevated above the equilibrium during at least some period of infection. This behavior is in agreement with available data [10] . We were also able to find the net reproductive rate and thus obtain a stability result for the disease-free equilibrium. While the current treatment of dengue infection is only supportive care in the form of hydration and pain relievers, we have explored theoretical approaches that might affect viral loads and reduce the severity of symptoms. Through simulations we have seen that two approaches seem promising. The first is the reduction of the infection rate of monocytes by viruses. The second and most effective theoretical treatment strategy is to reduce the number of new viruses, , produced by each infected monocyte.

We thank the Editor and the referee for their comments. This work was partially funded by the Marquette Fellowship at Loyola University New Orleans.

Thibodeaux, J.J. and Hennessey, M. (2016) A Within-Host Model of Dengue Infection with a Non- Constant Monocyte Production Rate. Applied Mathematics, 7, 2382-2393. http://dx.doi.org/10.4236/am.2016.718187