Mathematical Sciences https://doi.org/10.1007/s40096-018-0261-5
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ORIGINAL PAPER
Solving a modified nonlinear epidemiological model of computer viruses by homotopy analysis method Samad Noeiaghdam1
•
Muhammad Suleman2,3 • Hu¨seyin Budak4
Received: 9 April 2018 / Accepted: 29 August 2018 Ó The Author(s) 2018
Abstract The susceptible–infected–recovered model of computer viruses is investigated as a nonlinear system of ordinary differential equations by using the homotopy analysis method (HAM). The HAM is a flexible method which contains the auxiliary parameters and functions. This method has an important tool to adjust and control the convergence region of obtained solution. The numerical solutions are presented for various iterations, and the residual error functions are applied to show the accuracy of presented method. Several h-curves are plotted to demonstrate the regions of convergence, and the residual errors are obtained for different values of theses regions. Keywords Susceptible–infected–recovered model Modified epidemiological model Computer virus Homotopy analysis method h-Curve
Introduction In recent decades, using the computer systems has led to great changes in many fields of life. These changes are many important and wonderful where we cannot leave applying the computer systems. Computer viruses are a malicious software program that can be written with different aims. These softwares help the user to enter the victim computer without permission. The virus should never be considered to be harmless and remain in the system. There are several types of viruses that can be categorized according to their source, technique, file type that infects, where they are hiding, the type of damage they enter, the type of operating system or the design on which they are attacking. Several computer viruses such as & Samad Noeiaghdam
[email protected];
[email protected] 1
Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran
2
Faculty of Science, Jiangsu University, Zhenjiang, China
3
Department of Mathematics, Comsats Institute of Information Technology, Islamabad, Pakistan
4
Department of Mathematics, Faculty of Science and Arts, Du¨zce University, Du¨zce, Turkey
ILOVEYOU, Melissa, My Doom, Code Red, Sasser have been recognized. Also, in recent years, the Stuxnet virus targeted the Siemens hardware and software industry. Claims and statements allege that the virus was part of spy campaign to hit Iran’s nuclear plant, Natanz. These viruses have been able to infect thousands of computers and have hurt billions dollar in computers around the world. Thus, scientists start to work on finding methods to analyze, track, model and protect against viruses. Computer viruses are similar to biological viruses, and we can study it in two case, microscopic and macroscopic models. Therefore, many studies have proposed solutions that help us to understand how computer viruses operate. In order to control the growth and reproduction of computer viruses, many dynamical model have been presented [7, 18, 19, 29, 33–35, 40, 42]. In this study, the following modified SIR model of computer viruses dSðtÞ ¼ f1 kSðtÞIðtÞ dSðtÞ; dt dIðtÞ ¼ f2 þ kSðtÞIðtÞ eIðtÞ dRðtÞ; dt dRðtÞ ¼ f3 þ eIðtÞ dRðtÞ: dt
ð1Þ
is illustrated where the initial conditions are in the following form
123
Mathematical Sciences
Sð0Þ ¼ S0 ðtÞ ¼ S0 ; Ið0Þ ¼ I0 ðtÞ ¼ I0 ;
ð2Þ
Rð0Þ ¼ R0 ðtÞ ¼ R0 : In recent years, many applicable methods have been applied to solve the linear and nonlinear problems [13–16, 27, 28, 31, 39, 41]. Also, many numerical and semi-analytical methods presented to solve the nonlinear epidemiological model (1) such as multi-step homotopy analysis method [17], collocation method with Chebyshev polynomials [32] and variational iteration method [29]. In this paper, the HAM [22–26] is applied to solve the nonlinear system of Eq. (1). This method has many applications to solve various problems arising in the mathematics, physics and engineering [1–6, 8–12, 14–16, 20, 21, 30, 31, 36–38]. The h-curves are demonstrated to find the region of convergence. It is one of important abilities of HAM that the other methods do not have this ability. In order to show the efficiency and accuracy of presented approach, the residual error functions for different values of h; m are estimated. Several graphs of error functions are demonstrated to show the capabilities of method. List of functions, variables and initial assumptions of system (1) are presented in Table 1. R
Main idea Let LS ; LI and LR are the linear operators which are defined in the following form LS ¼
dS dI dR ; LI ¼ ; LR ¼ dt dt dt
ð3Þ
and for constant values c1 ; c2 and c3 we obtain LS ðc1 Þ ¼ 0; LI ðc2 Þ ¼ 0; LR ðc3 Þ ¼ 0:
ð4Þ
Now, the following homotopy maps can be defined as Fig. 1 h-curves of S(t), I(t) and R(t) for m ¼ 5; t ¼ 1
Table 1 List of parameters and functions
123
Parameters and functions
Meaning
Values
S(t) I(t)
Susceptible computers at time t Infected computers at time t
Sð0Þ ¼ 20 Ið0Þ ¼ 15
R(t)
Recovered computers at time t
Rð0Þ ¼ 10
f 1 ; f2 ; f3
Rate of external computers connected to the network
f1 ¼ f2 ¼ f3 ¼ 0
k
Rate of infecting for susceptible computer
k ¼ 0:001
e
Rate of recovery for infected computers
e ¼ 0:1
d
Rate of removing from the network
d ¼ 0:1
Mathematical Sciences
R
R
9
Fig. 2 h-curves of S(t), I(t) and R(t) for m ¼ 10; t ¼ 1
Fig. 3 h-curves of S(t), I(t) and R(t) for m ¼ 15; t ¼ 1
~ qÞ; Iðt; ~ qÞ S0 ðtÞ ~ qÞ; Rðt; ~ qÞ ¼ ð1 qÞLS Sðt; HS Sðt; ~ qÞ; Iðt; ~ qÞ; Rðt; ~ qÞ ; qhHS ðtÞNS Sðt; ~ qÞ; Iðt; ~ qÞ I0 ðtÞ ~ qÞ; Rðt; ~ qÞ ¼ ð1 qÞLI Iðt; HI Sðt; ~ qÞ; Iðt; ~ qÞ; Rðt; ~ qÞ ; qhHI ðtÞNI Sðt; ~ qÞ; Iðt; ~ qÞ R0 ðtÞ ~ qÞ; Rðt; ~ qÞ ¼ ð1 qÞLR Rðt; HR Sðt; ~ qÞ; Iðt; ~ qÞ; Rðt; ~ qÞ ; qhHR ðtÞNR Sðt;
operators, and finally NS ; NI and NR are the nonlinear operators which are defined as follows
ð5Þ where q 2 ½0; 1 is an embedding parameter, h 6¼ 0 is a convergence control parameter, HS ðtÞ; HI ðtÞ and HR ðtÞ are the auxiliary functions, LS ; LI and LR are the linear
123
Mathematical Sciences Table 2 Regions of convergence and the optimal values of h for m ¼ 5; 10; 15 and t ¼ 1 m
hS
hS
hI
I h
hR
hR
5
1:1 hS 0:8
0:99994
1:2 hI 0:8
1:00204
1:1 hR 0:9
0:99058
10
1:2 hS 0:7
1:0036
1:2 hI 0:7
1:0038
1:2 hR 0:7
1
15
1:3 hS 0:5
1
1:3 hI 0:6
1
1:3 hR 0:6
1
According to [22–24] the zero-order deformation, equations can be defined as
~ ~ qÞ; Iðt; ~ qÞ; Rðt; ~ qÞ ¼ oSðt; qÞ NS ½Sðt; dt ~ ~ ~ f1 þ kSðt; qÞIðt; qÞ þ dSðt; qÞ; ~ qÞ oIðt; f2 dt ~ qÞIðt; ~ qÞ þ eIðt; ~ qÞ þ dRðt; ~ qÞ; kSðt;
~ qÞ; Iðt; ~ qÞ; Rðt; ~ qÞ ¼ NI ½Sðt;
~ qÞ; Iðt; ~ qÞ; Rðt; ~ qÞ ¼ NR ½Sðt;
~ qÞ S0 ðtÞ qhHS ðtÞNS Sðt; ~ qÞ; Iðt; ~ qÞ; Rðt; ~ qÞ ¼ 0; ð1 qÞLS Sðt; ~ qÞ; Iðt; ~ qÞ I0 ðtÞ qhHI ðtÞNI Sðt; ~ qÞ; Rðt; ~ qÞ ¼ 0; ð1 qÞLI Iðt; ~ qÞ; Iðt; ~ qÞ R0 ðtÞ qhHR ðtÞNR Sðt; ~ qÞ; Rðt; ~ qÞ ¼ 0: ð1 qÞLR Rðt;
ð6Þ
ð7Þ ~ qÞ; Iðt; ~ qÞ and Now, we can write the Taylor series for Sðt; ~ Rðt; qÞ with respect to q in the following form
~ qÞ oRðt; f3 dt
~ qÞ þ dRðt; ~ qÞ: eIðt;
Table 3 Residual errors of S5 ðtÞ; I5 ðtÞ and R5 ðtÞ for different values of h t
h ¼ 1:2
h ¼ 1:1
h ¼ 1
hopt ¼ 0:99994
h ¼ 0:9
0.0
0.000736
0.000023
2:66454 1015
2:66454 1015
0.000023
0.2
0.00144214
0.0000696185
1:98321 1011
3:81375 1011
2:27788 10
6
0.000388735
0.4
0.00240313
0.00014723
4:38416 1010
7:35197 1010
7:10003 106
0.000128218
8:84562 106
0.0000577341
0.6
0.00366188
0.000265063
0.8
0.00526424
0.000433423
9
1:87637 10
9
1:93464 10
8
3:38906 10
9
6:74028 10
9
9:02851 10
11
1.0
0.00725902
0.000663706
1:1876 10
kEk
0.00725902
0.000663706
1:1876 108
t
h ¼ 1:2
h ¼ 1:1
hopt ¼ 1:00204
h ¼ 1
0.0
0.000704
0.000022
8:43769 1014
2:66454 1015
6:74028 109
9
10
h ¼ 0:8 0.000736
6
0.000180563
6:1317 10
1:58842 10
6
0.000250966
0.000023
0.000736
h ¼ 0:9
h ¼ 0:8
0.000022
0.000704 6
0.2
0.00079364
0.0000221048
1:19964 10
2:0886 10
9:04759 10
0.000553202
0.4
0.000675079
5:11674 106
1:10223 108
6:48732 109
9:76643 106
0.000345977
0.6
0.000291417
0.0000717513
8
4:78103 10
8
0.0000295926
0.0000980449
8
7
3:28701 10
0.8
0.000417872
0.000191176
4:72582 10
1:955 10
0.000046278
0.00017579
1.0
0.00151692
0.000378046
3:75195 109
5:78838 107
0.0000563829
0.000461654
kEk
0.00151692
0.000378046
4:72582 108
5:78838 107
0.0000563829
0.000704
t
h ¼ 1:2
h ¼ 1:1
h ¼ 1
hopt ¼ 0:99058
h ¼ 0:9
h ¼ 0:8
0.0
0.00016
5: 106
0
3:70868 1011
5: 106
0.00016
10
9
0.2
0.000761715
0.0000434312
5:69638 10
1:31142 10
0.000013481
0.000148655
0.4
0.00152906
0.0000988607
1:82284 108
1:09745 108
0.0000206331
0.000383918
0.6
0.00245803
0.000169652
1:38422 107
6:07365 108
0.0000172599
0.000546984
7
7
6
0.8
0.00354105
0.000252826
5:83309 10
1:10979 10
1.0
0.00476692
0.000343953
1:78012 106
1:40369 109
0.0000157769
0.000664199
kEk
0.00476692
0.000343953
1:78012 106
1:10979 107
0.0000206331
0.000664199
123
4:58752 10
0.000639632
Mathematical Sciences Table 4 Residual errors of S10 ðtÞ; I10 ðtÞ and R10 ðtÞ for different values of h t
h ¼ 1:2
h ¼ 1:1
hopt ¼ 1:0036
h ¼ 1
h ¼ 0:9
h ¼ 0:8
0.0
2:3552 107
2:30233 1010
1:02141 1013
9:14824 1014
2:30053 1010
2:3552 107
13
1:07914 10
13
11
13
7
0.2
9
7:98087 10
1:59804 10
9
6
1:03917 10
14
6:38727 106
5:71028 10
8
4:56784 10
1:45981 106
7:45872 108
1:67784 106
5:69842 10
11
1:85035 10
5:17876 10
9:81437 10
1:23901 10
0.6
3:60403 106
1:24299 108
1:38556 1013
1:22125 1013
7:26663 1012
0.8
6
8
14
1:66533 10
13
12
3:46834 10
13
2:47376 10
8
5:59552 10
13
8
2:30053 1010
2:3552 107
hopt ¼ 1:0038
h ¼ 1
h ¼ 0:9
h ¼ 0:8
h ¼ 0:7
2:19797 1010
1:23457 1013
9:14824 1014
2:19932 1010
10
13
9:81437 10
14
11
13
kEk
0.0000101995
4:26312 108
1:38556 1013
t
h ¼ 1:2
h ¼ 1:1
0.0
2:2528 107
0.2
7
1:79984 10
9
1:28342 10
13
0.4
3:0554 10
0.6 0.8
3:77689 10
2:17536 10
1:03695 10
1:51879 10
1.0
7:35477 106
3:95442 108
3:39728 1014
6
8
13
2:84757 10
1:32783 10
1:01252 10
1:52798 106
9:54972 109
1:32561 1013
1:11244 1013
6
8
13
13
0.0000119013
hopt ¼ 1
0.0
2:95245 106
5:11999 108
0.0000446347
1:16719 10
2:45188 10
0.6
0.0000746483
1:96985 106
3:30358 109
6
9
7:358 107
7
7
h ¼ 1:1
0.4
2:18038 107
11
4:17349 107
h ¼ 1:2
9
3:5494 1010
10
h ¼ 1:3
1:01934 10
4:78379 106
7:11569 1010
t
6
5:43913 10
13
4:14557 10
5:01628 10
3:81631 10
4:14557 1013
1:32783 10
9
9:5656 106
8
1:08779 10
6:45201 106
3:95442 10
5:00062 1011
10
7:11569 10
4:17349 10
0.0000129908
h ¼ 0:9
h ¼ 0:8
h ¼ 0:7
5:00069 1011
0 15
8:71525 10
14
10
1:82436 10
12
5:12 108
2:95245 106
1:17196 10
7
7
1:8454 10
3:94129 1014
3:91076 1010
1:62374 107
0.0000101862
8
0.0000107469
14
10
0.000108286
2:76035 10
1:49223 10
0.000142277
3:30389 106
6:33091 109
1:97065 1014
8:26158 1010
8:0793 108
0.000142277
6
9
14
10
7
1 X
~ qÞ ¼ S0 ðtÞ þ Sðt;
1:9984 10
3:94129 10
S~m ðtÞ ¼
m
Sm ðtÞq ;
m¼1
~ qÞ ¼ I0 ðtÞ þ Iðt;
1 X
m
Im ðtÞq ;
ð8Þ
I~m ðtÞ ¼
m¼1
~ qÞ ¼ R0 ðtÞ þ Rðt;
1 X
R~m ðtÞ ¼
Rm ðtÞqm ;
m¼1
where Sm ¼
m~ ~ qÞ 1 om Sðt; ; Im ¼ 1 o Iðt; qÞ ; m m m! oq m! oq q¼0 q¼0 ~ qÞ 1 om Rðt; : Rm ¼ m! oqm q¼0
7:79626 106
1:45817 10
1.0
6:33091 10
3:44612 106
1:15463 10
0.8
3:30389 10
0.0000129908
7
3:47132 10
7:35477 10
7
9:37903 10
2:2528 107
4:57945 10
kEk
kEk
3:44326 106
3:46834 1013
1:3145 10
0.0000203615
6:8329 10
4:19639 106 0.0000135813
4:26312 10
0.2
11
8
4:69374 10
0.0000101995
7
1:12355 10
4:58415 10
3:20961 10
1.0
1:8875 10
0.0000135813
8
0.4
6:30105 10
h ¼ 0:7
7:393 10
8:26158 10
6:72562 10 1:8454 10
S0 ðtÞ; S1 ðtÞ; . . .; Sm ðtÞ ;
I0 ðtÞ; I1 ðtÞ; . . .; Im ðtÞ ;
9:64522 106 0.0000107469
R0 ðtÞ; R1 ðtÞ; . . .; Rm ðtÞ :
Differentiating Eq. (7) m-times with respect to q, dividing by m! and putting q ¼ 0 the mth-order deformation equations can be obtained as follows LS ½Sm ðtÞ vm Sm1 ðtÞ ¼ hHS ðtÞ
