**2. Background**

#### **2.1 Biological background**

The initial response of the body to an acute biological stress, such as a bacterial infection, is an acute inflammatory response (Janeway et al. (2001)). The strategy of the HIS is to keep some resident macrophages on guard in the tissues to look for any signal of infection. When they find such a signal, the macrophages alert the neutrophils that their help is necessary. The cooperation between macrophages and neutrophils is essential to mount an effective defence, because without the macrophages to recruit the neutrophils to the location of infection, the neutrophils would circulate indefinitely in the blood vessels, impairing the control of huge infections.

The LPS endotoxin is a potent immunostimulant that can induce an acute inflammatory response comparable to that of a bacterial infection. After the lyse of the bacteria by the action of cells of the HIS, the LPS can be released in the host, intensifying the inflammatory response and activating some cells of the innate system, such as neutrophils and macrophages.

The LPS can trigger an inflammatory response through the interaction with receptors on the surface of some cells. For example, the macrophages that reside in the tissue recognize a bacterium through the binding of a protein, TLR4, with LPS. The commitment of this receptor activates the macrophage to phagocyte the bacteria, degrading it internally and secreting proteins known as cytokines and chemokines, as well as other molecules.

The inflammation of an infectious tissue has many benefits in the control of the infection. Besides recruiting cells and molecules of innate immunity from blood vessels to the location of the infected tissue, it increases the lymph flux containing microorganisms and cells that carry antigens to the neighbours' lymphoid tissues, where these cells will present the antigens to the lymphocytes and will initiate the adaptive response. Once the adaptive response is activated, the inflammation also recruits the effectors cells of the adaptive HIS to the location of infection.

#### **2.2 General-Purpose computation on Graphics Processing Units - GPGPUS**

NVIDIA's Compute Unified Device Architecture (CUDA)(NVIDIA (2007)) is perhaps the most popular platform in use for General-Purpose computation on Graphics Processing Units 2 Will-be-set-by-IN-TECH

process many streams simultaneously. This chapter describes the GPU-based implementation of the sensitivity analysis and also presents some of the sensitivity analysis results. Our experimental results showed that the parallelization was very effective in improving the

The remainder of this chapter is organized as follows. Section 2 includes the background necessary for understanding this chapter. Section 3 describes the mathematical model implemented. Section 4 describes the implementation of the GPU version of the sensitivity analysis. Section 5 presents some of the results of the sensitivity analysis and the speedup obtained. Section 7 presents related works. Our conclusions and plans of future works are

The initial response of the body to an acute biological stress, such as a bacterial infection, is an acute inflammatory response (Janeway et al. (2001)). The strategy of the HIS is to keep some resident macrophages on guard in the tissues to look for any signal of infection. When they find such a signal, the macrophages alert the neutrophils that their help is necessary. The cooperation between macrophages and neutrophils is essential to mount an effective defence, because without the macrophages to recruit the neutrophils to the location of infection, the neutrophils would circulate indefinitely in the blood vessels, impairing the control of huge

The LPS endotoxin is a potent immunostimulant that can induce an acute inflammatory response comparable to that of a bacterial infection. After the lyse of the bacteria by the action of cells of the HIS, the LPS can be released in the host, intensifying the inflammatory response

The LPS can trigger an inflammatory response through the interaction with receptors on the surface of some cells. For example, the macrophages that reside in the tissue recognize a bacterium through the binding of a protein, TLR4, with LPS. The commitment of this receptor activates the macrophage to phagocyte the bacteria, degrading it internally and secreting

The inflammation of an infectious tissue has many benefits in the control of the infection. Besides recruiting cells and molecules of innate immunity from blood vessels to the location of the infected tissue, it increases the lymph flux containing microorganisms and cells that carry antigens to the neighbours' lymphoid tissues, where these cells will present the antigens to the lymphocytes and will initiate the adaptive response. Once the adaptive response is activated, the inflammation also recruits the effectors cells of the adaptive HIS to the location

NVIDIA's Compute Unified Device Architecture (CUDA)(NVIDIA (2007)) is perhaps the most popular platform in use for General-Purpose computation on Graphics Processing Units

and activating some cells of the innate system, such as neutrophils and macrophages.

proteins known as cytokines and chemokines, as well as other molecules.

**2.2 General-Purpose computation on Graphics Processing Units - GPGPUS**

sensitivity analysis performance, yielding speedups up to 276.

presented in Section 8.

**2.1 Biological background**

**2. Background**

infections.

of infection.

Fig. 1. Relationship between the components.

(GPGPUs). CUDA includes C software development tools and libraries to hide the GPGPU hardware from programmers.

In CUDA, a parallel function is called kernel. A kernel is a function callable from the CPU and executed on the GPU simultaneously by many threads. Each thread is run by a *stream processor*. They are grouped into blocks of threads or just blocks. A set of blocks of threads form a grid. When the CPU calls a kernel, it must specify how many threads will be created at runtime. The syntax that specifies the number of threads that will be created to execute a kernel is formally known as the execution configuration, and is flexible to support CUDA's hierarchy of threads, blocks of threads, and grids of blocks.

Some steps must be followed to use the GPU: first, the device must be initialized. Then, memory must be allocated in the GPU and data transferred to it. The kernel is then called. After the kernel has finished, results must be copied back to the CPU.

#### **3. Mathematical model**

The model proposed in this chapter is based on a set of Partial Differential Equations (PDEs) originally proposed by Pigozzo et al. (2011). In the original work, a set of PDEs describe the dynamics of the immune response to LPS in a microscopic section of tissue. In particular, the interactions among antigens (LPS molecules), neutrophils and cytokines were modelled. In this chapter, a simplified model of the innate immune system using ODEs is presented to simulate the temporal behaviour of LPS, neutrophils, macrophages and cytokines during the first phase of the immune response. The main differences between our model and the original one (Pigozzo et al. (2011)) are: a) the current model does not consider the spatial dynamics of the cells and molecules and b) the macrophages in two stages of readiness, resting and activated, are introduced in the current model.

(*permeabilityN*) and the capacity of the tissue to support the entrance of neutrophils (*Nmax*),

Modelling the Innate Immune System 355

*dt* <sup>=</sup> <sup>−</sup>*μCHCH* + (*βCH*|*<sup>N</sup> <sup>N</sup>* <sup>+</sup> *<sup>β</sup>CH*|*AM AM*).*A*.(<sup>1</sup> <sup>−</sup> *CH*

The term *μCHCH* models the pro-inflammatory cytokine decay, where *μCH* is the decay rate. The term (*βCH*|*<sup>N</sup> <sup>N</sup>* <sup>+</sup> *<sup>β</sup>CH*|*AM AM*).*<sup>A</sup>* models the production of the pro-inflammatory cytokine by the neutrophils and activated macrophages, where *<sup>β</sup>CH*|*<sup>N</sup>* and *<sup>β</sup>CH*|*AM* are the rate of this

*RM* <sup>−</sup> *<sup>P</sup>min*

*sourceRM* <sup>=</sup> *permeabilityRM*.(*Mmax* <sup>−</sup> (*RM* <sup>+</sup> *AM*))

The term *permeabilityRM* models how permeability of the endothelium of the blood vessels to macrophages depends on the local concentration of cytokines. The term *μRMRM* models the

*dt* <sup>=</sup> <sup>−</sup>*μAM AM* <sup>+</sup> *<sup>λ</sup>RM*|*A*.*RM*.*<sup>A</sup>*

The term *μAMRM* models the activated macrophage apoptosis, where *μRM* is the rate of

The sensitivity analysis consists in the analysis of impacts caused by variations of parameters and initial conditions of the mathematical model against its dependent variables (Saltelli et al. (2008)). If a parameter causes a drastic change in the output of the problem, after suffering a minor change in its initial value, it is thought that this parameter is sensitive to the problem studied. Otherwise, this variable has little impact in the model. The sensitivity analysis is used to improve the understanding of the mathematical model as it allows us to identify input parameters that are more relevant for the model, i.e. the values of these parameters should be carefully estimated. In this chapter we use a brute force approach to exam the influence of the 19 parameters present in the equation and two of the initial conditions. A small change in the value of each parameter is done, and then the model is solved again for this new parameter set. This process is done many times, since all combinations of distinct values of parameters and initial conditions must be considered. We analyse the impact of changing one coefficient at a time. The parameters and initial conditions were adjusted from -100% to + 100% (in steps

*dt* <sup>=</sup> <sup>−</sup>*μRMRM* <sup>−</sup> *<sup>λ</sup>RM*|*A*.*RM*.*<sup>A</sup>* <sup>+</sup> *sourceRM*

*CH*(0) = <sup>0</sup> (3)

*RM* ). *CH*

*CH*+*keqch* <sup>+</sup> *<sup>P</sup>min*

*AM*(0) = <sup>0</sup> (5)

*RM*

(4)

*chInf* )

that can also represent the blood concentration of Neutrophils.

The dynamics of cytokine is presented in Equation 3.

production by neutrophils and macrophages, respectively.

Equation 4 presents the dynamics of the resting macrophages.

*permeabilityRM* = (*Pmax*

resting macrophage apoptosis, where *μRM* is the rate of apoptosis.

� *dAM*

Finally, the dynamics of activate macrophages is presented in Equation 5.

� *dCH*

⎧ ⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

apoptosis.

**4. Implementation**

*dRM*

*RM*(0) = 1

Figure 1 presents schematically the relationship between macrophages, neutrophils, proinflammatory cytokines and LPS. LPS cause a response in both macrophages and neutrophils, that recognize LPS and phagocyte them. The process of phagocytosis induces, in a rapid way, the apoptosis of neutrophils. This induction is associated with the generation of reactive oxygen species (ROS) (Zhang et al. (2003)). The resting macrophages become activated when they find LPS in the tissue. The pro-inflammatory cytokine is produced by both active macrophages and neutrophils after they recognize LPS. It induces an increase in the endothelial permeability allowing more neutrophils to leave the blood vessels and enter the infected tissue.

Our set of equations is given below, where *RM*, *AM*, *A*, *N* and *CH* represent the population of resting macrophages, activated macrophages, LPS, neutrophils and pro-inflammatory cytokines, respectively. The dynamics of LPS is modelled with Equation 1.

$$\begin{cases} \frac{dA}{dt} = -\mu\_A A - (\lambda\_{N|A}.N + \lambda\_{AM|A}.AM + \lambda\_{RM|A}.RM).A\\ A(0) = 20 \end{cases} \tag{1}$$

The term *μ<sup>A</sup> A* models the decay of LPS, where *μ<sup>A</sup>* is its decay rate. The term <sup>−</sup>(*λN*|*A*.*<sup>N</sup>* <sup>+</sup> *<sup>λ</sup>AM*|*A*.*AM* <sup>+</sup> *<sup>λ</sup>RM*|*A*.*RM*).*<sup>A</sup>* models the phagocytosis of LPS by macrophages and neutrophils, where *<sup>λ</sup>N*|*<sup>A</sup>* is the phagocytosis rate of neutrophils, *<sup>λ</sup>AM*|*<sup>A</sup>* is the phagocytosis rate of active macrophages, and *<sup>λ</sup>RM*|*<sup>A</sup>* is the phagocytosis rate of resting macrophages.

Neutrophils are modelled with Equation 2.

$$\begin{cases}permeability\_N = (P\_N^{\max} - P\_N^{\min}) \cdot \frac{CH}{CH + keepch} + P\_N^{\min} \\ source\_N = permeability\_N \cdot (N^{\max} - N) \\ \frac{dN}{dt} = -\mu\_N N - \lambda\_{A|N} A.N + source\_N \\ N(0) = 0 \end{cases} \tag{2}$$

The term *permeabilityN* uses a Hill equation (Goutelle et al. (2008)) to model how permeability of the endothelium of the blood vessels depends on the local concentration of cytokines. Hill equations are also used, for example, to model drug dose-response relationships (Wagner (1968)).

The idea is to model the increase in the permeability of the endothelium according to the concentration of the pro-inflammatory cytokines into the endothelium. In the Hill equation, *Pmax <sup>N</sup>* represents the maximum rate of increase of endothelium permeability to neutrophils induced by pro-inflammatory cytokines, *Pmin <sup>N</sup>* represents the minimum rate of increase of endothelium permeability induced by pro-inflammatory cytokines and *keqch* is the concentration of the pro-inflammatory cytokine that exerts 50% of the maximum effect in the increase of the permeability. The term *μ<sup>N</sup> N* models the neutrophil apoptosis, where *<sup>μ</sup><sup>N</sup>* is the rate of apoptosis. The term *<sup>λ</sup>A*|*<sup>N</sup> <sup>A</sup>*.*<sup>N</sup>* models the neutrophil apoptosis induced by the phagocytosis, where *<sup>λ</sup>A*|*<sup>N</sup>* represent the rate of this induced apoptosis. The term *sourceN* represents the source term of neutrophil, that is, the number of neutrophils that is entering the tissue from the blood vessels. This number depends on the endothelium permeability 4 Will-be-set-by-IN-TECH

Figure 1 presents schematically the relationship between macrophages, neutrophils, proinflammatory cytokines and LPS. LPS cause a response in both macrophages and neutrophils, that recognize LPS and phagocyte them. The process of phagocytosis induces, in a rapid way, the apoptosis of neutrophils. This induction is associated with the generation of reactive oxygen species (ROS) (Zhang et al. (2003)). The resting macrophages become activated when they find LPS in the tissue. The pro-inflammatory cytokine is produced by both active macrophages and neutrophils after they recognize LPS. It induces an increase in the endothelial permeability allowing more neutrophils to leave the blood vessels and enter

Our set of equations is given below, where *RM*, *AM*, *A*, *N* and *CH* represent the population of resting macrophages, activated macrophages, LPS, neutrophils and pro-inflammatory

*dt* <sup>=</sup> <sup>−</sup>*μ<sup>A</sup> <sup>A</sup>* <sup>−</sup> (*λN*|*A*.*<sup>N</sup>* <sup>+</sup> *<sup>λ</sup>AM*|*A*.*AM* <sup>+</sup> *<sup>λ</sup>RM*|*A*.*RM*).*<sup>A</sup>*

The term *μ<sup>A</sup> A* models the decay of LPS, where *μ<sup>A</sup>* is its decay rate. The term <sup>−</sup>(*λN*|*A*.*<sup>N</sup>* <sup>+</sup> *<sup>λ</sup>AM*|*A*.*AM* <sup>+</sup> *<sup>λ</sup>RM*|*A*.*RM*).*<sup>A</sup>* models the phagocytosis of LPS by macrophages and neutrophils, where *<sup>λ</sup>N*|*<sup>A</sup>* is the phagocytosis rate of neutrophils, *<sup>λ</sup>AM*|*<sup>A</sup>* is the phagocytosis rate of active macrophages, and *<sup>λ</sup>RM*|*<sup>A</sup>* is the phagocytosis rate of resting

*<sup>N</sup>* <sup>−</sup> *<sup>P</sup>min*

The term *permeabilityN* uses a Hill equation (Goutelle et al. (2008)) to model how permeability of the endothelium of the blood vessels depends on the local concentration of cytokines. Hill equations are also used, for example, to model drug dose-response relationships (Wagner

The idea is to model the increase in the permeability of the endothelium according to the concentration of the pro-inflammatory cytokines into the endothelium. In the Hill

increase of endothelium permeability induced by pro-inflammatory cytokines and *keqch* is the concentration of the pro-inflammatory cytokine that exerts 50% of the maximum effect in the increase of the permeability. The term *μ<sup>N</sup> N* models the neutrophil apoptosis, where *<sup>μ</sup><sup>N</sup>* is the rate of apoptosis. The term *<sup>λ</sup>A*|*<sup>N</sup> <sup>A</sup>*.*<sup>N</sup>* models the neutrophil apoptosis induced by the phagocytosis, where *<sup>λ</sup>A*|*<sup>N</sup>* represent the rate of this induced apoptosis. The term *sourceN* represents the source term of neutrophil, that is, the number of neutrophils that is entering the tissue from the blood vessels. This number depends on the endothelium permeability

*<sup>N</sup>* represents the maximum rate of increase of endothelium permeability to

*sourceN* <sup>=</sup> *permeabilityN*.(*Nmax* <sup>−</sup> *<sup>N</sup>*)

*dt* <sup>=</sup> <sup>−</sup>*μ<sup>N</sup> <sup>N</sup>* <sup>−</sup> *<sup>λ</sup>A*|*<sup>N</sup> <sup>A</sup>*.*<sup>N</sup>* <sup>+</sup> *sourceN*

*<sup>A</sup>*(0) = <sup>20</sup> (1)

*<sup>N</sup>* ). *CH*

*CH*+*keqch* <sup>+</sup> *<sup>P</sup>min*

*N*

*<sup>N</sup>* represents the minimum rate of

(2)

cytokines, respectively. The dynamics of LPS is modelled with Equation 1.

*permeabilityN* = (*Pmax*

the infected tissue.

macrophages.

(1968)).

equation, *Pmax*

� *dA*

Neutrophils are modelled with Equation 2. ⎧ ⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

*dN*

*N*(0) = 0

neutrophils induced by pro-inflammatory cytokines, *Pmin*

(*permeabilityN*) and the capacity of the tissue to support the entrance of neutrophils (*Nmax*), that can also represent the blood concentration of Neutrophils.

The dynamics of cytokine is presented in Equation 3.

$$\begin{cases} \frac{d\mathcal{C}H}{dt} = -\mu\_{\mathcal{C}H}\mathcal{C}H + (\mathcal{\beta}\_{\mathcal{C}H|N}N + \mathcal{\beta}\_{\mathcal{C}H|AM}AM).A.(1 - \frac{\mathcal{C}H}{\mathrm{ch}\ln f})\\ \mathcal{C}H(0) = 0 \end{cases} \tag{3}$$

The term *μCHCH* models the pro-inflammatory cytokine decay, where *μCH* is the decay rate. The term (*βCH*|*<sup>N</sup> <sup>N</sup>* <sup>+</sup> *<sup>β</sup>CH*|*AM AM*).*<sup>A</sup>* models the production of the pro-inflammatory cytokine by the neutrophils and activated macrophages, where *<sup>β</sup>CH*|*<sup>N</sup>* and *<sup>β</sup>CH*|*AM* are the rate of this production by neutrophils and macrophages, respectively.

Equation 4 presents the dynamics of the resting macrophages.

$$\begin{cases}
\text{permeability}\_{RM} = (P\_{RM}^{\max} - P\_{RM}^{\min}) \cdot \frac{\text{CH}}{\text{CH} + \text{keq} \, \text{ch}} + P\_{RM}^{\min} \\
\text{source}\_{RM} = \text{permeability}\_{RM} \cdot (M^{\max} - (RM + AM)) \\
\frac{dRM}{dt} = -\mu\_{RM} RM - \lambda\_{RM|A} \, RM.A + \text{source}\_{RM} \\
RM(0) = 1
\end{cases} \tag{4}$$

The term *permeabilityRM* models how permeability of the endothelium of the blood vessels to macrophages depends on the local concentration of cytokines. The term *μRMRM* models the resting macrophage apoptosis, where *μRM* is the rate of apoptosis.

Finally, the dynamics of activate macrophages is presented in Equation 5.

$$\begin{cases} \frac{dAM}{dt} = -\mu\_{AM}AM + \lambda\_{RM|A}.RM.A\\ AM(0) = 0 \end{cases} \tag{5}$$

The term *μAMRM* models the activated macrophage apoptosis, where *μRM* is the rate of apoptosis.

#### **4. Implementation**

The sensitivity analysis consists in the analysis of impacts caused by variations of parameters and initial conditions of the mathematical model against its dependent variables (Saltelli et al. (2008)). If a parameter causes a drastic change in the output of the problem, after suffering a minor change in its initial value, it is thought that this parameter is sensitive to the problem studied. Otherwise, this variable has little impact in the model. The sensitivity analysis is used to improve the understanding of the mathematical model as it allows us to identify input parameters that are more relevant for the model, i.e. the values of these parameters should be carefully estimated. In this chapter we use a brute force approach to exam the influence of the 19 parameters present in the equation and two of the initial conditions. A small change in the value of each parameter is done, and then the model is solved again for this new parameter set. This process is done many times, since all combinations of distinct values of parameters and initial conditions must be considered. We analyse the impact of changing one coefficient at a time. The parameters and initial conditions were adjusted from -100% to + 100% (in steps

The complete set of equations that has been simulated, including the initial values used, are

Modelling the Innate Immune System 357

*dt* <sup>=</sup> <sup>−</sup>*μ<sup>A</sup> <sup>A</sup>* <sup>−</sup> (*λN*|*A*.*<sup>N</sup>* <sup>+</sup> *<sup>λ</sup>AM*|*A*.*AM* <sup>+</sup> *<sup>λ</sup>RM*|*A*.*RM*).*<sup>A</sup>*

*<sup>N</sup>* ). *CH*

*RM* ). *CH*

*CH*+*keqch* <sup>+</sup> *<sup>P</sup>min*

*CH*+*keqch* <sup>+</sup> *<sup>P</sup>min*

*N*

*RM*

*chInf* )

(6)

*<sup>N</sup>* <sup>−</sup> *<sup>P</sup>min*

*RM* <sup>−</sup> *<sup>P</sup>min*

*sourceRM* <sup>=</sup> *permeabilityRM*.(*Mmax* <sup>−</sup> (*RM* <sup>+</sup> *AM*))

*dt* <sup>=</sup> <sup>−</sup>*μCHCH* + (*βCH*|*<sup>N</sup> <sup>N</sup>* <sup>+</sup> *<sup>β</sup>CH*|*AM AM*).*A*.(<sup>1</sup> <sup>−</sup> *CH*

It should be noticed that in this case two distinct initial values for *A*(0) will be used: *A*(0) =

The sensitivity analysis has shown that two parameters are relevant to the model: the capacity of the tissue to support the entrance of new neutrophils (*Nmax*) and the phagocytosis rate of

*Nmax* is the most sensitive parameter in the model. The capacity of the tissue to support the entrance of new neutrophils is directed related to the permeability of the endothelial cells, which form the linings of the blood vessels. If a positive adjustment is made in the parameter related to the permeability, then there are more neutrophils entering into the tissue. This larger amount of neutrophils into the tissue has many consequences: first, more cells are phagocyting, so the amount of LPS reduces faster. Second, a smaller amount of resting macrophages becomes active, because there is less LPS into the tissue. Third, a larger amount of cytokines are produced, since neutrophils are the main responsible for this production. If a negative adjustment is made, the inverse effect can be observed: with a smaller amount of neutrophils in the tissue, more resting macrophages become active. Also, a smaller amount of

Figures 2 to 6 illustrate this situation. It can be observed that the LPS decays faster when *Nmax*

*dt* <sup>=</sup> <sup>−</sup>*μRMRM* <sup>−</sup> *<sup>λ</sup>RM*|*A*.*RM*.*<sup>A</sup>* <sup>+</sup> *sourceRM*

*sourceN* <sup>=</sup> *permeabilityN*.(*Nmax* <sup>−</sup> *<sup>N</sup>*)

*dt* <sup>=</sup> <sup>−</sup>*μ<sup>N</sup> <sup>N</sup>* <sup>−</sup> *<sup>λ</sup>A*|*<sup>N</sup> <sup>A</sup>*.*<sup>N</sup>* <sup>+</sup> *sourceN*

*dt* <sup>=</sup> <sup>−</sup>*μAM AM* <sup>+</sup> *<sup>λ</sup>RM*|*A*.*RM*.*<sup>A</sup>*

presented by Equation 6:

⎧

*dA*

*A*(0) = 20|40

*permeabilityN* = (*Pmax*

*permeabilityRM* = (*Pmax*

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

*dN*

*dRM*

*dAM*

*dCH*

*RM*(0) = 1

*AM*(0) = 0

*CH*(0) = 0

*N*(0) = 0

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

20 and *A*(0) = 40.

LPS by neutrophils (*λN*|*A*).

cytokines are produced.

achieves its maximum value.

of 2%) of their initial values, except for some parameters, that were also adjusted from -100% to + 100%, but in steps of 20%. The combination of all different set of parameters and initial conditions give us a total of 450,000 system of ODEs that must be evaluated in this work.

The sequential code that implements the sensitivity analysis was first implemented in C. Then the code was parallelized using CUDA. The parallel code is based on the idea that each combination of distinct values of parameters and initial conditions can be computed independently by a distinct CUDA thread. The number of threads that will be used during computation depends on the GPU characteristics. In particular, the number of blocks and threads per block are chosen taking into account two distinct values defined by the hardware: a) the warp size and b) the maximum number of threads per block.

The forward Euler method was used for the numerical solution of the systems of ODEs with a time-step of 0.0001 days. The models were simulated to represent a total period equivalent to 5 days after the initial infection.
