Numerical and Chaotic Analysis of Proposed SIR Model

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1. Introduction

The continuous dynamical system is very rumor in several of applied sciences and engineering also computational mathematics [1] . in this paper, we will build a continuous dynamic system consisting of the classic three-dimensional SIR model, which is a model for studying and analyzing epidemic diseases [2] [3] [4] [5] and the known numerical method (Runge-Kutta) [5] [6] , with the fourth order [6] , sixth order [7] [8] and the seventh order [9] , where the model SIR is used with the 4^{th} order, 6^{th} order of the method (Runge-Kutta) Once and with the 7^{th} rank of the same method again to generate simulated data (hypothetical) by taking initial values (initial conditions) from the real data of the daily statistics of the epidemic disease (COVID-19) for a specific country of the world and using those resulting data for each rank in the chaos test of the outbreak disease By using the binary test method (0-1), which is one of the chaotic testing methods for dynamic systems, which has the advantage that it depends mainly on observations of time series data, and the test result is either close to zero (which is the regularity state) or close to one (which is the chaotic state of the system) [10] - [16] , The disease dynamic lamentation has been shown to be chaotic (Kcorr ≅ 1). All figures and drawings showing the behavior of the chaotic system have been included. Also, programs have been built in Matlab system to apply to all the aforementioned operations (from finding simulation data, testing chaos, etc.). A program is also build to indicate which of the three ranks is better to study the dynamic system in terms of finding the step size, the maximum error, the time limit for each step, and the total time it takes to solve (in seconds). The results of this mathematical process have been shown in a table. At the end, a summary is listed containing all work results.

2. SIR Model

It’s an epidemiologic mathematical model that does computing the theoretic numbers of individuals infected with a contagious disease in a closed population over times. This classic model his name is derived from the fact that they involve coupled equations relating the number of (susceptible (S), infected (I) and recovered (R)). SIR model is one of the simplest models that developed by (Kermack-Mckendrick) in (1927).

Description of SIR model:

$\begin{array}{l}\stackrel{\dot{}}{S}=\frac{\text{d}S}{\text{d}t}=-\beta SI\\ \stackrel{\dot{}}{I}=\frac{\text{d}I}{\text{d}t}=\beta SI-\gamma I\\ \stackrel{\dot{}}{R}=\frac{\text{d}R}{\text{d}t}=\gamma I\end{array}\}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{non-linearSystem}$ (1)

$N=S+I+R$ , $\stackrel{\dot{}}{S}+\stackrel{\dot{}}{I}+\stackrel{\dot{}}{R}=0$

where:

N: The total number of population.

S(t): The number of Susceptible people at time (t).

I(t): The number of Infected people at time (t).

R(t): The number of Recovered people at time (t).

$\begin{array}{l}\left(\beta =0.0845\right):\text{isatransmissionrate}\\ \left(\gamma =0.07\right):\text{isrecoveredrate}\end{array}\}\text{areconstantparametersofthemodelinIraq}$

where: $\beta =\mu +\gamma $ , $\gamma =1/D$ , (μ = 0.0145): is mortality rate in day, D = 14: is duration of disease time.

3. Basic Reproduction Number (R_{0})

Definition(R_{0}): Represent to the average number of new infections generated by each infected person, the high value of (R_{0}) means easy to transmit the disease, and the low value of (R_{0}) means difficult to transmit the disease. (R_{0}) is called threshold of disease, (the value of (R_{0}) assumes that no pre-existing immunity, i.e. it mean everyone is susceptible), where Ro = β/γ.

Lemma: If R_{0} > 1 then I(t) is increasing and the disease is epidemic, and if R_{0} < 1 then I(t) is decreasing and the disease is endemic, It is assumed in the absence of a vaccine, the entire population will be susceptible to infection, meaning that S ≅ N, so we divide β by N.

Proof: From the SIR model we have: $\text{d}I/\text{d}t=\left(\beta /N\right)\ast SI-\gamma I$ $\to $ $\text{d}I/\text{d}t=\left(\beta -\gamma \right)I$ $\to $ $\text{d}I/I=\left(\beta -\gamma \right)\text{d}t$ by integral of two hand sides: $\mathrm{ln}I=\left(\beta I-\gamma I\right)+C$ $\to $ $I\left(t\right)={\text{e}}^{\beta T\left(t\right)-\gamma I\left(t\right)}\cdot {\text{e}}^{C}$ , when $t=0$ $\to $ $I\left(0\right)={\text{e}}^{C}$ then $I\left(t\right)={\text{e}}^{\beta T\left(t\right)-\gamma I\left(t\right)}\cdot I\left(0\right)>0$ when $\text{d}I/\text{d}t>0$ then $\beta I-\gamma I>0$ $\beta I>\gamma I$ $\to $ $\beta /\gamma >1$ $\to $ ${R}_{0}>1$ .

4. The Numerical Method “Runge-Kutta”

To solve the differential equations, we will use the following Relationships:

$\text{d}{x}_{1}/\text{d}t={g}_{1}\left(t,{x}_{1},{x}_{2},\cdots ,{x}_{n}\right)$

$\text{d}{x}_{2}/\text{d}t={g}_{2}\left(t,{x}_{1},{x}_{2},\cdots ,{x}_{n}\right)$

$\vdots $

$\text{d}{x}_{n}/\text{d}t={g}_{n}\left(t,{x}_{1},{x}_{2},\cdots ,{x}_{n}\right)$

For more simply we write: $\text{d}y/\text{d}t=g\left(t,x\right)$

Where (x) is a vector for n-dimensions, this is not an independent equation. If we replace the right-hand side by g(x) we will get on an independent equation.

If the function (f) in the R.H. side is nonlinear, we will need numerical methods. The Runge-Kutta methods have the same precisions in solution as the Taylor expansions in any order, but there is no need for derivatives. We compute (x) at X_{ti}_{+1} = X_{ti} + h.

4.1. Runge-Kutta Method with 4^{th} Order

The 4^{th} order Runge-Kutta numerical method is one of the classic numerical methods used to solve the ordinary differential equations of nonlinear and time-related continuous dynamic systems with a number of iterations to obtain the best approximate value.

Description Method:

y_{n+1} = y_{n}+ h * (k1 + 2k2 + 2k3 + k4)/6.

where: Δt = h = t_{n+1} − t_{n} and:

k1 = f (t_{n}, y_{n});

k2 = f (t_{n} + h/2, y_{n} + h * k1/2);

k3 = f (t_{n} + h/2, y_{n} + h * k2/2);

k4 = f (t_{n} + h, y_{n} + h * k3).

From system (1) we have: dS/dt = f1, dI/dt = f2, dR/dt = f3, (So, Io, Ro) = (3 × 10^{6}, 100, 0) initial condition, to = 1,

K1 = h * f1(to, So, Io, Ro),

L1 = h * f2(to, So, Io, Ro),

M1 = h * f3(to, So, Io, Ro);

K2 = h * f1(to + h/2, So + K1/2, Io + L1/2, Ro + M1/2),

L2 = h * f2 (to + h/2, So + K1/2, Io + L1/2, Ro + M1/2),

M2 = h * f3(to + h/2, So + K1/2, Io + L1/2, Ro + M1/2);

K3 = h * f1(to + h/2, So + K2/2, Io + L2/2, Ro + M2/2),

L3 = h * f2(to + h/2, So + K2/2, Io + L2/2, Ro + M2/2),

M3 = h * f3(to + h/2, So + K2/2, Io + L2/2, Ro + M2/2);

K4 = h * f1(to + h, So + K3, Io + L3, Ro + M3),

L4 = h * f2(to + h, So + K3, Io + L3, Ro + M3),

M4 = h * f3(to + h, So + K3, Io + L3, Ro + M3);

S_{1} = So + h/6 × (K1 + 2K2 + 2K3 + K4),

I_{1} = Io + h/6 × (L1 + 2L2 + 2L3 + L4),

R_{1} = Ro + h/6 × (M1 + 2M2 + 2M3 + M4).

The process is repeated with (n) iterations by using Matlab program.

4.2. The Numerical Method “Runge-Kutta” of 6th Order

k1 = h * f(t(i); x(i), y(i), z(i)),

l1 = h * g(t(i), x(i), y(i), z(i)),

m1 = h * p(t(i), x(i), y(i), z(i));

k2 = h * f(x(i) + k1/3, y(i) + l1/3, z(i) + m1/3),

l2 = h * g(x(i) + k1/3, y(i) + l1/3, z(i) + m1/3),

m2 = h * p(x(i) + k1/3, y(i) + l1/3, z(i) + m1/3);

k3 = h * f(x(i) + 2 * k2/3, y(i) + 2 * l2/3, z(i) + 2 * m2/3),

l3 = h * g(x(i) + 2 * k2/3, y(i) + 2 * l2/3, z(i) + 2 * m2/3),

m3 = h * p(x(i) + 2 * k2/3, y(i) + 2 * l2/3, z(i) + 2 * m2/3);

k4 = h * f(x(i) + k1/12 + k2/3 − k3/12, y(i) + l1/12 + l2/3 − l3/12, z(i) + m1/12 + m2/3 − m3/12),

l4 = h * g(x(i) + k1/12 + k2/3 − k3/12, y(i) + l1/12 + l2/3 − l3/12, z(i) + m1/12 + m2/3 − m3/12),

m4 = h * p(x(i) + k1/12 + k2/3 − k3/12, y(i) + l1/12 + l2/3 − l3/12, z(i) + m1/12 + m2/3 − m3/12) k5 = h * f(x(i) + 25 * k1/48 − 55 * k2/24 + 35 * k3/48 + 15 * k4/8, y(i) + 25 * l1/48 − 55 * l2/24 + 35 * l3/48 + 15 * l4/8, z(i) + 25 * m1/48 − 55 * m2/24 + 35 * m3/48 + 15 * m4/8);

l5 = h * g(x(i) + 25 * k1/48 − 55 * k2/24 + 35 * k3/48 + 15 * k4/8, y(i) + 25 * l1/48 − 55 * l2/24 + 35 * l3/48 + 15 * l4/8, z(i) + 25 * m1/48 − 55 * m2/24 + 35 * m3/48 + 15 * m4/8),

m5 = h * p(x(i) + 25 * k1/48 − 55 * k2/24 + 35 * k3/48 + 15 * k4/8, y(i) + 25 * l1/48 − 55 * l2/24 + 35 * l3/48 + 15 * l4/8, z(i) + 25 * m1/48 − 55 * m2/24 + 35 * m3/48 + 15 * m4/8);

k6 = h * f(x(i) + 3 * k1/20 − 11 * k2/20 − k3/8 + k4/2 + k5/10, y(i) + 3 * l1/20 − 11 * l2/20 − l3/8 + l4/2 + l5/10, z(i) + 3 * m1/20 − 11 * m2/20 − m3/8 + m4/2 + m5/10),

l6 = h * g(x(i) + 3 * k1/20 − 11 * k2/20 − k3/8 + k4/2 + k5/10, y(i) + 3 * l1/20 − 11 * l2/20 − l3/8 + l4/2 + l5/10, z(i) + 3 * m1/20 − 11 * m2/20 − m3/8 + m4/2 + m5/10),

m6 = h * p(x(i) + 3 * k1/20 − 11 * k2/20 − k3/8 + k4/2 + k5/10, y(i) + 3 * l1/20 − 11 * l2/20 − l3/8 + l4/2 + l5/10, z(i) + 3 * m1/20 − 11 * m2/20 − m3/8 + m4/2 + m5/10);

k7 = h * f(x(i) − 261 * k1/260 + 33 * k2/13 + 43 * k3/156 − 118 * k4/39 + 32 * k5/195 + 80 * k6/39, y(i) − 261 * l1/260 + 33 * l2/13 + 43 * l3/156 − 118 * l4/39 + 32 * l5/195 + 80 * l6/39, z(i) − 261 * m1/260 + 33 * m2/13 + 43 * m3/156 − 118 * m4/39 + 32 * m5/195 + 80 * m6/39),

l7 = h * g(x(i) − 261 * k1/260 + 33 * k2/13 + 43 * k3/156 − 118 * k4/39 + 32 * k5/195 + 80 * k6/39, y(i) − 261 * l1/260 + 33 * l2/13 + 43 * l3/156 − 118 * l4/39 + 32 * l5/195 + 80 * l6/39, z(i) − 261 * m1/260 + 33 * m2/13 + 43 * m3/156 − 118 * m4/39 + 32 * m5/195 + 80 * m6/39),

m7 = h * p(x(i) − 261 * k1/260 + 33 * k2/13 + 43 * k3/156 − 118 * k4/39 + 32 * k5/195 + 80 * k6/39, y(i) − 261 * l1/260 + 33 * l2/13 + 43 * l3/156 − 118 * l4/39 + 32 * l5/195 + 80 * l6/39, z(i) − 261 * m1/260 + 33 * m2/13 + 43 * m3/156 − 118 * m4/39 + 32 * m5/195 + 80 * m6/39);

x(i + 1) = x(i) + h * (13 * k1 + 55 * k3 + 55 * k4 + 32 * k5 + 32 * k6 + 13 * k7)/200,

y(i + 1) = y(i) + h * (13 * l1 + 55 * l3 + 55 * l4 + 32 * l5 + 32 * l6 + 13 * l7)/200,

z(i + 1) = z(i) + h * (13 * m1 + 55 * m3 + 55 * m4 + 32 * m5 + 32 * m6 + 13 * m7)/200.

4.3. The Numerical Method “Runge-Kutta” of 7th Order

k1 = h * f(x(i), y(i), z(i)),

l1 = h * g(x(i), y(i), z(i)),

m1 = h * p(x(i), y(i), z(i));

k2 = h * f(t(i) + h/18, x(i) + h/18 * k1, y(i) + h/18 * l1),

l2 = h * g(t(i) + h/18, x(i) + h/18 * k1, y(i) + h/18 * l1),

m2 = h * p(t(i) + h/18, x(i) + h/18 * k1, y(i) + h/18 * l1);

k3 = h * f(t(i) + (3 * h)/36, x(i) + 1/60 * (4 * k1 + k2), y(i) + 1/60 * (4 * l1 + l2)),

l3 = h * g(t(i) + (3 * h)/36, x(i) + 1/60 * (4 * k1 + k2), y(i) + 1/60 * (4 * l1 + l2)),

m3 = h * p(t(i) + (3 * h)/36, x(i) + 1/60 * (4 * k1 + k2), y(i) + 1/60 * (4 * l1 + l2));

k4 = h * f(t(i) + (4 * h)/36, x(i) + 1/180 * (−181 * k1 + 171 * k2 + 130 * k3), y(i) + 1/180 * (−181 * l1 + 171 * l2 + 130 * l3)),

l4 = h * g(t(i) + (4 * h)/36, x(i) + 1/180 * (−181 * k1 + 171 * k2 + 130 * k3), y(i) + 1/180 * (−181 * l1 + 171 * l2 + 130 * l3)),

m4 = h * p(t(i) + (4 * h)/36, x(i) + 1/180 * (−181 * k1 + 171 * k2 + 130 * k3), y(i) + 1/180 * (−181 * l1 + 171 * l2 + 130 * l3));

k5 = h * f(t(i) + (5 * h)/36, x(i) + 1/180 * (−902 * k1 + 293 * k2 − 2040 * k3 + 30 * k4), y(i) + 1/180 * (−902 * l1 + 293 * l2 − 2040 * l3 + 30 * l4)),

l5 = h * g(t(i) + (5 * h)/36, x(i) + 1/180 * (−902 * k1 + 293 * k2 − 2040 * k3 + 30 * k4), y(i) + 1/180 * (−902 * l1 + 293 * l2 − 2040 * l3 + 30 * l4)),

m5 = h * p(t(i) + (5 * h)/36, x(i) + 1/180 * (−902 * k1 + 293 * k2 − 2040 * k3 + 30 * k4), y(i) + 1/180 * (−902 * l1 + 293 * l2 − 2040 * l3 + 30 * l4));

k6 = h * f(t(i) + h/6, x(i) + 1/24 * (−15 * k1 + 48 * k2 + 31 * k3 + k4 + k5), y(i) + 1/24 * (−15 * l1 + 48 * l2 + 31 * l3 + l4 + l5)),

l6 = h * g(t(i) + h/6, x(i) + 1/24 * (−15 * k1 + 48 * k2 + 31 * k3 + k4 + k5), y(i) + 1/24 * (−15 * l1 + 48 * l2 + 31 * l3 + l4 + l5)),

m6 = h * p(t(i) + h/6, x(i) + 1/24 * (−15 * k1 + 48 * k2 + 31 * k3 + k4 + k5), y(i) + 1/24 * (−15 * l1 + 48 * l2 + 31 * l3 + l4 + l5));

k7 = h * f(t(i) + (2 * h)/6, x(i) + 1/30 * (17 * k1 − 48 * k2 + 31 * k3 − k4 − k5 + 12 * k6), y(i) + 1/30 * (17 * l1 − 48 * l2 + 31 * l3 − l4 − l5 + 12 * l6)),

l7 = h * g(t(i) + (2 * h)/6, x(i) + 1/30 * (17 * k1 − 48 * k2 + 31 * k3 − k4 − k5 + 12 * k6), y(i) + 1/30 * (17 * l1 − 48 * l2 + 31 * l3 − l4 − l5 + 12 * l6)),

m7 = h * p(t(i) + (2 * h)/6, x(i) + 1/30 * (17 * k1 − 48 * k2 + 31 * k3 − k4 − k5 + 12 * k6), y(i) + 1/30 * (17 * l1 − 48 * l2 + 31 * l3 − l4 − l5 + 12 * l6));

k8 = h * f(t(i) + (3 * h)/6, x(i) + 1/80 * (192 * k1 − 528 * k2 + 341 * k3 − 11 * k4 − 11 * k5 + 32 * k6 + 25 * k7), y(i) + 1/80 * (192 * l1 − 528 * l2 + 341 * l3 − 11 * l4 − 11 * l5 + 32 * l6 + 25 * l7)),

l8 = h * g(t(i) + (3 * h)/6, x(i) + 1/80 * (192 * k1 − 528 * k2 + 341 * k3 − 11 * k4 − 11 * k5 + 32 * k6 + 25 * k7), y(i) + 1/80 * (192 * l1 − 528 * l2 + 341 * l3 − 11 * l4 − 11 * l5 + 32 * l6 + 25 * l7)),

m8 = h * p(t(i) + (3 * h)/6, x(i) + 1/80 * (192 * k1 − 528 * k2 + 341 * k3 − 11 * k4 − 11 * k5 + 32 * k6 + 25 * k7), y(i) + 1/80 * (192 * l1 − 528 * l2 + 341 * l3 − 11 * l4 − 11 * l5 + 32 * l6 + 25 * l7));

k9 = h * f(t(i) + (4 * h)/6, x(i) + 1/66 * (54 * k1 − 144 * k2 + 93 * k3 − 3 * k4 − 3 * k5 + 32 * k6 − 17 * k7 + 32 * k8), y(i) + 1/66 * (54 * l1 − 144 * l2 + 93 * l3 − 3 * l4 − 3 * l5 + 32 * l6 − 17 * l7 + 32 * l8)),

l9 = h * g(t(i) + (4 * h)/6, x(i) + 1/66 * (54 * k1 − 144 * k2 + 93 * k3 − 3 * k4 − 3 * k5 + 32 * k6 − 17 * k7 + 32 * k8), y(i) + 1/66 * (54 * l1 − 144 * l2 + 93 * l3 − 3 * l4 − 3 * l5 + 32 * l6 − 17 * l7 + 32 * l8)),

m9 = h * p(t(i) + (4 * h)/6, x(i) + 1/66 * (54 * k1 − 144 * k2 + 93 * k3 − 3 * k4 − 3 * k5 + 32 * k6 − 17 * k7 + 32 * k8), y(i) + 1/66 * (54 * l1 − 144 * l2 + 93 * l3 − 3 * l4 − 3 * l5 + 32 * l6 − 17 * l7 + 32 * l8));

k10 = h * f(t(i) + (5 * h)/6, x(i) + 1/3960 * (−22876 * k1 + 64464 * k2 − 41633 * k3 + 1343 * k4 + 1343 * k5 − 656 * k6 − 460 * k7 − 40 * k8 + 1815 * k9), y(i) + 1/3960 * (−22876 * l1 + 64464 * l2 − 41633 * l3 + 1343 * l4 + 1343 * l5 − 656 * l6 − 460 * l7 − 40 * l8 + 1815 * l9)),

l10 = h * g(t(i) + (5 * h)/6, x(i) + 1/3960 * (−22876 * k1 + 64464 * k2 − 41633 * k3 + 1343 * k4 + 1343 * k5 − 656 * k6 − 460 * k7 − 40 * k8 + 1815 * k9), y(i) + 1/3960 * (−22876 * l1 + 64464 * l2 − 41633 * l3 + 1343 * l4 + 1343 * l5 − 656 * l6 − 460 * l7 − 40 * l8 + 1815 * l9)),

m10 = h * p(t(i) + (5 * h)/6, x(i) + 1/3960 * (−22876 * k1 + 64464 * k2 − 41633 * k3 + 1343 * k4 + 1343 * k5 − 656 * k6 − 460 * k7 − 40 * k8 + 1815 * k9), y(i) + 1/3960 * (−22876 * l1 + 64464 * l2 − 41633 * l3 + 1343 * l4 + 1343 * l5 − 656 * l6 − 460 * l7 − 40 * l8 + 1815 * l9));

k11 = h * f(t(i) + h, x(i) + 1/902 * (16139 * k1 − 45120 * k2 + 29140 * k3 − 940 * k4 − 940 * k5 + 1828 * k6 − 769 * k7 + 2752 * k8 − 1980 * k9 + 792 * k10), y(i) + 1/902 * (16139 * l1 − 45120 * l2 + 29140 * l3 − 940 * l4 − 940 * l5 + 1828 * l6 − 769 * l7 + 2752 * l8 − 1980 * l9 + 792 * l10)),

l11 = h * g(t(i) + h, x(i) + 1/902 * (16139 * k1 − 45120 * k2 + 29140 * k3 − 940 * k4 − 940 * k5 + 1828 * k6 − 769 * k7 + 2752 * k8 − 1980 * k9 + 792 * k10), y(i) + 1/902 * (16139 * l1 − 45120 * l2 + 29140 * l3 − 940 * l4 − 940 * l5 + 1828 * l6 − 769 * l7 + 2752 * l8 − 1980 * l9 + 792 * l10)),

m11 = h * p(t(i) + h, x(i) + 1/902 * (16139 * k1 − 45120 * k2 + 29140 * k3 − 940 * k4 − 940 * k5 + 1828 * k6 − 769 * k7 + 2752 * k8 − 1980 * k9 + 792 * k10), y(i) + 1/902 * (16139 * l1 − 45120 * l2 + 29140 * l3 − 940 * l4 − 940 * l5 + 1828 * l6 − 769 * l7 + 2752 * l8 − 1980 * l9 + 792 * l10));

x(i + 1) = x(i) + (41 * k1 + 216 * k6 + 27 * k7 + 272 * k8 + 27 * k9 + 216 * k10 + 41 * k11)/840,

y(i + 1) = y(i) + (41 * l1 + 216 * l6 + 27 * l7 + 272 * l8 + 27 * l9 + 216 * l10 + 41 * l11)/840,

z(i + 1) = z(i) + (41 * m1 + 216 * m6 + 27 * m7 + 272 * m8 + 27 * m9 + 216 * m10 + 41 * m11)/840.

where: xn + 1 = xn + [(41k1 + 216k6 + 27k7 + 272k8 + 27k9 + 216k10 + 41k11)/840]. The results of the three methods are shown in Table 1.

Table 1. Shown the comparison between results of 4^{th}, 6^{th}, and 7^{th} order of R-K numerical method and get more accuracy.

5. Chaotic Analysis

To explain the chaotic state we will take the (0-1) test to know if the system is regular or chaotic.

The Binary Test (0-1)

Consider scalar observable $\varphi \left(k\right)$ :

${P}_{n}={\displaystyle {\sum}_{k=1}^{n}\varphi \left(k\right)\mathrm{cos}\left(kc\right)}$ , ${q}_{n}={\displaystyle {\sum}_{k=1}^{n}\varphi \left(k\right)\mathrm{sin}(\; k\; c\; )}$

where
$c\in \left(0,\pi \right)$ ,
$n=1,2,3,\cdots ,L$ from behavior of P_{n} and q_{n} can be computing the Mean Square Displacement (MSD) = M(n)

$M\left(n\right)={\mathrm{lim}}_{L\to \infty}\left(\frac{1}{L}\right){\displaystyle {\sum}_{k=1}^{L}\left[{\left(P\left(k+n\right)-P\left(k\right)\right)}^{2}+{\left({q}_{n}\left(k+n\right)-{q}_{n}\left(k\right)\right)}^{2}\right]}$

where $n=1,2,\cdots ,L/10$ .

$Vosc\left(n\right)={\left[E\left(\varphi \right)\right]}^{2}\times \frac{1-\mathrm{cos}\left(nc\right)}{1-\mathrm{cos}\left(c\right)}$ ,

where $E\left(\varphi \right)={\mathrm{lim}}_{L\to \infty}\left(\frac{1}{L}\right){\displaystyle {\sum}_{k=1}^{n}\varphi \left(k\right)}$ .

Then $D\left(n\right)=M\left(n\right)-Vosc\left(n\right)$ ,

$Kcorr=Kc={\mathrm{lim}}_{n\to \infty}\frac{\mathrm{log}Mc\left(n\right)}{\mathrm{log}\left(n\right)}$ .

Kc states:

Either the value of K ≅ 0 it is signifying to regular dynamics.

Or the value of K ≅ 1 it is signifying to chaotic dynamics, where $Kc\in \left[0,1\right]$ .

Remark: if the motion is torus then the dynamic system is regular (non-chaotic), and if it behaves like a Brownian motion then the dynamic system is said it chaotic.

the Chaotic of simulated data of the 4^{th} order for Iraq by using Zero-one test shown in Figure 1, the chaotic of simulated data of 6^{th} order for Iraq by using Zero-one test shown in Figure 2 and in Figure 3 shows the chaotic of simulated data of 7^{th} order by Zero-one test for Iraq.

Then the behavior of disease is chaotic (in Iraq). The results of the three methods are shown in Table 2.

Table 2. Shown the results of kcorr of 4^{th}, 6^{th} and 7^{th} order of R-Knumerical method for Iraq.

6. Graphical Analysis

Figure 1. Show a chaotic of simulated data by numerical method of 4^{th} order for Iraq. (a) log(M) versus log(t); (b) (M) versus (t); (c) (k) versus (c); (d) (p) versus (q).

Figure 2. Shown chaotic system of R-K 6^{th} order method. (a) log(M) versus log(t); (b) (M) versus (t); (c) (k) versus (c); (d) (p) versus (q).

Figure 3. Shown chaotic system of R-K 7^{th} order method for Iraq. (a) log(M) versus log(t); (b) M_{n} versur time (t); (c) K versus C; (d) P_{n} versus q_{n}.

7. Conclusions

We conducted a study on the error value when solving nonlinear problems and obtaining approximate values of the results as well as the time limit and the total execution time of the previously estimated error for the process of calculating a dynamical system consisting of ordinary differential equations and comparing the results which we obtained it.

In this paper, we dealt with the SIR mathematical model to study the characteristics of the epidemic disease (COVID-19), and We have used the numerical Runge-Kutta method of order 4^{th}, 6^{th}, and 7^{th} to obtain a comparison between the results in terms of the estimated error value, the time limits for each step and the total time taken to implement the process in the program. Our choice to the orders of the numerical method is to obtain a more accurate solution with the least error and the shortest time, and we note the difference by building a table of the results obtained showing us that. Initial values were used for the application in resolving the system which was obtained from statistics and data on Covid-19 for a specific population from among the world's population, which is (Iraq). The elementary values were applied to the composite system from the nonlinear SIR equations with the three-order numerical method (R-K) and it gave great benefit in the information. The binary test is used for separate analysis of deterministic dynamical systems and is also used to test the chaos of the dynamic disease system. And we have applied the test on (1-0) on the model for each of the 4^{th}, the 6^{th}, and the 7^{th} orders, and the result for all orders indicate that the behavior of the disease is chaotic (Kcorr of 4^{th} ord. = 0.912 ≅ 1), (Kcorr of 6^{th} ord. = 0.9212 ≅ 1) and (Kcorr of 7^{th} ord. = 0.9560 ≅ 1). The result with the application of the 7^{th} rank was better and more chaotic than the result of the application of the other ranks, Programs have been built in the Matlab system to perform all operations. Mathematical work through it and obtaining all the results and required values and figures illustrate our idea to provide organized scientific research that provides researchers with a good and useful idea.

Acknowledgements

The authors are very grateful to the University of Mosul/College of Computer Sciences and Mathematics for their provided facilities, which helped to improve the quality of this work.

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