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 Open Access
Numerical treatment of nonlinear model of virus propagation in computer networks: an innovative evolutionary Padé approximation scheme
 Javaid Ali^{1}Email author,
 Muhammad Saeed^{1},
 Muhammad Rafiq^{2} and
 Shaukat Iqbal^{3}
https://doi.org/10.1186/s1366201816721
© The Author(s) 2018
 Received: 17 January 2018
 Accepted: 6 June 2018
 Published: 20 June 2018
Abstract
This work proposes a novel mesh free evolutionary Padé approximation (EPA) framework for obtaining closed form numerical solution of a nonlinear dynamical continuous model of virus propagation in computer networks. The proposed computational architecture of EPA scheme assimilates a Padé approximation to transform the underlying nonlinear model to an equivalent optimization problem. Initial conditions, dynamical positivity and boundedness are dealt with as problem constraints and are handled through penalty function approach. Differential evolution is employed to obtain closed form numerical solution of the model by solving the developed optimization problem. The numerical results of EPA are compared with finite difference schemes like fourth order Runge–Kutta (RK4), ODE45 and Euler methods. Contrary to these standard methods, the proposed EPA scheme is independent of the choice of step lengths and unconditionally converges to true steady state points. An error analysis based on residuals witnesses that the convergence speed of EPA is higher than a globally convergent nonstandard finite difference (NSFD) scheme for smaller as well as larger time steps.
Keywords
 Nonlinear model
 Evolutionary computing
 Padé approximation
 Optimization
1 Introduction
A computer virus is a malicious code which executes harmful and unauthorized activities like erasing necessary files, accessing confidential data and personal information like passwords, account numbers, contact lists etc. Depending on the way of propagation, functioning and damaging the systems/users, malwares are classified into various categories. These include computational viruses, computer worms, Trojans, Rootkits, spyware, logic bombs and so on [1, 2]. Dissemination of computer viruses to other connected systems bears a high resemblance to the behavior of biological viruses [3–5]; therefore, various models of computer virus propagation have been proposed by using an epidemiological analog [6–9].
Transformation of the dynamics of computer virus propagation into mathematical language is an effective methodology to understand and analyze the spreading behavior of viruses. The mathematical prototypes related to the characteristics of variables, parameters and the functional relations governing the dynamics of the virus propagation classify the model as deterministic, stochastic, continuous, discrete, global or individual [2]. Over the years several compartmental models have been evolved. These involve susceptibleinfectedsusceptible (SIS) models [10, 11], susceptibleinfectedrecovered (SIR) [12–14], susceptibleinfectedrecoveredsusceptible (SIRS) [15, 16], susceptibleexposedinfectedrecovered (SEIR) [17, 18], SEIQS (quarantined class included) [19], SEIRS [20, 21], SEIQRS [22], SEIRSV (including vaccinated subpopulation) [23] models and so on.
Mathematical toolbased numerical analysis of these epidemiological models is an essential part of investigations for acquiring better knowledge of their evolution, impact and the deriving mechanisms, especially when the analytical solution is not available. A profound understanding of the model helps in imposing precautionary measures and evaluating their effectiveness in preventing the networks from infections.
The studies conducted by Rafiq et al. [24] on a nonlinear model of virus propagation in a computer network proposed by Mei Peng [25] have exposed the divergence behaviors of RK4 and Euler methods for certain step lengths. The similar behaviors of RK4 and Euler schemes have also been highlighted in [26] and [27]. In these studies a globally convergent nonstandard finite difference (NSFD) scheme proposed by Mickens [28] has successfully been applied to the said model. To the best of the author’s knowledge, till date, the nonlinear model of virus propagation in a computer network has no analytical solution. The important factors which require further research are: (i) finding an analytical solution or constructing alternative efficient convergent numerical schemes, (ii) analysis of the convergence speeds, and (iii) the error analyses of numerical schemes, at least in the common cases when they all converge.
Over the recent years, many modern metaheuristics have been proposed to cope with the most sophisticated problems by transforming them to optimization problems. Metaheuristic algorithms are inspired by natural phenomena like evolution [29, 30], swarm behaviors [31, 32], food foraging behavior [33, 34], sport strategies [35], water dynamics [36, 37] etc. For more detailed studies one may consult the survey articles [35, 38]. Metaheuristicsbased approaches of solving differential equations belong to the class of nonstandard mesh free methods. The applications of these heuristics to differential equations can be found in [39–43], but their applications to epidemic models are very rare.
 (i)
Construction of an equivalent optimization problem by exploiting interpolation and extrapolation strengths of Padé approximation.
 (ii)
Preservation of positivity, boundedness and initial conditions agreement by defining problem constraints.
 (iii)
Construction of fitness/objective function, an essential requirement for evolutionary computing, by use of penalty function approach.
 (iv)
Implementation of differential evolution (DE) to optimize the constructed fitness function.
 (v)
Evolvement of unconditionally convergent closed form numerical solution of nonlinear model of virus propagation in computer networks.
The rest of the paper is organized as follows. In Sect. 2, relative basic concepts are revisited. Section 2 elaborates the proposed framework of EPA scheme for numerical treatment of nonlinear dynamical continuous model of virus propagation in computer networks. In Sect. 5, analyses of the results are presented. In the end, conclusion and some future directions are given.
2 Related concepts
2.1 Padé approximation
2.2 Differential evolution: the evolutionary algorithm
2.3 Penalty function
3 Mathematical model of virus propagation in computer networks
At any time ‘t’ the state variables of the model are defined by:
\(S ( t )\): susceptible computers; \(E ( t )\): exposed computers; \(I ( t )\): infected computers; \(R ( t )\): recovered computers.

N: The total population of computers in the network.

p: Rate at which antivirus recovers susceptible computers.

k: Rate at which antivirus recovers exposed computers.

a: Rate at which antivirus cannot cure exposed computers.

\(\beta_{1}\): Contact rate of susceptible with infected computers.

\(\beta_{2}\): Contact rate of susceptible with exposed computers.

μ: Removal rate of a computer from the network.

r: Recovery rate of infected computers that are cured.
Values of the model parameters for two points of equilibriums
Equilibrium point  Model parameters  

α  \(\beta_{1}\)  \(\beta_{1}\)  N  p  r  k  μ  
VE  0.01  0.7  0.8  100  0.5  0.6  0.4  0.02 
VFE  0.01  0.002  0.003  100  0.5  0.6  0.4  0.02 
4 Evolutionary Padé approximation numerical (EPA) scheme
The architecture of the proposed evolutionary computingbased Padé approximation scheme involves the four main steps which are presented in the following.
4.1 Construct residual functional based on Padé approximation
4.2 Formation of problem constraints
4.3 Employing penalty function approach
4.4 Use of differential evolution for optimization process
 1.
Generate a population of K solutions (\(\boldsymbol{x}_{j} \in \mathbb{R}^{3 ( M+N )}\); \(1\leq j\leq K \)) randomly.
 2.
Evaluate the fitness \(\varphi_{j} =\varphi ( \boldsymbol{x}_{j} )\) of each solution. Preserve the best solution with the smallest objective function value. Set\(T=0\).
 3.
Set \(T=T+1\).
 4.
For each of \(j=1, 2, 3, \ldots,K\), choose three distinct solutions \(\boldsymbol{x}_{A}\), \(\boldsymbol{x}_{B}\) and \(\boldsymbol {x}_{C}\) from the population excluding \(\boldsymbol{x}_{j}\). Set \(\boldsymbol{y}= \boldsymbol{x}_{j}\).
 5.For each of the dimensions \(i=1,2,3,\ldots,3(M+N)\), alter the ith coordinate according to$$y_{i} = \textstyle\begin{cases} x_{Ai} +F\times ( x_{Bi}  x_{Ci} )& \mbox{if }\mathit{rand}< \mathit {CR}, \\ x_{ji} &\mbox{otherwise}. \end{cases} $$
 6.
If \(\varphi ( \boldsymbol{y} ) < \varphi_{j}\) then \(\boldsymbol{x}_{j} \leftarrow\boldsymbol{y}\), otherwise discard y.
 7.
Update the best solution.
 8.
If \(T>\mbox{Number of Allowed Iterations}\), then terminate by preserving the best solution, otherwise start next iteration from step 3.
In step 5 the symbol rand denotes a random number in the interval \((0, 1)\), F is a differential constant and CR is crossover fraction.
5 Numerical results
Four parameters of DE algorithm have been set: population \(\mbox{size}=50\); \(\mathit{CR}=0.9\); \(F=0.5\) and the \(\mbox{maximum number of iterations}=2000\). The order of the Padé approximation is set as \(( N,M ) =(2, 2)\). The parameter \(q_{\mathrm{max}}\) is set to be 2000. The value of each penalty factor is set to be \(L_{q} = 10^{10}\) for all q.
Optimized coefficients of Padé approximate solutions
i  (sign(coefficient))×log(coefficient)  

\(a_{i}\)  \(b_{i}\)  \(c_{i}\)  \(d_{i}\)  \(e_{i}\)  \(f_{i}\)  
VE  1  20.5658  20.3369  19.696  15.817  −19.0657  −19.3137 
2  21.019  20.7901  18.1203  14.2412  −18.6749  −18.923  
VFE  1  36.4774  31.9115  −13.0156  −41.8388  −14.5164  −41.3819 
2  −38.6245  −34.0586  −6.66374  −43.6549  −5.88668  −44.8167 
5.1 Convergence analysis
5.2 Error analysis
To describe the dynamics of system (1) accurately, a necessary condition for a numerical solution is to satisfy the system (1) for all of the time steps. This section presents the error analysis by evaluating absolute residuals ((6) to (8)) of the numerical solutions found by EPA, RK4, Euler and NSFD. The derivatives of the EPA solution are calculated analytically whereas a forward difference scheme is used to approximate the derivatives of numerical solutions of EU, RK4 and NSFD.
Absolute residuals of numerical solutions for \(h=0.01\)
Method  t  VE  VFE  

\(\vert \varepsilon_{1} \vert \)  \(\vert \varepsilon _{2} \vert \)  \(\vert \varepsilon_{3} \vert \)  \(\vert \varepsilon_{1} \vert \)  \(\vert \varepsilon_{2} \vert \)  \(\vert \varepsilon_{3} \vert \)  
RK4  0.01  3.7E02  3.7E02  5.5E03  2.4E−02  1.5E−01  2.0E03 
0.40  1.5E−04  1.4E−01  6.0E03  2.1E−02  1.0E−01  1.3E03  
0.80  3.3E−04  9.5E−02  5.4E03  1.9E−02  6.9E−02  8.8E03  
1.20  4.3E−04  6.3E−02  5.0E03  1.7E−02  4.7E−02  5.9E02  
1.60  4.6E−04  4.2E−02  4.8E03  1.6E−02  3.3E−02  4.1E02  
1.99  4.3E−04  2.8E−04  4.7E03  1.6E−02  2.4E−02  2.9E02  
Euler  0.01  2.3E−13  6.8E−13  6.0E03  3.2E−13  2.2E−13  2.0E03 
0.40  1.7E−15  1.4E−13  6.0E03  2.4E−13  1.1E−13  1.3E03  
0.80  3.6E−15  4.3E−13  5.4E03  2.9E−13  3.2E−14  8.8E02  
1.20  3.9E−15  3.3E−14  5.0E03  7.8E−14  3.6E−14  5.9E02  
1.60  1.9E−15  3.0E−13  4.8E03  4.5E−13  5.7E−14  4.0E02  
1.99  5.6E−16  5.0E−14  4.7E03  1.7E−13  7.9E−14  2.8E02  
NSFD  0.01  7.3E02  7.3E02  4.8E03  1.1E−01  3.5E−01  1.9E03 
0.40  2.0E−01  4.8E−01  6.0E03  9.0E−02  2.5E−01  1.3E03  
0.80  1.6E−01  3.5E−01  5.4E03  7.7E−02  1.8E−01  8.9E02  
1.20  1.2E−01  2.5E−01  5.5E03  6.5E−02  1.3E−01  6.0E02  
1.60  9.6E−02  1.8E−01  4.8E03  5.6E−02  9.1E−02  4.1E02  
1.99  7.3E−02  1.3E−01  4.7E03  4.8E−02  6.7E−02  2.9E02  
EPA  0.01  2.2E − 08  9.5E − 08  1.3E − 08  1.9E − 09  1.2E − 11  2.1E − 10 
0.40  1.5E − 12  1.2E − 12  6.6E − 14  9.5E − 13  7.1E − 14  9.3E − 14  
0.80  2.3E − 13  2.1E − 14  2.8E − 14  6.5E − 13  1.3E − 14  3.5E − 13  
1.20  1.8E − 14  5.7E − 14  1.1E − 14  7.8E − 14  2.1E − 15  1.9E − 15  
1.60  4.6E − 14  2.8E − 14  4.4E − 15  7.1E − 15  2.2E − 15  6.1E − 17  
1.99  5.8E − 14  4.3E − 14  1.7E − 15  1.4E − 14  3.8E − 15  1.1E − 15 
Absolute residuals for \(h=0.1\)
Method  t  VE  VFE  

\(\vert \varepsilon_{1} \vert \)  \(\vert \varepsilon _{2} \vert \)  \(\vert \varepsilon_{3} \vert \)  \(\vert \varepsilon_{1} \vert \)  \(\vert \varepsilon_{2} \vert \)  \(\vert \varepsilon_{3} \vert \)  
RK−4  0.10  5.9E83  5.9E83  5.9E83  2.3E−01  1.3E00  1.6E02 
4.00  NaN  NaN  NaN  8.0E−02  4.8E−02  5.5E00  
8.00  NaN  NaN  NaN  1.7E−02  3.6E−03  5.0E−01  
12.0  NaN  NaN  NaN  2.8E−03  3.4E−04  5.3E−02  
16.0  NaN  NaN  NaN  4.1E−04  3.1E−05  5.2E−03  
19.9  NaN  NaN  NaN  5.9E−05  3.04E−06  5.3E−04  
Euler  0.10  0.0E00  0.0E00  3.8E03  2.1E−14  3.6E−15  1.6E02 
4.00  NaN  NaN  NaN  2.8E−14  4.4E−16  4.5E00  
8.00  NaN  NaN  NaN  2.8E−14  6.9E−17  4.1E−01  
12.0  NaN  NaN  NaN  3.6E−14  0.0E00  4.1E−02  
16.0  NaN  NaN  NaN  2.1E−14  2.2E−19  3.8E−03  
19.9  NaN  NaN  NaN  7.1E−14  2.7E−20  3.6E−04  
NSFD  0.10  6.0E02  6.0E02  6.4E02  9.8E−01  3.0E00  1.7E02 
4.00  1.8E−01  2.6E−01  4.7E02  2.2E−01  1.7E−01  7.9E00  
8.00  1.1E−02  1.2E−02  4.7E02  4.3E−02  1.5E−02  8.5E−02  
12.0  8.5E−04  8.6E−03  4.8E02  7.1E−03  1.6E−03  9.8E−02  
16.0  7.4E−05  7.3E−05  4.8E02  1.1E−03  1.6E−04  1.1E−02  
19.9  7.1E−06  6.9E−06  4.8E02  1.7E−04  1.7E−05  1.7E−05  
EPA  0.10  1.5E − 11  1.3E − 10  8.5E − 12  1.9E − 11  2.7E − 13  2.0E − 12 
4.00  4.9E − 14  1.4E − 14  1.7E − 16  2.1E − 14  4.3E − 15  1.8E − 15  
8.00  2.8E − 14  7.1E − 15  8.6E − 16  1.8E − 14  2.9E − 15  1.2E − 15  
12.0  2.8E − 14  3.6E − 15  1.6E − 15  1.1E − 14  2.1E − 15  8.6E − 16  
16.0  6.8E − 15  5.3E − 15  1.9E − 15  1.8E − 14  1.6E − 15  6.7E − 16  
19.9  2.8E − 14  8.0E − 15  2.3E − 15  7.1E − 15  1.3E − 15  5.5E − 16 
It can be observed from Tables 3 and 4 that the numerical solutions found by EPA scheme for both steady state points satisfy the governing equations with high accuracies as compared to RK4, Euler and NSFD schemes. The convergence of RK4, Euler and NSFD schemes occurs at least after 200 time steps.
6 Conclusion

The evolutionary Padé approximation technique is successfully developed and implemented to the model of virus propagation in computer network.

EPA yielded good approximations of state variables which satisfy the governing equations with high accuracy.

The initial conditions and preservation of positivity and bondedness of the solution were efficiently handled through constraints and the penalty function approach.

The EPA produced a closed form numerical solution of the model having no analytical solution.

The obtained solution possesses very fast convergence, surpassing NSFD.

An advantageous aspect of the proposed framework is that the solution found by EPA is valid for several values of step lengths and needs no resimulation for changed step length. It is analogous to consumption of less computational efforts.

The comparison of the tables and figures demonstrates that the solutions obtained from EPA are found to be in good agreement with NSFD particular.

The error analysis shows that residual errors of EPA solution remain very low at each time step as compared to RK4, Euler and NSFD schemes.

It is also observed that the Euler, RK4 and ODE45 type finite difference schemes are not equipped with specific tools to preserve essential properties like positivity, boundedness and dynamical consistency of real world physical models. On the other hand EPA preserves all of these vital properties of the underlying model.
Declarations
Acknowledgements
We would like to express sincere thanks to the reviewers for their highly insightful and valuable suggestions concerning our paper.
Availability of data and materials
All of the necessary data and the implementation details have been included in the manuscript.
Funding
Not applicable.
Authors’ contributions
The authors have achieved equal contributions. All authors read and approved the manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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