**9.3 The SIR (susceptible-infected-recovered) model**

The SIR is a simple mathematical model of epidemic [24], which divides the (fixed) population of *N* individuals into three" compartments" which may vary as a function of time *t*.

*S t*ð Þ are those susceptible but not yet infected with the disease,

*I t*ð Þ is the number of infectious individuals,

*R t*ð Þ are those individuals who have recovered from the disease and now have immunity to it,

the parameter *λ* describes the effective contact rate of the disease,

the parameter *μ* is the mean recovery rate.

The SIR model describes the change in the population of each of these compartments in terms of two parameters:

$$\begin{cases} \frac{dS(t)}{dt} = -\lambda \frac{S(t)I(t)}{N},\\ \frac{dI(t)}{dt} = \lambda \frac{S(t)I(t)}{N} - \mu I(t),\\ \frac{dR(t)}{dt} = \mu I(t), \end{cases} \tag{32}$$

*Invariants for a Dynamical System with Strong Random Perturbations DOI: http://dx.doi.org/10.5772/intechopen.96235*

and its restrictsion is

Let us consider a classical model

with an invariant *u t*, **y** � �.

*u t*ð Þ¼ , **x**ð Þ*t u*ð Þ¼ 0, **x**ð Þ 0 *C*:

**y**ð Þ¼ 0 **y**0*:*

8 ><

>:

8 >>><

>>>:

tions for PCP1:

function of time *t*.

immunity to it,

**74**

8 ><

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

>:

*dx*1ðÞ¼ *t a*1ð Þ *t*, **x**ð Þ*t dt* þ *b*1ð Þ *t*, **x**ð Þ*t d*wðÞþ*t* Ð

*dx*2ðÞ¼ *t a*2ð Þ *t*, **x**ð Þ*t dt* þ *b*2ð Þ *t*, **x**ð Þ*t d*wðÞþ*t* Ð

*dx*3ðÞ¼ *t a*3ð Þ *t*, **x**ð Þ*t dt* þ *b*3ð Þ *t*, **x**ð Þ*t d*wðÞþ*t* Ð

Hence, the stochastic model has a representation

*dy*1ðÞ¼ *t a*1 *t*, **y**ð Þ*t* � �*dt* þ *b*1 *t*, **y**ð Þ*t* � �*d*wð Þþ*t* Ð

*dy*2ðÞ¼ *t a*2 *t*, **y**ð Þ*t* � �*dt* þ *b*2 *t*, **y**ð Þ*t* � �*d*wðÞþ*t* Ð

*dy*3ðÞ¼ *t a*3 *t*, **y**ð Þ*t* � �*dt* þ *b*3 *t*, **y**ð Þ*t* � �*d*wðÞþ*t* Ð

a differential equations system and having an invariant function.

*S t*ð Þ are those susceptible but not yet infected with the disease,

the parameter *λ* describes the effective contact rate of the disease,

*dS t*ð Þ *dt* ¼ �*λ*

8 >>>>>><

>>>>>>:

*dI t*ð Þ *dt* ¼ *λ*

*dR t*ð Þ

*dt* ¼ *μI t*ð Þ,

**9.3 The SIR (susceptible-infected-recovered) model**

*I t*ð Þ is the number of infectious individuals,

the parameter *μ* is the mean recovery rate.

ments in terms of two parameters:

8 ><

>:

*dy*1ðÞ¼ *t F*1 *t*, **y**ð Þ*t* � �*dt*, *dy*2ðÞ¼ *t F*2 *t*, **y**ð Þ*t* � �*dt*, *dy*3ðÞ¼ *t F*3 *t*, **y**ð Þ*t* � �*dt*,

Further, we determine complementary function which is unit of control func-

*s*1 *t*, **y**ð Þ*t* � � ¼ *a*1 *t*, **y**ð Þ*t* � � � *F*1 *t*, **y**ð Þ*t* � �, *s*2 *t*, **y**ð Þ*t* � � ¼ *a*2 *t*, **y**ð Þ*t* � � � *F*2 *t*, **y**ð Þ*t* � �, *s*3 *t*, **y**ð Þ*t* � � ¼ *a*3 *t*, **y**ð Þ*t* � � � *F*3 *t*, **y**ð Þ*t* � �*:*

Finally, we have constructed stochastic analogue for classical model described by

The SIR is a simple mathematical model of epidemic [24], which divides the (fixed) population of *N* individuals into three" compartments" which may vary as a

*R t*ð Þ are those individuals who have recovered from the disease and now have

The SIR model describes the change in the population of each of these compart-

*S t*ð Þ*I t*ð Þ

*S t*ð Þ*I t*ð Þ *N* ,

*N* � *μI t*ð Þ,

Then we construct the GSDE system, taking into account the equality

(28)

(29)

(30)

(31)

(32)

*g*1ð Þ *t*, **x**ð Þ*t* , *γ ν*ð Þ *dt*, *dγ* ,

*g*2ð Þ *t*, **x**ð Þ*t* , *γ ν*ð Þ *dt*, *dγ* ,

*g*3ð Þ *t*, **x**ð Þ*t* , *γ ν*ð Þ *dt*, *dγ :*

*g*1 *t*, **y**ð Þ*t* , *γ* � �*ν*ð Þ *dt*, *dγ* ,

*g*2 *t*, **y**ð Þ*t* , *γ* � �*ν*ð Þ *dt*, *dγ* ,

*g*3 *t*, **y**ð Þ*t* , *γ* � �*ν*ð Þ *dt*, *dγ* ,

$$I(t) + I(t) + R(t) = N.\tag{33}$$

Let the model with strong perturbation be

$$\begin{cases} dS(t) = \left(-\lambda \frac{S(t)I(t)}{N} + s\_1(t, S(t), I(t), R(t))\right) dt + \\ \qquad + b\_1(t, S(t), I(t), R(t)) d\mathbf{w}(t) + \int\_{\mathbb{R}} \mathbf{g}\_1(t, S(t), I(t), R(t), \gamma) \nu(dt, d\gamma), \\\ dI(t) = \left(\lambda \frac{S(t)I(t)}{N} - \mu I(t) + s\_2(t, S(t), I(t), R(t))\right) dt + \\ \qquad + b\_1(t, S(t), I(t), R(t)) d\mathbf{w}(t) + \int\_{\mathbb{R}} \mathbf{g}\_2(t, S(t), I(t), R(t), \gamma) \nu(dt, d\gamma), \\\ dR(t) = (\mu S(t) + s\_3(t, S(t), I(t), R(t))) dt + \\ \qquad + b\_1(t, S(t), I(t), R(t)) d\mathbf{w}(t) + \int\_{\mathbb{R}} \mathbf{g}\_3(t, S(t), I(t), R(t), \gamma) \nu(dt, d\gamma), \end{cases} \tag{34}$$

and

$$u(t, \mathcal{S}(t), I(t), R(t)) = \mathcal{S}(t) + I(t) + R(t) - N \equiv \mathbf{0}.\tag{35}$$

Suppose that the function *u t*ð Þ¼ , *x*, *y*, *z x* þ *y* þ *z* � *N* is a first integral, *v t*ð Þ¼ , *x*, *y*, *z* 2*e*�*t* þ *x* and *h t*ð Þ¼ , *x*, *y*, *z y* are complementary functions, and *q t*ð Þ¼ , *x*, *y*, *z x* is arbitrary function. The initial condition is: *x*ð Þ¼ 0 1, *y*ð Þ¼ 0 0, *z*ð Þ¼ 0 0. Then constructed differential equations system has the form

$$
\begin{bmatrix} dx(t) \\ dy(t) \\ dz(t) \end{bmatrix} = \begin{bmatrix} 2e^{-t} \\ 0 \\ -2e^{-t} \end{bmatrix} dt + \begin{bmatrix} 0 \\ \varkappa(t) \\ -\varkappa(t) \end{bmatrix} d\mathbf{w}(t) + \int\_{R\_{\gamma}} \begin{bmatrix} 0 \\ \varkappa(t)\gamma \\ -\varkappa(t)\gamma \end{bmatrix} \nu(dt, d\gamma). \tag{36}
$$

Let us simulate a numerical solution of Eg.(36), where *N* ¼ 1 (for example). **Figure 1** shows simulation for system without jumps, the **Figure 2** shows the processes with jumps.

**Figure 1.** *Numerical solution for Eq.(36) without jumps.*

• During the process, the environment does not change in favor of one species,

Let us note: *N*1ð Þ*t* is the number of prey, and *N*2ð Þ*t* is the number of some predator, *ε*1, *ε*2, *η*1 and *η*2 are positive real parameters describing the interaction of

The populations change through time according to the pair of equations:

*dN*1ðÞ¼ *t N*1ð Þ*t* ð Þ *ε*1 � *η*1*N*2ð Þ*t dt*, *dN*2ðÞ¼� *t N*2ð Þ*t* ð Þ *ε*2 � *η*2*N*1ð Þ*t dt:*

*η*2*N*1ð Þ*t* ¼ *CNε*1

2 ð Þ*t e*

*Rγ*

*Rγ*

*η*2*x*1ð Þ*t* � *Cxε*1

Let us assume that *ε*1 ¼ 2, *ε*2 ¼ 1, *η*1 ¼ *η*2 ¼ 1, and *C* ¼ 1, and initial condition is

2 ð Þ*t e*

�*z*2ð Þ *t*,*x*,*y*,*γ z*1ð Þ *t*, *x*, *y*, *γ z*2ð Þ *t*, *x*, *y*, *γ* ð Þ *z*2ð Þ� *t*, *x*, *y*, *γ* 2 ,

1 ð Þ *t*, *x*, *y*, *γ* � �*:*

*D t*ð Þþ , *x t*ð Þ, *y t*ð Þ *E t*ð Þ , *x t*ð Þ, *y t*ð Þ " #*dt*þ

3 7 5 *d*wð Þ*t* , (38)

(40)

(42)

�*η*1*N*2ð Þ*t* , (39)

�*η*1*x*2ð Þ*t :* (41)

*g*1ð Þ *t*, *x*1ð Þ*t* , *x*2ð Þ*t* , *γ ν*ð Þ *dt*, *dγ* ,

*g*2ð Þ *t*, *x*1ð Þ*t* , *x*2ð Þ*t* , *γ ν*ð Þ *dt*, *dγ* ,

and genetic adaptation is inconsequential.

*Invariants for a Dynamical System with Strong Random Perturbations*

�

*N*�*ε*2 1 ð Þ*t e*

We can introduce the stochastic model as a form

*dx*1ðÞ¼ *t* ð Þ *ε*1*x*1ð Þ�*t η*1*x*1ð Þ*t x*2ðÞþ*t s*1ð Þ *t*, *x*1ð Þ*t* , *x*2ð Þ*t dt*þ

*dx*2ðÞ¼ � *t* ð Þ *ε*2*x*2ðÞþ*t η*2*x*1ð Þ*t x*2ðÞþ*t s*2ð Þ *t*, *x*1ð Þ*t* , *x*2ð Þ*t dt*þ

1 ð Þ*t e*

*x*ð Þ¼ 0 *y*ð Þ¼ 0 1. The function *u t*ð Þ¼ , *x*, *y x*�1*ex* � *y*2*e*�2*y* is a first integral, *h t*ð Þ¼ , *x*, *y y* � *x* þ *e*�*t* and *q t*ð Þ¼ , *x*, *y x* are complementary functions.

�*z*1ð Þ *t*,*x*,*y*,*γ* 1 � *z*�1

*dy t*ð Þ " # ¼ *A t*ð Þþ , *x t*ð Þ, *y t*ð Þ *B t*ð Þþ , *x t*ð Þ, *y t*ð Þ *C t*ð Þ , *x t*ð Þ, *y t*ð Þ

*x t*ð Þ*e*�*y t*ð Þ *y*2 ð Þ ðÞ�*t* 2*y t*ð Þ

*x t*ð Þ

*ex t*ð Þ *x t*ð Þ � *e*

Then, the constructed SDE system includes only Wiener perturbation:

We cannot find an analytical solution of the differential equations system

þ*b*1ð Þ *t*, *x*1ð Þ*t* , *x*2ð Þ*t d*w1ð Þþ*t* Ð

þ*b*2ð Þ *t*, *x*1ð Þ*t* , *x*2ð Þ*t d*w2ðÞþ*t* Ð

*u t*ð Þ¼ , **x**ð Þ*t x*�*ε*2

• Predators have limitless appetite.

*DOI: http://dx.doi.org/10.5772/intechopen.96235*

Eq. (38) has the invariant function

*x*1ð Þ¼ 0 *N*1, *x*2ð Þ¼ 0 *N*2,

*∂z*1ð Þ *t*, *x*, *y*, *γ*

*∂z*2ð Þ *t*, *x*, *y*, *γ*

*∂γ* ¼ *e*

*∂γ* ¼ �*e*

þ

2 6 4

the two species.

where *C* ¼ *const*.

with condition

8 >><

>>:

*dx t*ð Þ

**77**

8

>>>>>>>>>><

>>>>>>>>>>:

**Figure 2.** *Numerical solution for Eq.(36) with jumps.*

In such a way we could use the system of differential equations

$$\begin{cases} d\boldsymbol{y}\_1(t) = 2\boldsymbol{e}^{-t}dt, \\\\ d\boldsymbol{y}\_2(t) = \boldsymbol{y}\_1(t)d\mathbf{w}(t) + \int \boldsymbol{\eta}\mathbf{y}\_1(t)\boldsymbol{\nu}(dt, d\boldsymbol{\eta}), \\\\ d\boldsymbol{y}\_3(t) = -2\boldsymbol{e}^{-t}dt + \boldsymbol{\eta}\_1(t)d\mathbf{w}(t) - \int \boldsymbol{\eta}\mathbf{y}\_1(t)\boldsymbol{\nu}(dt, d\boldsymbol{\eta}), \\\\ \boldsymbol{y}(0) = \boldsymbol{y}\_0, \end{cases} \tag{37}$$

as initial step for construction of stochastic SIR-model. A good choice of complementary functions *v t*ð Þ , *x*, *y*, *z* and *h t*ð Þ , *x*, *y*, *z* allows us to obtain such coefficients that ensure that the solution f g *x t*ð Þ, *y t*ð Þ, *z t*ð Þ of the differential equations system satisfy some reasonable limitations.

#### **9.4 The predator–prey model**

The Lotka - Volterra equations or the predator–prey equations used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey.

The Lotka - Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:

Let us note: *N*1ð Þ*t* is the number of prey, and *N*2ð Þ*t* is the number of some predator, *ε*1, *ε*2, *η*1 and *η*2 are positive real parameters describing the interaction of the two species.

The populations change through time according to the pair of equations:

$$\begin{cases} dN\_1(t) = N\_1(t)(\varepsilon\_1 - \eta\_1 N\_2(t))dt, \\ dN\_2(t) = -N\_2(t)(\varepsilon\_2 - \eta\_2 N\_1(t))dt. \end{cases} \tag{38}$$

Eq. (38) has the invariant function

$$N\_1^{-\varepsilon\_1}(t)e^{\eta\_2 N\_1(t)} = \text{CN}\_2^{\varepsilon\_1}(t)e^{-\eta\_1 N\_2(t)},\tag{39}$$

where *C* ¼ *const*.

In such a way we could use the system of differential equations

*Advances in Dynamical Systems Theory, Models, Algorithms and Applications*

*γy*1ð Þ*t ν*ð Þ *dt*, *dγ* ,

*γy*1ð Þ*t ν*ð Þ *dt*, *dγ* ,

(37)

*dt* þ *y*1ð Þ*t d*wðÞ�*t* Ð

as initial step for construction of stochastic SIR-model. A good choice of complementary functions *v t*ð Þ , *x*, *y*, *z* and *h t*ð Þ , *x*, *y*, *z* allows us to obtain such coefficients that ensure that the solution f g *x t*ð Þ, *y t*ð Þ, *z t*ð Þ of the differential equations system

The Lotka - Volterra equations or the predator–prey equations used to describe the dynamics of biological systems in which two species interact, one as a predator

The Lotka - Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey

• The food supply of the predator population depends entirely on the size of the

*dt*,

*dy*2ðÞ¼ *t y*1ð Þ*t d*wðÞþ*t* Ð

• The prey population finds ample food at all times.

• The rate of change of population is proportional to its size.

*dy*1ðÞ¼ *t* 2*e*�*t*

8 >>>>>>><

*Numerical solution for Eq.(36) with jumps.*

**Figure 2.**

>>>>>>>:

satisfy some reasonable limitations.

**9.4 The predator–prey model**

and the other as prey.

prey population.

populations:

**76**

*dy*3ðÞ¼� *t* 2*e*�*t*

*y*ð Þ¼ 0 *y*0,

We can introduce the stochastic model as a form

$$\begin{cases} d\mathbf{x}\_{1}(t) = (\mathbf{e}\_{1}\mathbf{x}\_{1}(t) - \eta\_{1}\mathbf{x}\_{1}(t)\mathbf{x}\_{2}(t) + s\_{1}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t)))dt + \\ \qquad \quad + b\_{1}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t))d\mathbf{w}\_{1}(t) + \int\_{\mathbb{R}} \mathbf{g}\_{1}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t),\boldsymbol{\chi})\boldsymbol{\nu}(dt,d\boldsymbol{\chi}), \\\\ d\mathbf{x}\_{2}(t) = (-\varepsilon\_{2}\mathbf{x}\_{2}(t) + \eta\_{2}\mathbf{x}\_{1}(t)\mathbf{x}\_{2}(t) + s\_{2}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t)))dt + \\ \qquad \quad + b\_{2}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t))d\mathbf{w}\_{2}(t) + \int\_{\mathbb{R}} \mathbf{g}\_{2}(t,\mathbf{x}\_{1}(t),\mathbf{x}\_{2}(t),\boldsymbol{\chi})\boldsymbol{\nu}(dt,d\boldsymbol{\chi}), \\\\ \mathbf{x}\_{1}(0) = N\_{1}, \qquad \quad \mathbf{x}\_{2}(0) = N\_{2}, \end{cases} (40)$$

with condition

$$u(t, \mathbf{x}(t)) = \mathfrak{x}\_1^{-\varepsilon\_2}(t)e^{\eta\_2 \mathbf{x}\_1(t)} - \mathbf{C} \mathfrak{x}\_2^{\varepsilon\_1}(t)e^{-\eta\_1 \mathbf{x}\_2(t)}.\tag{41}$$

Let us assume that *ε*1 ¼ 2, *ε*2 ¼ 1, *η*1 ¼ *η*2 ¼ 1, and *C* ¼ 1, and initial condition is *x*ð Þ¼ 0 *y*ð Þ¼ 0 1. The function *u t*ð Þ¼ , *x*, *y x*�1*ex* � *y*2*e*�2*y* is a first integral, *h t*ð Þ¼ , *x*, *y y* � *x* þ *e*�*t* and *q t*ð Þ¼ , *x*, *y x* are complementary functions.

We cannot find an analytical solution of the differential equations system

$$\begin{cases} \frac{\partial \mathbf{z}\_1(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma})}{\partial \boldsymbol{\gamma}} = e^{-x\_2(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma})} \mathbf{z}\_1(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma}) \mathbf{z}\_2(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma}) (\mathbf{z}\_2(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma}) - \mathbf{2}),\\ \frac{\partial \mathbf{z}\_2(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma})}{\partial \boldsymbol{\gamma}} = -e^{-x\_1(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma})} \left(1 - \mathbf{z}\_1^{-1}(t, \mathbf{x}, \mathbf{y}, \boldsymbol{\gamma})\right). \end{cases}$$

Then, the constructed SDE system includes only Wiener perturbation:

$$
\begin{aligned}
\begin{bmatrix} dx(t) \\ dy(t) \end{bmatrix} &= \begin{bmatrix} A(t, \mathbf{x}(t), y(t)) + B(t, \mathbf{x}(t), y(t)) + C(t, \mathbf{x}(t), y(t)) \\ & D(t, \mathbf{x}(t), y(t)) + E(t, \mathbf{x}(t), y(t)) \end{bmatrix} dt + \\ &+ \begin{bmatrix} \mathbf{x}(t)e^{-\mathbf{y}(t)}(y^2(t) - 2\mathbf{y}(t)) \\ \mathbf{e}^{\mathbf{x}(t)} - \mathbf{e}^{\mathbf{x}(t)} \end{bmatrix} d\mathbf{w}(t), \end{aligned} \tag{42}
$$

where

*A t*ð Þ¼ , *x t*ð Þ, *y t*ð Þ 0*:*5*e* �*y t*ð Þ*x t*ð Þ *y*2 ðÞ�*t* 2*y t*ð Þ*e* �*y t*ð Þ 2 , *B t*ð Þ¼ , *x t*ð Þ, *y t*ð Þ 0*:*5*e* �*y t*ð Þ *x t*ð Þ*e x t*ð Þ � *e x t*ð Þ*x*�1 ð Þ*t* 2 � 4*y t*ð Þþ *y*2 ð Þ*t* , *C t*ð Þ¼� , *x t*ð Þ, *y t*ð Þ *e*�*t e*�*y t*ð Þ *y*2 ð Þ ð Þ�*t* 2*y t*ð Þ *ex t*ð Þ*x*�1ðÞ�*t* 2*y t*ð Þ*e*�*y t*ð Þ � *ex t*ð Þ*x*�2ðÞþ*t y*2ð Þ*t e*�*y t*ð Þ , *D t*ð Þ¼ , *x t*ð Þ, *y t*ð Þ *e*�*t ex t*ð Þ *x*�1ðÞ�*t x*�2 ð Þ ð Þ*t ex t*ð Þ*x*�1ðÞ�*t* 2*y t*ð Þ*e*�*y t*ð Þ � *ex t*ð Þ*x*�2ðÞþ*t y*2ð Þ*t e*�*y t*ð Þ , *E t*ð Þ¼� , *x t*ð Þ, *y t*ð Þ 0*:*5*e x t*ð Þ*e* �*y t*ð Þ *y*2 ðÞ�*t* 2*y t*ð Þ 1 � *x*�1 ð Þþ*t x t*ðÞ� 2 þ 2*x*�1 ð Þ*t :* (43)

Finally, we have the stochastic Lotka Volterra model associated to (38)

*N*1ð Þ*t e*�*N*2ð Þ*t N*2

*Invariants for a Dynamical System with Strong Random Perturbations*

*eN*1ð Þ*t N*1ð Þ*t* � *e*

¼ *A t*ð Þþ , *N t*ð Þ *B t*ð Þþ , *N t*ð Þ *C t*ð Þ , *N t*ð Þ *D t*ð Þþ , *N t*ð Þ *E t*ð Þ , *N t*ð Þ " #*dt*þ

2ðÞ�*t* 2*N*2ð Þ*t* � �

3 7 5 *d*wð Þ*t* , (44)

*N*1ð Þ*t*

where *A t*ð Þ , *N t*ð Þ , *B t*ð Þ , *N t*ð Þ , *C t*ð Þ , *N t*ð Þ , *D t*ð Þ , *N t*ð Þ , *E t*ð Þ , *N t*ð Þ are determined by

**Figures 3** and **4** show two realizations for numerical solution of Eq. (44). Another examples of a differential equation system construction and models see

The invariant method widens horizons for constructing and researching into mathematical models of real systems with the invariants that hold out under any

(*N t*ðÞ¼ ð Þ *N*1ð Þ*t* , *N*2ð Þ*t* ):

Eq.(43).

in [25–29].

**10. Conclusion**

**Author details**

**79**

Elena Karachanskaya

Far-Eastern State Transport University, Khabarovsk, Russia

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to<EMAIL_ADDRESS>
provided the original work is properly cited.

strong random disturbances.

*dN*1ð Þ*t dN*2ð Þ*t*

þ

2 6 4

" #

*DOI: http://dx.doi.org/10.5772/intechopen.96235*

**Figure 3.** *Numerical simulation 1 for solution of Eq.(44).*

**Figure 4.** *Numerical simulation 2 for solution of Eq.(44).*

*Invariants for a Dynamical System with Strong Random Perturbations DOI: http://dx.doi.org/10.5772/intechopen.96235*

Finally, we have the stochastic Lotka Volterra model associated to (38) (*N t*ðÞ¼ ð Þ *N*1ð Þ*t* , *N*2ð Þ*t* ):

$$\begin{aligned} \begin{bmatrix} dN\_1(t) \\ dN\_2(t) \end{bmatrix} &= \begin{bmatrix} A(t, N(t)) + B(t, N(t)) + C(t, N(t)) \\ D(t, N(t)) + E(t, N(t)) \end{bmatrix} dt + \\ &+ \begin{bmatrix} N\_1(t)e^{-N\_2(t)} \left(N\_2^2(t) - 2N\_2(t)\right) \\ e^{N\_1(t)} - e^{N\_1(t)} \end{bmatrix} d\mathbf{w}(t), \end{aligned} \tag{44}$$

where *A t*ð Þ , *N t*ð Þ , *B t*ð Þ , *N t*ð Þ , *C t*ð Þ , *N t*ð Þ , *D t*ð Þ , *N t*ð Þ , *E t*ð Þ , *N t*ð Þ are determined by Eq.(43).

**Figures 3** and **4** show two realizations for numerical solution of Eq. (44).

Another examples of a differential equation system construction and models see in [25–29].


**12. Microbiome-immunity crosstalk mechanisms and COVID-19**

The microbiome is composed of a great number of microorganisms which in the human body are found in the gut and also on the skin and other mucosal microenvironments. Studies have shown that microbiome is active participant in a host functions like immunity and other metabolism activities. Other functions which have a microbiome impact are nutrition end products and the circadian clock operation [28–30]. Studies have shown that lung microbiome can modify the risk and consequences of COVID-19 by activating the immune response. On the same line, other suggestions show that bacterial co-infections as well as the gut-lung cross talk may be important players in COVID-19 disease prognosis. Recent studies have established that COVID-19 patients with GI complications experience more respiratory distress when compared to the patients without GI complications [31, 32]. Interestingly, obesity, *diabetes mellitus*, cardiovascular diseases and old age-related disorders have been related to weak microbiome communities and the same category of patients have also been more vulnerable to COVID-19 and SARS CoV-2 attacks [33–36]. Even in gut microbiome, COVID-19 patients were found to be deficient to beneficial commensals like *Eubacterium ventriosum, Faecalibacterium prausnitzii,* and *Roseburia* these microbes highly correlate to disease severity [37–39].

### **13. Recommendations**

The recommendations from this crucial evaluation of the immune system role of the microbiome is that in future we should not over look natural mechanisms which have established an equilibrium with the immune system. This should also involve the educating of masses on how to cultivate the natural human microbiome defense systems as physical barriers. The training and boosting of the immune system both in children and adults respectively should regionally be evaluated. This is because increased traveling and climate change cannot be assumed to be of little consequences in pandemic control strategies in light of the microbiome roles.


**3. Incidence**

Addressing the diagnosis of a dysmorphic newborn is similar to the diagnosis of systemic diseases – it relies on analyzing the family history and on performing a meticulous examina-

The steps to be taken after identifying a neonatal dysmorphism are to confirm the diagnosis through cytogenetic testing via molecular techniques (in order to confirm/exclude a genetic

After many years spent 'looking after little patients', we hereby discuss a number of anomalies and abnormal physical characteristics, isolated or associated, together with the genetic

Since the neonatologists are the first to evaluate the neonates, they must be familiar with various major and minor dysmorphisms. The diagnosis of a syndrome depends on good clinical skills, knowledge of phenotypic features of various syndromes and the experience of the examiner.

Dysmorphism [1] is a morphological anomaly of a structure, a deviation from the norm, and can be classified as major or minor. Major abnormalities may be surgical, medical or cosmetic, and they may be markers for other malformations too. Minor anomalies do not have significant surgical or cosmetic importance, though many genetic syndromes can be recognized

Anomalies may occur through three mechanisms [2], each having different implications for

• *The malformative mechanism* causes structural defects, resulting from an inherently abnormal development process, a primary error in morphogenesis. Malformations include congenital heart, lips, and palate abnormalities. These types of malformations are most

○ The malformation sequence results from a single primary malformation, as is the case

○ Malformative syndrome results from several different biological errors during

• *The deforming mechanism* is an anomaly resulting from the action of prenatal mechanical forces on normal fetal structures. The femur, the fingers (that become overlapped) and the head (that grows into an unusual shape) can be affected. Deformations are rarely genetic,

• *The disruptive mechanism* causes structural defects resulting from the destruction or interruption of normal intrinsic tissue, such as limbs reduction in amniotic band sequence or certain types of intestinal atresia due to vascular insufficiency [3]. The anomalies are rarely

caused by a genetic condition and unlikely to occur in a future pregnancy.

commonly associated with a genetic disease or a genetic predisposition.

tion of signs and expressions, in an effort to identify a syndrome [1].

syndromes in which they can be included.

504 Congenital Anomalies - From the Embryo to the Neonate

based on basis of minor anomalies.

**2. Mechanisms of occurrence**

with lumbar neural tube defects.

and the recurrence risk is usually low.

morphogenesis.

the diagnosis:

etiology), followed by family counseling by the neonatologist-geneticist team.

The incidence of congenital abnormalities is approximately 10% of total admissions in neonatal intensive care units (NICUs). Many of them have underlying genetic syndromes. Worldwide, around 7.9 million children (6% of births worldwide) are born with congenital anomalies [4] annually.

Minor anomalies, the subject of this chapter, appear to be isolated more frequently. About 15% of neonates are diagnosed with one minor anomaly (**Table 1**). About 71% of them are found in head, neck and hands. Among neonates diagnosed with an isolated minor anomaly, 3% have a major associated abnormality.

**4. Classification**

and gastrointestinal.

*4.1.2. Aplasia cutis congenita*

about 3 out of 10,000 births [8].

**Figure 1.** Aplasia cutis congenita.

lies [6].

**4.1. Minor head and throat anomalies**

Asymmetric crying facies (ACF) is a minor abnormality, characterized by lowering the corner of the mouth on the unaffected side when crying or sketching a grimace. This is caused by the congenital absence of the anguli oris depressant muscle. The ACF neonates show both nasolabial folds with normal, symmetrical depth and do have the normal ability to lift their forehead and close both eyes. This anomaly must be distinguished from facial nerve paralysis, which is less common [5]. In 20–70% of cases, ACF is associated with other congenital abnormalities, the most common being head/neck, cardiovascular, musculoskeletal, genitourinary

The Neonate with Minor Dysmorphisms http://dx.doi.org/10.5772/intechopen.71902 507

Once this anomaly has been identified, genetic testing is recommended (FISH test or chromosomal microarray comparative genomic hybridization) because ACF is especially associated with 22q11 deletion syndrome (also known as velocardiofacial or DiGeorge syndrome). In this syndrome, the facial dysmorphism coexist with structural heart anomalies, long fingers/limbs, thymus aplasia/hypoplasia and kidney abnormalities. The postnatal follow-up protocol recommends close monitoring of growth and development, evaluation of thyroid and parathyroid function, immunological, hearing and ophthalmic evaluation, echocardiography, renal ultrasonography and the treatment of possibly associated anoma-

Aplasia cutis congenita (ACC) is the congenital absence of the skin, and may occur on any part of the body. It affects the scalp in 70–80% of the cases (**Figure 1**), either as solitary lesions or associated with skull and dura mater defects [2, 7]. Aplasia cutis congenita is a rare anomaly in neonates. Over 500 cases have been reported since the first description, by Cordon in 1767. Due to the unreported cases, their real incidence is unknown. An estimate of the incidence is

*4.1.1. Asymmetric crying facies*

**Table 1.** Minor anomalies seen in various systems.

0.8% of neonates have two minor anomalies associated, and 11% of them have a major associated abnormality.

The presence of three or more minor abnormalities is rare (about 0.5%), and in most cases (90%), neonates also associate a major malformation.


**Precisely Controllable Synthesized Nanoparticles for Surface Enhanced Raman Spectroscopy Surface Enhanced Raman Spectroscopy**

**Precisely Controllable Synthesized Nanoparticles for**

DOI: 10.5772/intechopen.73086

Meng Xu and Jiatao Zhang Additional information is available at the end of the chapter

Meng Xu and Jiatao Zhang

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/intechopen.73086

#### **Abstract**

[30] Albiter E, Valenzuela MA, Alfaro S, Valverde-Aguilar G, Martinez-Pallares FM.

[31] Goodman DW. "Catalytically active Au on Titania": Yet another example of a strong metal support interaction (SMSI)? Catalysis Letters. 2005;**99**(1-2):1-4. DOI: 1011-372X/

[33] Pisarek M, Krajczewski J, Wierzbicka E, Hołdyński M, Sulka GD, Nowakowski R, Kudelski A, Janik-Czachor M. Influence of the silver deposition method on the activity of platforms for chemometric surface-enhanced Raman scattering measurements:

[34] Zuo C, Jagodzinski PW. Surface-enhanced Raman scattering of pyridine using different metals: Differences and explanation based on selective formation of α-Pyridyl on metal surfaces. The Journal of Physical Chemistry. B. 2005;**109**:1788-1793. DOI: 10.1021/

[35] DY W, Li JF, Ren B, Tian ZQ. Electrochemical surface-enhanced Raman spectroscopy of nanostructures. Chemical Society Reviews. 2008;**37**:1025-1041. DOI: 10.1039/b707872m

[36] Michaels AM, Nirmal M, Brus LE. Surface enhanced Raman spectroscopy of individual Rhodamine 6G molecules on large Ag Nanocrystals. Journal of the American Chemical

[37] Liu YC, ChCh Y, Sheu SF. Low concentration rhodamine 6G observed by surfaceenhanced Raman scattering on optimally electrochemically roughened silver substrates.

[38] Dieringer JA, Wustholz KL, Masiello DJ, Camden JP, Kleinman SL, Schatz GC, Van Duyne RP. Surface-enhanced Raman excitation spectroscopy of a single Rhodamine 6G molecule. Journal of the American Chemical Society. 2009;**131**:849-854. DOI: 10.1021/

[39] Pristinski D, Tan S, Erol M, Du HSS.In situ SERS study of Rhodamine 6G adsorbed on individually immobilized Ag nanoparticles. Journal of Raman Specroscopy. 2006;**37**:762-770.

Journal of Materials Chemistry. 2006;**16**:3546-3551. DOI: 10.1039/b609417a

Biomolecular Spectroscopy. 2017;**182**:124-129. DOI: 10.1016/j.saa.2017.04.005

nanopore arrays. Spectrochimica Acta – Part A: Molecular and

tural and photoactivity properties. Journal of Saudi Chemical Society. 2015;**19**:563-573.

: Metal precursor effect on the struc-

(110) interface. Surface Science.

Photocatalytic deposition of Ag nanoparticles on TiO2

[32] Lopez N, Nørskov JK. Theoretical study of the Au/TiO2

Society. 1999;**121**:9932-9939. DOI: 10.1021/ja992128q

2002;**515**:175-186 S00 39-6028 (02) 01873- 3

DOI: 10.1016/j.jscs.2015.05.009

05/0100-0001/0

54 Raman Spectroscopy

Silver films on ZrO2

jp0406363

ja8080154

DOI: 10.1002/jrs.1496

Surface-enhanced Raman scattering (SERS) is a powerful technique for trace molecular detection because of its ultrahigh molecular structure sensitivity and unique fingerprinting spectra. The morphology, size and structure of the plasmonic nanoparticles seriously influence the Raman scattering intensity of sample. In this chapter, we focus on the influence of nanoparticle morphology. By tailoring the plasmonic properties of anisotropic Au, Ag nanoparticles and generating electromagnetic "hot spots" of SERS active substrate, the SERS intensity can be seriously influenced. We also focus on providing a general introduction to understand the main parameters of anisotropic noble metal nanoparticles of SERS performance.

**Keywords:** surface enhanced Raman spectroscopy (SERS), localized surface plasmon resonances (LSPRs), noble metal nanoparticles, core/shell nanoparticles, controllable synthesis


**3. Statistical analysis**

To achieve the study, we have used individual medical files, and, for instruments of interpreting data, we have used a series of statistic applications including both the descriptive methods and the analytic ones; the data are centralized in the database EXCEL and SPSS and processed with the available statistic functions, compatible and adequate to the type of the collected data (at nominal, regular and reporting level).

The descriptive statistics, the distribution of the frequencies and the comparisons have been done by means of the program SPSS Statistics, ver. 21; the data of the different parameters are presented as average ± standard deviation (SD). The correlations between the parameters, using the correlation coefficients (Pearson, Spearman, and Kendall) have also been evaluated. The comparisons among the groups under study have been made by means of the Student and Anova tests. All of them have been statistically significant at the level of p1 = 0,05 and p2 = 0,01.


**3.1 Daily energy expenditure structure**

Unlike basal metabolism, which is the minimum level of body energy expenditure, average daily metabolism includes the sum of all expenditures associated with the realization of various body functions. Food processing and digestion, thermoregulation and muscle activity are the most power-consuming functions. Unfortunately, there is almost no data in literature on the energy value of mental activity (not taking into consideration the indirect calculations by Holliday, 1971).

Fig. 3. Age changes in contribution of various organs and tissues into the structure of human basal metabolism (After: Holliday, 1971; Kornienko, 1979; modified)

Energy Metabolism in Children and Adolescents 127

The rate of heat production is considerably increased after food consumption, despite of lack of muscle activity, and it remains elevated for 2-3 hours (depending on the structure of food and other factors). Though the thermogenic effect of food is a known phenomenon and has been studied since the end of the 19th century, there is still no single opinion about its reasons and occurrence mechanisms. The simplest explanation, which states that extra energy production is required to activate a motor function of the gastrointestinal tract, has not been borne out by experiments: the thermogenic effect of glucose cannot be recovered in patients with diabetes and lab animals, even though carbohydrate is absorbed to the

Today the most probable reason considered for the thermogenic effect of food is the effect of enterohormones produced by the duodenum epithelium. For example, it has been proven that the lack of these hormones in blood results in a lower body temperature, meaning heat production (Ugolev et al, 1976). But even in this case it is unclear which particular tissue accounts for extra heat production. Recently brown adipose tissue has been considered as the reason (Himms-Hagen, 1989; Nedergaard & Cannon, 2010), which, according to recent data, is preserved in adults (Nedergaard et al., 2007) and maintains substrate homeostasis

A unique systematic research of age changes in the thermogenic effect of food substance (glucose) in school children was made in the laboratory of I.A.Kornienko (Kornienko et al, 1984). A standard test was used to evaluate glucose tolerance during this study: glucose was taken orally on an empty stomach in quantity proportional to mass (1g/kg). Content of glucose in blood in such a probe is usually increased during 30 minutes after intake and gradually normalizes within 2-3 hours. As glucose increases, with some delay (about 0.5 hours) oxidation processes start in the body, and in 3 hours the oxygen consumption levels return to the primary level. Total intensification of energy production with age for children

The given data proves that reactivity of oxygen metabolism decreases with age, meaning efficiency of mechanisms providing homeostasis increases. The difference in age dynamics of thermogenic glucose effect for boys and girls is the most interesting phenomenon. It is known that adult women, on average, have a better tolerance to glucose than men (Korkushko &

Age, years Boys Girls

Table 1. Total thermogenic effect of glucose (per oral 1 g/kg) for school children for 3 hours

In the same research (Son'kin et al., 1975) it was proved that glucose put into the body depends considerably on their body constitution: children of 11-12 years with a low fat content in the body of no less 1/3 introduced glucose is oxidized in process of a thermogenic response to its putting into the body, while children with a high fat content a thermogenic response to input of glucose is considerably less. Similar results for adult

7 – 8 2.125 + 0.16 (n=18) 1.825 + 0.14 (n=23) 11 – 12 1.255 + 0.10 (n=22) 1.365 + 0.11 (n=21) 15 – 17 0.585 + 0.06 (n=16) 1.060 + 0.08 (n=14)

of 7-8 up to 15-17 years is considerably reduced, especially for boys (Table 1).

Orlov, 1974).

(Son'kin et al., 2010).

#### **3.2 Functional range**

Various body tissues can change their metabolic activity to a different degree, ranging from rest to maximum functional activity. It depends on the organization of metabolic paths in cells that form the corresponding tissue. Based on content and activity data of vital energy ferments in body tissues, Demin (1983) calculated the hypothetic values of minimum and maximum metabolic activity for the liver, brain and muscles of a young man (Fig. 4). Characteristics of the functional range in skeletal muscles obtained using this method are close to the actual measured maximum energy expenditure (Son'kin, 1990; Kornienko, et al, 2000). As seen in fig. 3, muscles have both the highest potential for metabolic activity and the widest functional range. It should be noted that the maximum level of functional activity is carried out through the anaerobic metabolic processes, not limited by the possibilities of mitochondrial oxidation. The functional range value of various tissues can be affected differently depending on age. According to Demin (1983), this value for the brain is at its peak in childhood; for the liver it stays more or less the same at all periods of postnatal ontogenesis; for skeletal muscles it considerably increases from birth to the end of puberty.

Fig. 4. Comparative characteristic of functional range in various tissues of human body (calculation for a young man) (After: Demin, 1985; modified)

The intensity of body functions in a child is much higher than of an adult. The rate of children's basal metabolism is 1.5-2 times higher, but the maximum activity level is considerably lower than that of an adult. This results in a smaller functional range, and that makes the child body's existence more stressful.

The high intensity of energy metabolism in children becomes particularly obvious when you look at the fact that the child's body reacts with higher intensity to an impact equal in power, demonstrating a higher lability of autonomic systems and metabolic processes. These differences are well known, when talking about muscle activity (Åstrand, 1952; Kornienko et al, 2000). The fact that similar differences are evident in metabolic reactions to other functional loads, in particular food load, is less known.

Various body tissues can change their metabolic activity to a different degree, ranging from rest to maximum functional activity. It depends on the organization of metabolic paths in cells that form the corresponding tissue. Based on content and activity data of vital energy ferments in body tissues, Demin (1983) calculated the hypothetic values of minimum and maximum metabolic activity for the liver, brain and muscles of a young man (Fig. 4). Characteristics of the functional range in skeletal muscles obtained using this method are close to the actual measured maximum energy expenditure (Son'kin, 1990; Kornienko, et al, 2000). As seen in fig. 3, muscles have both the highest potential for metabolic activity and the widest functional range. It should be noted that the maximum level of functional activity is carried out through the anaerobic metabolic processes, not limited by the possibilities of mitochondrial oxidation. The functional range value of various tissues can be affected differently depending on age. According to Demin (1983), this value for the brain is at its peak in childhood; for the liver it stays more or less the same at all periods of postnatal ontogenesis; for skeletal muscles it considerably increases

Fig. 4. Comparative characteristic of functional range in various tissues of human body

The intensity of body functions in a child is much higher than of an adult. The rate of children's basal metabolism is 1.5-2 times higher, but the maximum activity level is considerably lower than that of an adult. This results in a smaller functional range, and that

The high intensity of energy metabolism in children becomes particularly obvious when you look at the fact that the child's body reacts with higher intensity to an impact equal in power, demonstrating a higher lability of autonomic systems and metabolic processes. These differences are well known, when talking about muscle activity (Åstrand, 1952; Kornienko et al, 2000). The fact that similar differences are evident in metabolic reactions to

(calculation for a young man) (After: Demin, 1985; modified)

other functional loads, in particular food load, is less known.

makes the child body's existence more stressful.

**3.2 Functional range**

from birth to the end of puberty.

#### **3.3 Specific – Dynamic (thermogenic) effect of food**

The rate of heat production is considerably increased after food consumption, despite of lack of muscle activity, and it remains elevated for 2-3 hours (depending on the structure of food and other factors). Though the thermogenic effect of food is a known phenomenon and has been studied since the end of the 19th century, there is still no single opinion about its reasons and occurrence mechanisms. The simplest explanation, which states that extra energy production is required to activate a motor function of the gastrointestinal tract, has not been borne out by experiments: the thermogenic effect of glucose cannot be recovered in patients with diabetes and lab animals, even though carbohydrate is absorbed to the bloodstream and is extracted with urine (Lusk, 1919).

Today the most probable reason considered for the thermogenic effect of food is the effect of enterohormones produced by the duodenum epithelium. For example, it has been proven that the lack of these hormones in blood results in a lower body temperature, meaning heat production (Ugolev et al, 1976). But even in this case it is unclear which particular tissue accounts for extra heat production. Recently brown adipose tissue has been considered as the reason (Himms-Hagen, 1989; Nedergaard & Cannon, 2010), which, according to recent data, is preserved in adults (Nedergaard et al., 2007) and maintains substrate homeostasis (Son'kin et al., 2010).

A unique systematic research of age changes in the thermogenic effect of food substance (glucose) in school children was made in the laboratory of I.A.Kornienko (Kornienko et al, 1984). A standard test was used to evaluate glucose tolerance during this study: glucose was taken orally on an empty stomach in quantity proportional to mass (1g/kg). Content of glucose in blood in such a probe is usually increased during 30 minutes after intake and gradually normalizes within 2-3 hours. As glucose increases, with some delay (about 0.5 hours) oxidation processes start in the body, and in 3 hours the oxygen consumption levels return to the primary level. Total intensification of energy production with age for children of 7-8 up to 15-17 years is considerably reduced, especially for boys (Table 1).

The given data proves that reactivity of oxygen metabolism decreases with age, meaning efficiency of mechanisms providing homeostasis increases. The difference in age dynamics of thermogenic glucose effect for boys and girls is the most interesting phenomenon. It is known that adult women, on average, have a better tolerance to glucose than men (Korkushko & Orlov, 1974). Possibly, the given data reflects the formation of such sex differences.

Table 1. Total thermogenic effect of glucose (per oral 1 g/kg) for school children for 3 hours of observation (kcal/kg, M±m) (After: Kornienko et al., 1984)

In the same research (Son'kin et al., 1975) it was proved that glucose put into the body depends considerably on their body constitution: children of 11-12 years with a low fat content in the body of no less 1/3 introduced glucose is oxidized in process of a thermogenic response to its putting into the body, while children with a high fat content a thermogenic response to input of glucose is considerably less. Similar results for adult persons are described in press of the last years (Nedergaard & Cannon, 2010).

Energy Metabolism in Children and Adolescents 129

the child's body constitution, therefore it is activated in the last turn, if temperature of the

High activity of special mechanisms of thermogenesis in infants is connected not only with small size and large relative surface increasing heat insulation, not only with low heat insulation of cutaneous coverings, but with a relatively low level of basal metabolism, which has been noted before in this paper. Within the first year of life all these parameters are changing and the chemical thermoregulation activity is reduced. For a child of 5—6 months the importance of physical thermoregulation is considerably increased; it makes the temperature threshold and latent period of an interscapular brown adipose tissue activation almost double compared with the same parameters for infants 1-2 months old (Gohblit et al,

Under usual conditions the child older than 3 years old has a high value of heat flow in relation to the body surface unit, and heat insulation of cutaneous coverings is low, therefore children's skin is practically always warm. Even at the age of 4.5—5 years for girls and 5.5—6 years for boys the body heat insulation is very low: 0.226 + 0.003 °C • m2 • hour/kcal, not changed a lot compared to infants. Their mechanisms of physical thermoregulation are poorly developed. Therefore, if such a child is in conditions of room temperature (+ 20°С) in underwear and T-shirt, in 80 cases of 100 his thermoregulatory heat

Intensification of growth processes at the age of 5-7 years results in accelerating the length and surface area of extremities, providing a regulated heat exchange of the body with the environment. It is, in turn, results in the fact that from the age of 5.5— 6 years (it is especially visible for girls) the thermoregulation function is considerably changed. The body heat insulation is increased, and the chemical thermoregulation activity is substantially reduced. This method of body temperature regulation is more efficient and it becomes predominant in further development with age. In girls this transformation of

At the age of 10 years for girls and 11 years for boys quicker growth processes and considerably lower rate of basal metabolism which are typical for them are observed again. According to thermoregulation conditions, this age can be marked out as a crucial period: physical thermoregulation is activated again, with chemical thermoregulation becoming less

The next stage of thermoregulation development is during pubescence, becoming apparent in the frustration of the forming functional system. For 11—12-year old girls and 13-year old boys, despite the continuous decrease in the resting metabolism rate, there is no corresponding adjustment of vascular regulation. Worse heat insulation facilities of covering tissues result in the fact that, notwithstanding the age tendency, the critical temperature shifts to higher values with the temporarily growing role of chemical thermoregulation most teenagers (up to 80%) enlarge their heat production even under slight cooling

Distinct sex differences in dynamics of thermoregulation development are seen during pubescence (Kornienko & Gohblit, 1988). Parallel to decrease in basal metabolism, for girls after the second stage of pubescence (according to Tanner), heat insulation properties are rapidly increased, and the function of physical regulation is restored in full. By the age of 16 this process is usually over, and all thermoregulation parameters reach the values typical for adults. The same tendencies exist in boys, but by the age of 16 years, processes forming

important. For boys these changes are distinctly expressed at the age of 12 years.

body core falls down despite the processes (Kornienko, 1979).

1975; Kornienko, 1979).

conditions effects.

production is activated (Kornienko, 1979).

thermoregulation happens, on average, one year earlier than boys.

Results obtained in such studies as well as other data about thermogenic effect of food cast doubt on the validity of widely used calculations of caloric food value. All such calculations do not take into account energy expenditure on digestion of food substances which are known to take from 1/5 up to 1/3 caloric value of the taken food substance. The problem is complicated by the fact that fats and proteins have a greater thermogenic effect than carbohydrates (Kassirsky, 1934), while mixed products have a smaller thermogenic effect than the total thermogenic effects of food substances they contain (Forbes & Swift, 1944). It is proven that liquid food, with similar calorie value has a less specifically dynamic effect than solids (Habas & Macdonald, 1998). We think that mechanical calculations of food caloricity based on the caloric equivalent of proteins, fats and carbohydrates in it which is widely used in clinic and health-improving systems, including paediatric practice (Morgan, 1980; Young et al. 1991; Schmelzle et al., 2004) need to be corrected.

#### **3.4 Thermoregulation development in ontogenesis**

Thermoregulation, support of constant temperature in the body core is determined by two basic processes: heat production and heat dissipation. Heat production (thermogenesis) depends primarily on the rate of metabolic processes, while heat dissipation is defined by heat insulation provided by cutaneous coverings, vascular reactions, active outer respiration and perspiration. Because of this, thermogenesis is considered a mechanism of chemical thermoregulation, and heat dissipation regulation – a mechanism of physical thermoregulation. Both these processes change with age, as well as their role in providing a constant body temperature.

As a result of laws of physics, increase in mass and body absolute dimensions reduces the contribution of chemical thermoregulation. Thus, the value of thermoregulation heat production for newborn children makes about 0.5 kcal/kg • hour • °C, and for adults – 0.15 kcal/kg• hour•°C.

A newborn child, if temperature of the environment lowers, can enlarge heat production to adult levels - to 4 kcal/kg•hour. But because of lower heat insulation (0.15 °C•m2•hour/kcal) the chemical thermoregulation range of a newborn is small—no more than 5°.

At that it should be accounted that the critical temperature level (Th), switching thermogenesis for a healthy newborn is 33°C, by the adult period it falls down to 27—23°C. But in clothes with heat insulation usually making 2.5 CLO, or 0.45 °C •m2•hour/kcal, Th value falls down to 20°C, therefore a child in his usual clothes at room temperature is in a thermoneutral environment, meaning that in these conditions a child requires no extra expenditure to support body temperature.

If the temperature falls down below threshold values (for instance, during the change of a child's clothes), mechanisms of extra heat production switch on. For a child they are mainly, "nonshivering thermogenesis", localized in metabolically active tissues – liver and brown adipose tissue (Brück, 1970; Kornienko, 1979). Researches of the latest years have revealed that an acute short-term cooling of adults also results in activation of nonshivering thermogenesis in brown adipose tissue (Nedergaard et al., 2007; Son'kin et al., 2010), which is proved to be preserved for most adults residing in a moderate climatic zone (Nedergaard & Cannon, 2010). Another mechanism of thermogenesis is a cold-induced muscle tremor which is usually observed in adults when the cooling effect is strengthened or prolonged. For children this physiological mechanism turns out inefficient due to particular features of

Results obtained in such studies as well as other data about thermogenic effect of food cast doubt on the validity of widely used calculations of caloric food value. All such calculations do not take into account energy expenditure on digestion of food substances which are known to take from 1/5 up to 1/3 caloric value of the taken food substance. The problem is complicated by the fact that fats and proteins have a greater thermogenic effect than carbohydrates (Kassirsky, 1934), while mixed products have a smaller thermogenic effect than the total thermogenic effects of food substances they contain (Forbes & Swift, 1944). It is proven that liquid food, with similar calorie value has a less specifically dynamic effect than solids (Habas & Macdonald, 1998). We think that mechanical calculations of food caloricity based on the caloric equivalent of proteins, fats and carbohydrates in it which is widely used in clinic and health-improving systems, including paediatric practice (Morgan,

Thermoregulation, support of constant temperature in the body core is determined by two basic processes: heat production and heat dissipation. Heat production (thermogenesis) depends primarily on the rate of metabolic processes, while heat dissipation is defined by heat insulation provided by cutaneous coverings, vascular reactions, active outer respiration and perspiration. Because of this, thermogenesis is considered a mechanism of chemical thermoregulation, and heat dissipation regulation – a mechanism of physical thermoregulation. Both these processes change with age, as well as their role in providing a

As a result of laws of physics, increase in mass and body absolute dimensions reduces the contribution of chemical thermoregulation. Thus, the value of thermoregulation heat production for newborn children makes about 0.5 kcal/kg • hour • °C, and for adults – 0.15

A newborn child, if temperature of the environment lowers, can enlarge heat production to adult levels - to 4 kcal/kg•hour. But because of lower heat insulation (0.15 °C•m2•hour/kcal) the chemical thermoregulation range of a newborn is small—no more

At that it should be accounted that the critical temperature level (Th), switching thermogenesis for a healthy newborn is 33°C, by the adult period it falls down to 27—23°C. But in clothes with heat insulation usually making 2.5 CLO, or 0.45 °C •m2•hour/kcal, Th value falls down to 20°C, therefore a child in his usual clothes at room temperature is in a thermoneutral environment, meaning that in these conditions a child requires no extra

If the temperature falls down below threshold values (for instance, during the change of a child's clothes), mechanisms of extra heat production switch on. For a child they are mainly, "nonshivering thermogenesis", localized in metabolically active tissues – liver and brown adipose tissue (Brück, 1970; Kornienko, 1979). Researches of the latest years have revealed that an acute short-term cooling of adults also results in activation of nonshivering thermogenesis in brown adipose tissue (Nedergaard et al., 2007; Son'kin et al., 2010), which is proved to be preserved for most adults residing in a moderate climatic zone (Nedergaard & Cannon, 2010). Another mechanism of thermogenesis is a cold-induced muscle tremor which is usually observed in adults when the cooling effect is strengthened or prolonged. For children this physiological mechanism turns out inefficient due to particular features of

1980; Young et al. 1991; Schmelzle et al., 2004) need to be corrected.

**3.4 Thermoregulation development in ontogenesis**

constant body temperature.

expenditure to support body temperature.

kcal/kg• hour•°C.

than 5°.

the child's body constitution, therefore it is activated in the last turn, if temperature of the body core falls down despite the processes (Kornienko, 1979).

High activity of special mechanisms of thermogenesis in infants is connected not only with small size and large relative surface increasing heat insulation, not only with low heat insulation of cutaneous coverings, but with a relatively low level of basal metabolism, which has been noted before in this paper. Within the first year of life all these parameters are changing and the chemical thermoregulation activity is reduced. For a child of 5—6 months the importance of physical thermoregulation is considerably increased; it makes the temperature threshold and latent period of an interscapular brown adipose tissue activation almost double compared with the same parameters for infants 1-2 months old (Gohblit et al, 1975; Kornienko, 1979).

Under usual conditions the child older than 3 years old has a high value of heat flow in relation to the body surface unit, and heat insulation of cutaneous coverings is low, therefore children's skin is practically always warm. Even at the age of 4.5—5 years for girls and 5.5—6 years for boys the body heat insulation is very low: 0.226 + 0.003 °C • m2 • hour/kcal, not changed a lot compared to infants. Their mechanisms of physical thermoregulation are poorly developed. Therefore, if such a child is in conditions of room temperature (+ 20°С) in underwear and T-shirt, in 80 cases of 100 his thermoregulatory heat production is activated (Kornienko, 1979).

Intensification of growth processes at the age of 5-7 years results in accelerating the length and surface area of extremities, providing a regulated heat exchange of the body with the environment. It is, in turn, results in the fact that from the age of 5.5— 6 years (it is especially visible for girls) the thermoregulation function is considerably changed. The body heat insulation is increased, and the chemical thermoregulation activity is substantially reduced. This method of body temperature regulation is more efficient and it becomes predominant in further development with age. In girls this transformation of thermoregulation happens, on average, one year earlier than boys.

At the age of 10 years for girls and 11 years for boys quicker growth processes and considerably lower rate of basal metabolism which are typical for them are observed again. According to thermoregulation conditions, this age can be marked out as a crucial period: physical thermoregulation is activated again, with chemical thermoregulation becoming less important. For boys these changes are distinctly expressed at the age of 12 years.

The next stage of thermoregulation development is during pubescence, becoming apparent in the frustration of the forming functional system. For 11—12-year old girls and 13-year old boys, despite the continuous decrease in the resting metabolism rate, there is no corresponding adjustment of vascular regulation. Worse heat insulation facilities of covering tissues result in the fact that, notwithstanding the age tendency, the critical temperature shifts to higher values with the temporarily growing role of chemical thermoregulation most teenagers (up to 80%) enlarge their heat production even under slight cooling conditions effects.

Distinct sex differences in dynamics of thermoregulation development are seen during pubescence (Kornienko & Gohblit, 1988). Parallel to decrease in basal metabolism, for girls after the second stage of pubescence (according to Tanner), heat insulation properties are rapidly increased, and the function of physical regulation is restored in full. By the age of 16 this process is usually over, and all thermoregulation parameters reach the values typical for adults. The same tendencies exist in boys, but by the age of 16 years, processes forming

Energy Metabolism in Children and Adolescents 131

(MOC) rises proportionally to the muscle mass growth (Kornienko et al., 2000), but it lacks information about qualitative changes in muscle energy supply of children and teenagers

Newborn 0,9 0,6 - - 5 – 7 4,6 6,7 1,01 151 9 5,2 10,5 1,8 171 11 6,6 11,6 2,1 181 14 4,8 21,2 3,5 165 20 4,5 25,0 4,1 164 36-40 3,7 28,3 3,9 138 Table 2. Age changes in cytochrome **a** content in thigh muscle and human aerobic capacity

Age dynamics of cytochrome **a** content – a terminal site of the oxidative chain – in human skeletal muscles (Kornienko, 1979; Demin et al., 1987) is given in Table 2. Calculations of the estimated value for specific MOC (per 1 kg of somatic muscle mass) are shown. As obvious from the given data, the highest cytochrome **a** concentration is registered in skeletal muscles for boys 9-11 years old. It is also proved by data of electron microscopic researches (Kornienko, 1979; Kornienko et al, 1987): the number of mitochondria in relation to the area of myofibrils for 11-year old boy is considerably more than in an adult man (Table 3). It is remarkable, that, according to data of morphologists, a capillary network in the muscles of extremities turns out to be more developed in children of 9-11 years (Topol'sky, 1951), which is the age when there is the highest content of oxidizing ferments in the muscle tissue. Thus, an age development of the energy production aerobic source in skeletal muscles does not happen monotonously, but gets the expressed maximum during prepubescence (Fig.5). These conditions have a

Muscle mass, kg (after: Holliday, 1971)

MOC, l/min (after: Åstrand, 1952)

MOC, ml/kg muscle mass

(Son'kin & Tambovtseva, 2011).

(After: Kornienko, 1979)

Age, years Cytochrome **а**

concentration, nM/g raw mass of muscle tissue

Fig. 5. Dynamics of aerobic capacity indices in schoolboys

considerable effect on the functioning of the muscle energy supply system.

mechanisms of physical thermoregulation are incomplete. Only in youths after pubescence do thermoregulation facilities reach their final level. Increase in tissue heat insulation to the level of 1.1 CLO allows to function without activating the chemical thermoregulation (meaning extra heat production) even when the environment temperature falls down by 10-15 degrees below thermoneutral. Such body reaction is naturally more economical and efficient.

The given data prove that in the process of postnatal ontogenesis the primary line of the system development providing temperature homeostasis is indirect (Falk, 1998). At each stage of individual organism development there is a complex dependence of thermoregulation active mechanisms on growth and development, the rate of metabolic processes and conditions of some autonomic functions. It is this dependence that determines a primary activity of physical or chemical thermoregulation mechanisms, providing temperature homeostasis at the corresponding stage of development.


**9.7 Quota sampling**

Quota sampling is a non-random sampling method in which participants are selected based on predetermined characteristics such that the total sample will have the same distribution of characteristics as the larger population. Selection of participants who meet certain characteristics, such as gender, age, fitness, etc. Scientists will also go back over the preferred sample population to guarantee that subgroups are represented to the same degree as those in the wider population.


**6.5 Calculation of properties**

To find elastic constants of bone, fundamental elasticity equations were used assuming transverse isotropy of bone. Measurement of longitudinal and lateral

### **Figure 2.**

*Test sample, cubes of water Buffalo female (Bubalusbubalis), cut out of test specimen.*

*Mechanical Properties and Elasticity Model for Bovine Hard Tissue DOI: http://dx.doi.org/10.5772/intechopen.98410*

deformations facilitates the calculation of strains in respective directions. Stresses are evaluated from loads applied.

Sample calculation of Goat Male

Orthotropic material stiffness matrix constant of Goat Male (*Capra aegagrushircus*)

$$C\_{ijkl} = \begin{cases} 9.08 & \mathbf{1.81} & \mathbf{1.85} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{1.81} & 9.08 & \mathbf{1.85} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{1.85} & \mathbf{1.85} & \mathbf{13.4} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{3.57} & \mathbf{0} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{3.57} & \mathbf{0} \\\\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{3.6} \end{cases} \tag{1}$$

Orthotropic material stiffness matrix constant of Water Buffalo Female (*Bubalusbubalis*) (**Table 1**)

### **Table 1.** *Mechanical properties derived from testing.*

$$C\_{ijkl} = \begin{cases} 5.19 & 1.12 & 1.2 & 0 & 0 & 0 \\ 1.12 & 5.19 & 1.2 & 0 & 0 & 0 \\ 1.2 & 1.2 & 8.28 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1.09 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1.09 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2 \end{cases} \tag{2}$$

Orthotropic material stiffness matrix constant of Water buffalo Male (*Bubalusbubalis*)

$$\mathbf{C}\_{ijkl} = \begin{Bmatrix} 6.06 & 1.27 & 1.49 & 0 & 0 & 0 \\ 1.27 & 6.06 & 1.49 & 0 & 0 & 0 \\ 1.49 & 1.49 & 12.56 & 0 & 0 & 0 \\ 0 & 0 & 0 & 2.33 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2.33 & 0 \\ 0 & 0 & 0 & 0 & 0 & 2.4 \end{Bmatrix} \tag{3}$$

## **7. Discussion**

This experimental study shows that Young's Modulus and Poisson's Ratio can be determined from load–displacement relation. When tested using UTM, All bone samples show much higher modulus in the axial direction (Ey) than the other two transverse directions, this is due to the transversely isotropic behavior of bone. This is attributed to the parallel orientation of grains along the longitudinal direction. In the other two transverse directions, Modulus was nearly the same. This is consistent with the values obtained by other researchers, Reilly and Burstein [15], Kulkarni and Sathe [16], R. Shahar [17]. Hence orthotropic nature of the material is exhibited by long bone. It is seen that the values of compression strain, Young's modulus, Poisson's ratio, and shear modulus are higher for Water Buffalo Male (Bubalusbubalis) than that of female showing gender difference. This may be attributed to lower bone density in females due to hormone secretion.

The results obtained here for Poisson's ratios of femur bone fall within a narrow range. The values found on our experiments for goat male 0.17 to 0.19, for Water Buffalo Female (Bubalusbubalis) 0.19 to 0.22, and for Water Buffalo Male (Bubalusbubalis) 0.22 to 0.23. These values are in good agreement with 0.20 to 0.22 reported by Kulkarni and Sathe [16], however, the range is different than 0.12 to0.63 reported by Pithioux et al. [18].

The values of compression strength of femur bones tested in this study show variation in three groups of samples. The compressive strength of the Goat male (Capra aegagrushircus) femur bone ranges from 92 MPa to 100 MPa in the longitudinal direction. The compression strength of the Goat Male (Capra aegagrushircus) femur bone reported in this work is 97 MPa. A previous study reported a similar value of compressive strength of femur bone in Ovis (sheep) 90 Mpa, Erickson et al. [19]. The compressive strength of Goat male (Capra aegagrushircus) femur bone ranges from 40 MPa - 52 MPa in the transverse direction.

Considering the above results following conclusions may be drawn:

1.Diameter of the diaphysis of Water Buffalo Female (Bubalusbubalis) femur bone was seen to be greater than Water Buffalo Male (Bubalusbubalis) femur bone but the stiffness constant shows the higher valves in Water Buffalo Male (Bubalusbubalis) than Water Buffalo Female (Bubalusbubalis) because in Water Buffalo Male (Bubalusbubalis) femur bone grains are closely spaced that is more compact cortical bone and less spongy but in Water Buffalo Female (Bubalusbubalis) femur bone grains are relatively sparsely spaced i.e. It also exhibits more cancellous bone and more spongy nature as compared to male bone sample, this difference is attributed may be due to hormonal effect.