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In this paper we modify previous models to develop a new model of within-host dengue infection without the assumption that monocyte production is constant. We show that this new model exhibits behavior not seen in previous models. We then proceed by obtaining an expression for the net reproductive rate of the virus and thus establish a stability result. We also perform a sensitivity analysis to test various treatment strategies and find that two strategies might be fruitful. One is the reduction of the infection rate of monocytes by viruses and the other, more effective, theoretical approach is to reduce the number of new viruses per infected monocyte.

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 [

The World Health Organization estimates that nearly 50 million infections occur annually in over 100 countries [

The incubation period of the virus in an infected host ranges from 5 to 10 days [

There have been many mathematical studies of dengue infection. Of those, relatively few [

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 [

In [

where

where

have chosen the function

It is also known that M-CSF production increases as a result of susceptible cells being infected [

The infection of susceptible monocytes depends on the successful invasion rate

is assumed constant as

It is assumed that the immune cells are produced at a constant rate

current level of infection

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,

found in [

All model parameters are assumed to be positive.

We will focus on the disease-free 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

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

Proof. Recall that all parameter values are positive. Upon inspection, we can clearly see that all

which leads to the result. W

By substituting

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

In this section we provide numerical simulations of different theoretical treatment techniques. We were able to find all parameters in the model in literature [

where

The rest of the parameter values are given in

As previously mentioned, this model displays dynamics of the monocyte population that are more in agreement with the data in [

We see that in both models with constant monocyte production the monocyte levels are never higher than the equilibrium. The new model with dynamic monocyte production does demonstrate this behavior and agrees quite well with the data in [

There are several theoretical approaches to treating the disease. For example, the illness

Parameter | Value | Parameter | Value |
---|---|---|---|

9.175 | 0.5 | ||

146.66 | 0.05 | ||

0.333 | 0.5 | ||

0.0027 | 20 | ||

0.002 | 0.8 | ||

11.09 | 0.0265 | ||

0.01 | 0.03 |

might be less severe if the death rate of the infected cells,

We see that while this type of treatment appears successful in reducing the viral and infected cell loads, it also prolongs the infection. Still, it seems like a promising approach. We can now compare this scenario with the previously mentioned increase in

It can be seen here that the model has nearly no sensitivity to the parameter

Again the model reacts very little to adjusting this parameter suggesting that increasing the free viral death rate is not a useful strategy. Another logical approach is to reduce the number of new viruses produced by an infected monocyte,

One can see the most drastic reaction in this case. By reducing

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 [

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