Utilizing fractional derivatives and sensitivity analysis in a random framework: a model-based approach to the investigation of random dynamics of malware spread

Abstract

In this study, an ordinary-deterministic equation system modeling the spread dynamics of malware under mutation is analyzed with fractional derivatives and random variables. The original model is transformed into a system of fractional-random differential equations (FRDEs) using Caputo fractional derivatives. Normally distributed random variables are defined for the parameters of the original system that are related to the mutations and infections of the nodes in the network. The resulting system of FRDEs is simulated using the predictor-corrector method based fde12 algorithm and the forward fractional Euler method (ffEm) for various values of the model components such as the standard deviations, orders of derivation, and repetition numbers. Additionally, the sensitivity analysis of the original model is investigated in relation to the random nature of the components and the basic reproduction number ($R_{0}$) to underline the correspondence of random dynamics and sensitivity indices. Both the normalized forward sensitivity indices (NFSI) and the standard deviation of $R_{0}$ with random components give matching results for analyzing the changes in the spread rate. Theoretical results are backed by the simulation outputs on the numerical characteristics of the fractional-random model for the expected number of infections and mutations, expected timing of the removal of mutations from the network, and measurement of the variability in the results such as the coefficients of variation. Comparison of the results from the original model and the fractional-random model shows that the fractional-random analysis provides a more generalized perspective while facilitating a versatile investigation with ease and can be used on different models as well.

1 Introduction

Mathematical modeling of real-life events has been an important research area for mathematical sciences for decades. Modeling daily phenomena such as financial situations, spread of diseases, and solar radiation has been instrumental in understanding the dynamics of these events. Dating back to Al-Khwarizmi’s contributions to the mathematical handling of trade and inheritance problems around the ninth century and Daniel Bernoulli’s mathematical investigation on smallpox inoculation in eighteenth century, mathematical modeling is still an important tool for analyzing many systems or events that affect everyday life. Several types of equations are used in these models according to the nature of the problem and the information available to the researcher. The use of differential equations is one of the most popular approaches in current mathematical modeling studies.

The use of ordinary differential equations for modeling applications has been extensively studied for different aspects of the models. Studies on the local/global stability of the models, numerical or approximate analysis of the solutions, and formulation of improvements of existing models for a more detailed modeling of events are within the center of attention in such studies. The use of time delays, stochastic noise, fuzzy numbers, and integro-differential components in models consisting of differential equations enable a wide range of modeling applications. However, the most popular approach in recent modeling studies is the use of fractional calculus.

Fractional calculus provides a generalization of ordinary calculus by making use of noninteger ordered derivatives. The transition from ordinary derivation to fractional derivation has several advantages in modeling, such as the ability to model memory effect in biological systems. A smooth transition between derivatives of integer orders enables a more accurate modeling or real-life systems such as diffusion processes [1], image processing [2], and viscoelastic mechanics [3]. Studies on nonlinear analysis, partial differential equations, and various other topics in applied mathematics are carried out within a more detailed and generalized approach using tools of fractional calculus [4–8]. Several definitions of fractional derivation such as the Riemann–Liouville derivative, Caputo–Fabrizio derivative, Atangana–Baleanu derivative, etc. have their own advantages that facilitate a more precise modeling study. Recent studies within the last decade have utilized a broader approach to studying ordinary differential equations (ODE) by the use of fractional differential equations (FDE). The use of FDEs in modeling studies has been a popular area for many researchers [9, 10]. An ordinary differential equation $x'(t) = f(t,x(t))$ defines the rate of change of the variable $x(t)$ at any time, while the equation $x(t) = f(t,x(t))$ provides information about the current state (i.e., derivative of order zero) of the variable. Fractional derivatives of noninteger order $0<\alpha <1$ enable the analysis of the transition between these two equations. As $\alpha \rightarrow 0$, the equation transitions from modeling the change of the variable to its future state to modeling its current state. This phenomenon is among the general interpretation that fractional derivatives can be used to incorporate memory in the equation [11]. In contrast to ODEs, fractional differential equations provide a more generalized approach (with the ability to use arbitrary orders of derivation) and the ability to take into account the dependence on past values [11].

Another tool that provides a more generalized analysis is to use random components in the models. Mathematical models that describe events and systems in nature are built upon detailed investigations of the problems that are analyzed. However, in most cases, the random nature of several components in these problems are neglected for a deterministic modeling approach. For instance, most of the mathematical models for the transmission of COVID-19 have been investigated on a deterministic level [12–14]. Although it is known that some components of these models, such as the death rates, rates for infected individuals to be quarantined, or recovery rates, may vary between regions or stages of the pandemic, these studies use deterministic quantities for the analysis. While ignoring the random nature of these components enables a more theoretical analysis of the equation systems, an analysis of the variability in certain dynamics of the problems being modeled is not possible within the deterministic framework.

In this study, an ordinary-deterministic mathematical model will be analyzed with fractional derivatives and random components for a more generalized modeling approach. Many of the equation systems that exist in the literature can be used to present this methodology; however, a mathematical model for the spread of malware among networks will be used for this study. Malware, such as Trojan horses and computer viruses, is software that is mostly used to steal personal or government data. This malware spreads in a network in a similar way to how viruses spread within populations. Hence, models of disease transmission are frequently used to investigate the spread of malware within networks. Some of the recent studies in this area can be exemplified as follows: Guillen and del Rey used a SCIRS type compartmental model to investigate malware spread on wireless sensor networks [15]. Masood et al. modeled the spread and attack of the Stuxnet virus in a network of critical control infrastructure [16]. Ahn et al. analyzed the spread of Stuxnet-style autonomous vehicle malware [17]. Abazari et al. investigated the effects of antimalware software in cloud environment [18].

The improvements in smart technology and the fact that the use of smart devices has become widespread among individuals have increased the number of studies in this area. Thus, the recent trend of using fractional calculus has also been applied in modeling malware spread within networks. Sabir et al. analyzed a computer virus propagation model using fractional derivatives [19]. Zarin et al. used Haar wavelet collocation methods for a fractional computer virus model [20]. Achar et al. used fractional derivatives for analyzing worm transmission in wireless sensor networks [21]. Xu et al. studied bifurcation of a fractional delayed malware model in social networks [22]. The number of examples from the literature on modeling studies utilizing fractional components [9, 10] can be increased. Although many similar applications can be found for the use of fractional derivatives of malware propagation in networks, only a small fraction of these studied have been found to contain random components. There are some studies such as those of Anwar et al., Arif et al., and Raza et al. that use the stochastic approach for modeling uncertainty [23–25]. However, a fractional system containing random variables provides a swifter and straightforward modeling opportunity and can be applied by using almost all the ordinary-deterministic models in the literature.

This study aims to present an accessible and uncomplicated modeling methodology for malware propagation in networks that generalizes the existing popular approach of using ordinary-deterministic models. The use of fractional derivatives and random components will provide a more accurate and extensive modeling analysis, which considers the variability of results, the memory effect dynamics, and future expectations for spread dynamics as well as results that can be found with ordinary-deterministic models. Main contributions of this study can be listed as follows.

• 1. An existing ordinary-deterministic model will be transformed into a system of fractional-random differential equations (FRDEs) to analyze random malware spread dynamics when the parameters modeling transmission rates for new malware infections and mutations vary.

• 2. Caputo fractional derivatives and random components with normal distribution will be used for a fundamental approach, which can be modified using other definitions of fractional derivatives and probability distributions suitable for the problem being investigated.

• 3. Numerical characteristics of the fractional-random model such as expected number of new infections, expected time for the removal of malware from the network, and the coefficient of variation (CV) for the number of susceptible nodes in the network will be presented among other random dynamics.

• 4. Sensitivity analysis of the basic reproduction number ($R_{0}$) will be analyzed together with the standard deviation of the randomized $R_{0}$ to emphasize the correspondence of these two measures for analyzing the uncertainty in spread dynamics. Results for the positivity and invariance for the original model will be handled with respect to the new randomized parameters.

• 5. Results for the CV and and randomness of $R_{0}$ will be used along with the expectations for the compartments of the model to highlight the more generalized perspective achieved by using a system of FRDEs instead of ODEs. Simulation results will be given for various orders of derivation, CV for the random parameters, numbers of repetition and by using various methods to analyze the models.

Results show that the use of FRDEs provides alternative solutions that successfully replace the results of the ODEs, as well as providing outputs to analyze the variability of the solutions with respect to changes in the parameters (which model spread dynamics). The proposed approach based on a randomized system for studying uncertainty in relation to sensitivity analysis utilizes tools of fractional calculus and numerical analysis and forms a robust basis for modeling the deviations within a more detailed framework. This approach can be used for models in other fields such as engineering, epidemiology, and even social sciences for a more generalized modeling approach.

2 The deterministic model

The model given in [26] will be used to give a model-based presentation of the relationship between the sensitivity and randomness analyses. The equation system given by Ren et al. consists of four ordinary-deterministic differential equations with nonlinear components. This system enables a novel approach to modeling malware spread in networks by the use of additional compartments and parameters to represent mutations, which separates it from similar models in the literature [15, 16].

The ordinary-deterministic model is given as follows [26]:

$$\begin{aligned} &\frac{d}{dt} S(t)=b-\beta S(t)M(t)-\lambda S(t)I(t)-\mu S(t), \\ &\frac{d}{dt} E(t)=\beta S(t)M(t)+\lambda S(t)I(t)+\delta M(t)-( \varepsilon +\mu )E(t), \\ &\frac{d}{dt} I(t)=\varepsilon E(t)+\chi M(t)-(\eta +\theta +\mu )I(t), \\ &\frac{d}{dt} M(t)=\eta I(t)-(\gamma +\delta +\chi +\mu )M(t). \end{aligned}$$

(1)

Here, $S(t)$, $E(t)$, $I(t)$, and $M(t)$ denote the number of susceptible, latent (exposed), infected, and mutated nodes at any unit time t, respectively. The original model consists of five compartments and extends the existing literature by using an additional compartment for mutated malware. It is a generalization of the popular $SEIR$ type compartmental model with the additional M-compartment; and like these models, the recovered nodes do not affect the other compartments. Hence the four compartment model (1) will be analyzed for an equivalent and concise investigation.

The parameter descriptions and values presented in Table 1 have been obtained from the referred study [26]. The model will be investigated with initial conditions $S(0)=S_{0}$, $E(0)=E_{0}$, $I(0)=I_{0}$, and $M(0)=M_{0}$ that properly define the initial state of the network, such that $S_{0},E_{0},I_{0},M_{0} \geq 0$.

Table 1 Parameters of the model along with their definitions and values

The positivity of the solutions of (1) are shown below for positively defined initial conditions $S_{0},E_{0},I_{0},M_{0} \geq 0$.

Theorem 2.1

[27, 28] If the initial values $S(0)=S_{0}$, $E(0)=E_{0}$, $I(0)=I_{0}$, and $M(0)=M_{0}$ are positive such that $S_{0},E_{0},I_{0},M_{0} > 0$, then $\forall t>0$, the solutions of the deterministic model (1) are positive.

Proof

The trivial case where the initial conditions are all zero, i.e., $S_{0}=E_{0}=I_{0}=M_{0}=0$ is apparent. If the positive parts in the right-hand side of the first equation in (1) are neglected, the equation becomes separable as follows:

$$\begin{aligned} \frac{d}{dt} S(t)=b-\beta S(t)M(t)-\lambda S(t)I(t)-\mu S(t) & \Rightarrow \frac{dS}{dt} \geq -(\beta M+\lambda I+\mu )S \\ &\Rightarrow \frac{dS}{S} \geq -(\beta M+\lambda I+\mu ) dt \\ &\Rightarrow \int _{S_{0}}^{S(t)} \frac{dS}{S} \geq \int _{0}^{t} -( \beta M+\lambda I+\mu ) dt \\ &\Rightarrow S(t) \geq S_{0} e^{-(\beta M+\lambda I+\mu )t}, \forall t \geq 0. \end{aligned}$$

(2)

Similarly, for the rest of the model, one gets $\forall t \geq 0$:

$$\begin{aligned} & \frac{dE}{dt} \geq - (\varepsilon + \mu ) E \Rightarrow \int _{E_{0}}^{E(t)} \frac{dE}{E} \geq \int _{0}^{t} -(\varepsilon + \mu ) dt \Rightarrow E(t) \geq E_{0} e^{- (\varepsilon + \mu )t}, \end{aligned}$$

(3)

$$\begin{aligned} & \frac{dI}{dt} \geq - (\eta + \theta + \mu ) I \Rightarrow \int _{I_{0}}^{I(t)} \frac{dI}{I} \geq \int _{0}^{t} -(\eta + \theta + \mu ) dt \Rightarrow I(t) \geq I_{0} e^{- (\eta + \theta + \mu )t}, \\ & \frac{dM}{dt} \geq - (\gamma + \delta + \chi + \mu ) M \Rightarrow \int _{M_{0}}^{M(t)} \frac{dM}{M} \geq \int _{0}^{t} -(\gamma + \delta + \chi + \mu ) dt \\ &\hphantom{\frac{dM}{dt} \geq - (\gamma + \delta + \chi + \mu ) M}\Rightarrow M(t) \geq M_{0} e^{- (\gamma + \delta + \chi + \mu )t}. \end{aligned}$$

(4)

 □

The positive invariance has been examined in the referred study by Ren et al. [26].

3 Randomness and sensitivity analysis

In this section, the parameters of the deterministic model (1) will be transformed into normal (Gaussian) random variables to investigate the random dynamics of the model. The random variables will be defined such that they deviate around their deterministic values given in Table 1 following the normal distribution. Hypothetically, various other probability distributions can be used to randomize the deterministic model. However, as the central limit theorem suggests that sample means with sufficiently many elements are approximately normally distributed as the sample size grows (regardless of the probability distribution of the population), the methodology will be presented by using normal distribution.

3.1 The random model

The original model given in [26] assumes that the parameters are constants for the analysis of the spread. This assumption prevents the analysis of the changes in the course of the spread in relation to the deviations in the malware transmission dynamics represented with these parameters given in Table 1. In this section, all of the parameters will be randomized to make an overall random analysis, whereas only certain parameters related to mutations and infections of the nodes will be randomized to investigate the random dynamics of transmission with mutations and infections. The transition from a system of ordinary differential equations to random differential equations is based on the definition below, which has been applied to numerous systems in the literature [27, 29].

Definition 3.1

[30] Consider a deterministic ordinary differential equation $X' (t)=f(X(t),Y(t),A,t)$ for $t\in T$ with the initial value $X(t_{0})=X_{0}$. This equation can be transformed into a random differential equation by randomizing various parts such as i) the initial values, ii) the nonhomogeneous terms, and iii) the parameters. The coefficient (which corresponds to the parameters of the deterministic model) of the ordinary differential equation A can be transformed into a random variable to obtain a random differential equation

$$ X' (t)=f(X(t),Y(t),A^{*},t), t\in T, X(t_{0})=X_{0}. $$

The random variable $A^{*}$ can be arbitrarily distributed for an accurate modeling of the problem under investigation. The appropriate probability distribution for modeling the uncertainty of the random component can be determined by analyzing real data related to the problem.

The random model can be obtained by transforming the deterministic parameters η, β, λ, γ, δ, χ, μ, ε, b, and θ to randomized parameters $\eta ^{*}$, $\beta ^{*}$, $\lambda ^{*}$, $\gamma ^{*}$, $\delta ^{*}$, $\chi ^{*}$, $\mu ^{*}$, $\varepsilon ^{*}$, $b^{*}$, and $\theta ^{*}$:

$$\begin{aligned} &\frac{d}{dt} S(t)=b^{*}-\beta ^{*} S(t)M(t)-\lambda ^{*} S(t)I(t)- \mu ^{*} S(t), \\ &\frac{d}{dt} E(t)=\beta ^{*} S(t)M(t)+\lambda ^{*} S(t)I(t)+\delta ^{*} M(t)-(\varepsilon ^{*}+\mu ^{*})E(t), \\ &\frac{d}{dt} I(t)=\varepsilon ^{*} E(t)+\chi ^{*} M(t)-(\eta ^{*}+ \theta ^{*}+\mu ^{*})I(t), \\ &\frac{d}{dt} M(t)=\eta ^{*} I(t)-(\gamma ^{*}+\delta ^{*}+\chi ^{*}+ \mu ^{*})M(t). \end{aligned}$$

(5)

Here, the random parameters are defined as independent and identically distributed random variables that can be simulated using the statistics and machine learning toolbox in MATLAB.

$$\begin{aligned} & \eta ^{*} \sim N(\eta ,\sigma _{1}^{2}),\: \beta ^{*} \sim N(\beta , \sigma _{2}^{2}),\: \lambda ^{*} \sim N(\lambda ,\sigma _{3}^{2}),\: \gamma ^{*} \sim N(\gamma ,\sigma _{4}^{2}),\: \delta ^{*} \sim N( \delta ,\sigma _{5}^{2}), \\ & \chi ^{*} \sim N(\chi ,\sigma _{6}^{2}),\: \mu ^{*} \sim N(\mu , \sigma _{7}^{2}),\: \varepsilon ^{*} \sim N(\varepsilon ,\sigma _{8}^{2}), \: b^{*} \sim N(b,\sigma _{9}^{2}),\: \theta ^{*} \sim N(\theta , \sigma _{10}^{2}), \end{aligned}$$

(6)

where $\sigma _{i},\:i=\overline{1,10}$, denote the standard deviations of random variables. The initial conditions of the random system are identical to the initial values used for the deterministic model (1).

Corollary 3.2

The positivity of solutions under the random parameters can be shown similarly using Theorem 2.1. Note that the random variable $p \in \{\eta ^{*}, \beta ^{*}, \lambda ^{*}, \gamma ^{*}, \delta ^{*}, \chi ^{*}, \mu ^{*}, \varepsilon ^{*}, b^{*}, \theta ^{*}\}$ is a real-valued measurable function $p:\Omega \rightarrow \mathbb{R}$ meaning that although its exact value is uncertain, its values are known to be real numbers, i.e., $p\in \mathbb{R},\forall p$. Hence, if the conditions of Theorem 2.1hold, then it can be shown that $\forall t \geq 0$,

$$\begin{aligned} & S(t) \geq S_{0} e^{-(\beta ^{*} M+\lambda ^{*} I+\mu ^{*})t},\:E(t) \geq E_{0} e^{- (\varepsilon ^{*} + \mu ^{*})t}, \\ & I(t) \geq I_{0} e^{- (\eta ^{*} + \theta ^{*} + \mu ^{*})t}, \: M(t) \geq M_{0} e^{- (\gamma ^{*} + \delta ^{*} + \chi ^{*} + \mu ^{*})t}. \end{aligned}$$

(7)

This means that the solutions of random model (5) are also positive if the initial conditions are defined positive such that $S_{0},E_{0},I_{0},M_{0} > 0$.

Corollary 3.3

The feasible region Ω defined by $\Omega = \{(S,E,I,M)\in \mathbb{R}_{+}^{4}:S+E+I+M \leq b/\mu \}$ and the initial conditions $S_{0},E_{0},I_{0},M_{0} \geq 0$ are positively invariant for the original model (1) as shown in Ren et al. [26]. Using this result, it is clear that the positive invariance holds for random parameters $b^{*} \sim N(b,\sigma _{9}^{2})$ and $\mu ^{*} \sim N(\mu ,\sigma _{7}^{2})$ since $b^{*},\mu ^{*}\in \mathbb{R}$ as stated in Corollary 3.2.

The positivity of solutions and positive invariance under randomness guarantee the meaningfulness of the solutions since the variables of the system (S, E, I, M) are numbers of nodes under the corresponding compartment and hence cannot be negative.

3.2 Sensitivity analysis

The sensitivity of the basic reproduction number to the changes in the parameters is one way of quantifying the randomness of the event. The normalized forward sensitivity index (NFSI) can be used to measure how changes in transmission components (represented as parameters) affect the transmission rate (quantified by the basic reproduction number $R_{0}$).

Definition 3.4

[27, 31, 32] The normalized forward sensitivity index of a variable z that depends differentiably on a parameter p is defined by

$$ \Upsilon _{p}^{z} = \frac{\partial z}{\partial p} \times \frac{p}{z}. $$

(8)

The basic reproduction number was given as follows in the referred study [26]:

$$ R_{0} = \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu )) \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))}. $$

(9)

The sensitivity analysis of the basic reproduction number (9) was not studied in detail by Ren et al. in [26]. The sensitivity of reproduction to only the changes in the mutation rate was given shortly to comment on the infection dynamics using the partial derivative $\partial R_{0} / \partial \eta$. Here, the normalized forward sensitivity index (NFSI) of each parameter is calculated to investigate the relationship between the randomness of the parameters and the model behavior. Using (8), we obtain

$$\begin{aligned} \Upsilon _{\eta}^{R_{0}} =& \frac{\partial R_{0}}{\partial \eta} \times \frac{\eta}{R_{0}} = \Bigg[ \frac{\beta \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \\ & - \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu ))\epsilon b (\delta \mu + (\gamma + \mu ) (\epsilon + \mu ))}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))^{2}} \Bigg] \times \frac{\eta}{R_{0}} = 0.1849, \end{aligned}$$

(10)

$$\begin{aligned} \Upsilon _{\beta}^{R_{0}} =& \frac{\partial R_{0}}{\partial \beta} \times \frac{\beta}{R_{0}} \\ =& \Bigg[ \frac{\eta \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \Bigg] \times \frac{\beta}{R_{0}} \\ =& 0.3254, \end{aligned}$$

(11)

$$\begin{aligned} \Upsilon _{\lambda}^{R_{0}} =& \frac{\partial R_{0}}{\partial \lambda} \times \frac{\lambda}{R_{0}} \\ =& \Bigg[ \frac{(\gamma + \delta + \chi + \mu ) \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \Bigg] \times \frac{\lambda}{R_{0}} \\ =& 0.6746, \end{aligned}$$

(12)

$$\begin{aligned} \Upsilon _{\gamma}^{R_{0}} =& \frac{\partial R_{0}}{\partial \gamma} \times \frac{\gamma}{R_{0}} = \Bigg[ \frac{\lambda \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \\ & - \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu ))\epsilon b (\eta (\epsilon + \mu ) (\epsilon + \mu ) (\theta + \mu ))}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))^{2}} \Bigg] \times \frac{\gamma}{R_{0}} \\ =& -0.0051 , \end{aligned}$$

(13)

$$\begin{aligned} \Upsilon _{\delta}^{R_{0}} =& \frac{\partial R_{0}}{\partial \delta} \times \frac{\delta}{R_{0}} = \Bigg[ \frac{\lambda \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \\ & - \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu ))\epsilon b (\eta \mu + (\epsilon + \mu ) (\theta + \mu ))}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))^{2}} \Bigg] \times \frac{\delta}{R_{0}} \\ =& -0.0094, \end{aligned}$$

(14)

$$\begin{aligned} \Upsilon _{\chi}^{R_{0}} =& \frac{\partial R_{0}}{\partial \chi} \times \frac{\chi}{R_{0}} = \Bigg[ \frac{\lambda \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \\ & - \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu ))\epsilon b (\epsilon + \mu ) (\theta + \mu )}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))^{2}} \Bigg] \times \frac{\chi}{R_{0}} \\ =& -0.0973, \end{aligned}$$

(15)

$$\begin{aligned} \Upsilon _{\epsilon}^{R_{0}} =& \frac{\partial R_{0}}{\partial \epsilon} \times \frac{\epsilon}{R_{0}} = \Bigg[ \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu ))}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \\ & - \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu ))\epsilon b (\epsilon (\gamma + \mu ) + (\theta + \mu )(\gamma + \delta + \chi + \mu )}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))^{2}} \Bigg] \times \frac{\epsilon}{R_{0}} \\ =& 0.8757 , \end{aligned}$$

(16)

$$\begin{aligned} \Upsilon _{b}^{R_{0}} =& \frac{\partial R_{0}}{\partial b} \times \frac{b}{R_{0}} \\ =& \Bigg[ \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu ))\epsilon}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \Bigg] \times \frac{b}{R_{0}} = 1 , \end{aligned}$$

(17)

$$\begin{aligned} \Upsilon _{\theta}^{R_{0}} =& \frac{\partial R_{0}}{\partial \theta} \times \frac{\theta}{R_{0}} \\ =& \Bigg[ - \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu ))\epsilon b (\epsilon + \mu )(\gamma + \delta + \chi + \mu )}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))^{2}} \Bigg] \times \frac{\theta}{R_{0}} \\ =& -0.3581. \end{aligned}$$

(18)

The normalized forward sensitivity indices (NFSI) of each parameter are shown in Fig. 1. The NFSI is a “normalized” measure, as the name suggests, and its values are typically within the interval $[-1,1]$. However, in some cases, $\Upsilon _{p}^{R_{0}} \notin [-1,1]$ may correspond to issues related to the valuation of the parameter p and its relation with the basic reproduction number $R_{0}$. Note that the NFSI for μ can be calculated as $\Upsilon _{\mu}^{R_{0}} = -2.5907$. It can be seen in (9) that $R_{0}$ contains $\mu ^{4}$ in its denominator, showing that there is a highly nonlinear relationship between this parameter and the basic reproduction number.

Figure 1

Normalized forward sensitivity indices

The NFSI measures the relation between the variability of the parameters and the basic reproduction number [27, 31]. Specifically, $\Upsilon _{b}^{R_{0}} = 1$ suggests that $R_{0}$ has a very strong positive relation with the parameter b and, for instance, a 10% increase in the value of b would cause a 10% increase in the value of $R_{0}$. Similarly, $\Upsilon _{\theta}^{R_{0}} = -0.3581$ suggests a weak negative relation between θ and $R_{0}$. This value means that a 10% increase in the value of θ would trigger a decrease around $1/3$ of the same percentage. This measure of correlated change between the parameters and the basic reproduction number $R_{0}$ provides a significant advantage for monitoring spread dynamics. For instance, the value of the parameter b is used as 0.5 in the original study, and doubling its value to obtain $b=1$ means that the value of the basic reproduction number would also experience the same change since $\Upsilon _{b}^{R_{0}} = 1$. Similarly, changing the value of b to 0.25 would mean a 50% decrease and the effect of this change on $R_{0}$ could also be analyzed solely by using $\Upsilon _{b}^{R_{0}}$.

3.3 Analyzing randomness of transmission with standard deviations

The effects of uncertainty in the parameters on the spread rate can also be analyzed through the deviations in the parameters. An argument can be presented through the investigation of the numerical characteristics of the basic reproduction number containing random variables. For instance, if the parameter b is randomized in (9) as $b^{*} \sim N(b,\sigma _{9}^{2})$, the expected value and the variance of the new randomized $(R_{0}^{*})_{b}$ are obtained as follows:

$$\begin{aligned} &(R_{0}^{*})_{b} = \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu )) \epsilon}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} b^{*} \\ \Rightarrow &E((R_{0}^{*})_{b}) = \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu )) \epsilon}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} E(b^{*}) = R_{0} \\ \Rightarrow &Var((R_{0}^{*})_{b}) = \Bigg( \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu )) \epsilon}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \Bigg)^{2} Var(b^{*}). \end{aligned}$$

(19)

Assume that $b^{*}$ is assigned a 5% coefficient of variation (or relative standard deviation), i.e., $Var(b^{*})=\frac{b^{2}}{20^{2}}$ if $b^{*} \sim N(b,\sigma _{9}^{2})$ for $\sigma _{9} = b/20$. This yields

$$\begin{aligned} Var((R_{0}^{*})_{b}) =& \Bigg( \frac{(\beta \eta + \lambda (\gamma + \delta + \chi + \mu )) \epsilon}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \Bigg)^{2} \frac{b^{2}}{20^{2}} \\ =& \frac{1}{20^{2}} R_{0}^{2}. \end{aligned}$$

(20)

If the basic reproduction number is randomized by transforming the variable λ to a random variable $\lambda ^{*}$ such that $\lambda ^{*} \sim N(\lambda ,\sigma _{3}^{2})$ with a similar coefficient of variation of 5%, i.e., $\sigma _{3} = \lambda /20$, then one gets

$$\begin{aligned} (R_{0}^{*})_{\lambda} =& \frac{(\beta \eta + \lambda ^{*} (\gamma + \delta + \chi + \mu )) \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \\ \Rightarrow E((R_{0}^{*})_{\lambda}) =& E \Bigg[ \Bigg( \frac{\beta \eta \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \Bigg) \\ &+ \Bigg( \frac{(\gamma + \delta + \chi + \mu ) \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \Bigg) \lambda ^{*} \Bigg] \\ =& R_{0}. \end{aligned}$$

(21)

Similar operations for the variance yield

$$\begin{aligned} Var((R_{0}^{*})_{\lambda}) =& \Bigg( \frac{(\gamma + \delta + \chi + \mu ) \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \Bigg)^{2} Var(\lambda ^{*}) \\ =& \frac{1}{20^{2}} \Bigg( \frac{\lambda (\gamma + \delta + \chi + \mu ) \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \Bigg)^{2}. \end{aligned}$$

(22)

The same procedure can also be used with $\beta ^{*} \sim N(\beta ,\sigma _{2}^{2})$ for $\sigma _{2} = \beta /20$. If $(R_{0}^{*})_{\beta}$ is the randomized basic reproduction number containing the random parameter $\beta ^{*}$, then its expected value is similarly obtained as $E((R_{0}^{*})_{\beta}) = R_{0}$. The variance becomes

$$\begin{aligned} Var((R_{0}^{*})_{\beta}) =& \Bigg( \frac{\eta \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \Bigg)^{2} Var(\beta ^{*}) \\ =& \frac{1}{20^{2}} \Bigg( \frac{\beta \eta \epsilon b}{\mu (\eta \delta \mu + \eta (\gamma + \mu ) (\epsilon + \mu ) + (\epsilon + \mu ) (\theta + \mu ) (\gamma + \delta + \chi + \mu ))} \Bigg)^{2}. \end{aligned}$$

(23)

The probability density functions $R_{0}$ containing the random parameters $b^{*}$, $\lambda ^{*}$, and $\beta ^{*}$ are shown in Fig. 2. Note that identical expected values for $(R_{0}^{*})_{b}$, $(R_{0}^{*})_{\lambda}$, and $(R_{0}^{*})_{\beta}$ result in identical positioning of peak values, whereas the varying standard deviation changes the shape of the curves (Fig. 2).

Figure 2

Probability density function of $R_{0}$ with random parameters

The random parameters $b^{*}$, $\lambda ^{*}$, and $\beta ^{*}$ are defined to be normally distributed. Using the results for the expected values and variances alone would provide valuable information for their random characteristics. For instance, it is known that the confidence interval for the expected value of $b^{*}$ would contain approximately 99.7% of all values of the random parameter, assuming three standard deviations are used for the interval.

3.4 Comparison of sensitivity indices and standard deviation of $R_{0}$

The standard deviations of the basic reproduction number $R_{0}$ containing only one of the random parameters $b^{*}$, $\lambda ^{*}$, and $\beta ^{*}$, shown as ($std(R_{0}^{*})_{p}$), are given in Table 2 along with the normalized forward sensitivity indices of these parameters ($\Upsilon _{p}^{R_{0}}$).

Table 2 Standard deviations of $R_{0}$ with random parameters and their NFSI with a comparison of the ratio of these values

The definition of $R_{0}$ (Eq. (9)) can be analyzed to show that the parameters $b^{*}$, $\lambda ^{*}$, and $\beta ^{*}$ are linearly related with $R_{0}$. Hence, when $R_{0}$ is analyzed in random conditions by randomizing a single parameter at a time (using the same relative standard deviation for each parameter), it is seen that the ratio of NFSI and the standard deviations of random $R_{0}$ are almost constant for these parameters:

$$ 100 \times \frac{\Upsilon _{b}^{R_{0}}}{std(R_{0}^{*})_{b}} = 100 \times \frac{\Upsilon _{\lambda}^{R_{0}}}{std(R_{0}^{*})_{\lambda}} = 100 \times \frac{\Upsilon _{\beta}^{R_{0}}}{std(R_{0}^{*})_{\beta}} \approx 0.4\%. $$

(24)

Both the NFSI of the parameters and the standard deviations $std(R_{0}^{*})_{b}$, $std(R_{0}^{*})_{\lambda}$, and $std(R_{0}^{*})_{\beta}$ are means to analyzing the variations in $R_{0}$ caused by the changes in b, λ, and β. The correspondence between the functionality of these measures is the reason between the similarity in these ratios, which are shown in the Fig. 3.

Figure 3

NFSI - standard deviation ratios for $\beta ^{*}$, $\lambda ^{*}$ and $b^{*}$

4 The fractional-random model and methods

Ordinary derivatives used in (1) can be transformed to fractional derivatives for a more detailed approach. Some of the most essential definitions needed for the transition from ODEs to FRDEs are given below.

Definition 4.1

[33] Let $n = \lceil \alpha \rceil$. Then the Caputo fractional derivative operator is given as follows:

Here, $\lceil n \rceil$ is the ceiling function and denotes the smallest integer that is larger than n.

Caputo’s definition enables fractional order derivation for $n-1 < \alpha < n$, which also coincides with ordinary derivation for integer α. The reason for using Caputo’s fractional derivation definition is the fact that this definition does not need fractional ordered initial values, making it a suitable choice for the transition from the original set of initial values problems modeling the initial state of the network. A fractional model containing random parameters can be obtained from the original model for modeling random infection and mutation dynamics. The fractional-random model with random parameters (or the system of FRDEs) is given below:

Here, random variables are used for parameters concerning new infections; a random mutation rate for infected nodes $\eta ^{*}$ and random transmission rates from susceptible to exposed through infection by mutated $(\beta ^{*})$ and infected $(\lambda ^{*})$ nodes. The initial conditions are given as $S(0)=S_{0}$, $E(0)=E_{0}$, $I(0)=I_{0}$, and $M(0)=M_{0}$. Another approach for fractional-random modeling could also be used for the case when the initial values are random variables.

4.1 Methods

The fractional-random model (25) will be analyzed numerically using the forward fractional Euler method (ffEm) and fde12 algorithm (fde12.m) of MATLAB. The results from both methods will be compared with the numerical characteristics of the random model and the solutions of the original model (1) obtained in MATLAB. In order to present forward fractional Euler method, consider the following fractional initial value problem:

where $j=0,1,\ldots ,m-1$ and $m-1<\alpha <m$, $m\in \mathbb{Z}^{+}$. This Caputo type fractional IVP can be analyzed through the fractional counterpart of Euler method which approximates $\left [D_{0,t}^{-\alpha}f\left (t,u\left (t\right )\right )\right ]_{t=t_{n+1}}$ [34]. The forward fractional Euler method (ffEm) is given as follows [34, 35]:

$$ u_{n+1}=\sum _{j=0}^{m-1}{\frac{t_{n+1}^{j}}{j!}u_{0}^{(j)}}+\Delta t^{ \alpha}\sum _{j=0}^{n}{b_{j,n+1}f\left (t_{j},u_{j}\right )}, $$

and

$$ b_{j,n+1}=\frac{1}{\Gamma \left (\alpha +1\right )}\left [\left (n-j+1 \right )^{\alpha}-\left (n-j\right )^{\alpha}\right ]. $$

Note that for $\alpha =1$, this scheme reduces to the classical version of the Euler method [34].

The fde12.m function in MATLAB is a popular tool that is widely used to solve initial value problems for nonlinear fractional differential equations [36, 37]. The function is an implementation of the predictor-corrector method of Adams–Bashforth–Moulton given in [37]. This function has recently been used in many applications such as the simulation of Darcy–Forchheimer hybrid nanoliquid flow [38] and the optimal control of respiratory syncytial virus infection [39].

5 Results

The fractional-random model (25) will be simulated in MATLAB for the random parameters. The system will be analyzed numerically for N repetitions and numerical characteristics of the models will be obtained. Various orders of derivation, repetition numbers and standard deviations for the random components will be used, which will be specified for each simulation. The outline of the simulation algorithm can be given as follows:

• 1. The deterministic model with ordinary derivatives is transformed into a fractional differential equation system using Caputo fractional derivatives.

• 2. The parameters of the fractional model are transformed into random variables with normal distributions to obtain a system of fractional-random differential equations.

• 3. N samples of the fractional-random model are produced using statistics and machine learning toolbox in MATLAB to simulate the random components in the model.

• 4. The resulting fractional models are analyzed using the fde12 function and the forward fractional Euler method (ffEm) in MATLAB for the solutions.

• 5. The set of solutions, containing N repetitions for each compartment, are used to obtain the numerical characteristics of the fractional-random model.

5.1 Comparison of deterministic and fractional-random results

The original model (1) consisting of deterministic differential equations with ordinary derivatives, the fractional-random model (25), and the random model with ordinary derivatives (or (1) with $\eta ^{*}$, $\beta ^{*}$, and $\lambda ^{*}$)—which will be referred to as the random model—have been analyzed in MATLAB numerically and the solution curves have been combined in a single graph (Fig. 4).

Figure 4

Solution curves of systems (1), (25), and the random model ($S(t)$: black, $E(t)$: green, $I(t)$: blue, and $M(t)$: red curves)

The curves have been obtained by solving the deterministic system with ode45 solver in MATLAB, simulating the random model with randomized parameters over the same deterministic solver and the fractional-random model (25) with fde12 and ffEm, all with a time step of $\Delta t = 0.1$. The extreme values of the solutions are given in Table 3. The values in the table for random models are the expected values obtained from average solutions. The extreme values obtained in all of the simulations have been marked on the curves with colors matching the curves for which the extreme points belong throughout the study. Values belonging to overlapping extreme value markers can be found in the tables that contain the results for the corresponding simulation.

Table 3 Comparison of simulation results for deterministic and random models using $\sigma = 5$, $N = 2^{15}$, $\Delta t = 0.1$, and $\alpha = 0.98$ (for fde12) and $\alpha = 0.99$ (for ffEm)

Note that a fractional-order derivative of $\alpha =0.98$ was used for analyzing the fractional-model (25) with fde12 and $\alpha =0.99$ was used for ffEm. All of the random models were simulated with $N=2^{15}$ repetitions, unless stated otherwise. The random parameters were simulated as $\eta ^{*} \sim N(\eta ,\sigma _{1}^{2})$, $\beta ^{*} \sim N(\beta , \sigma _{2}^{2})$, $\lambda ^{*} \sim N(\lambda ,\sigma _{3}^{2})$ such that $\sigma _{i},\:i=1,2,3$, are defined as

$$ \eta ^{*} \sim N\left ( \eta ,\left ( \frac{\sigma \eta}{20}\right ) ^{2} \right ) , \beta ^{*} \sim N\left ( \beta ,\left ( \frac{\sigma \beta}{20}\right ) ^{2}\right ) , \lambda ^{*} \sim N \left ( \lambda ,\left ( \frac{\sigma \lambda}{20}\right ) ^{2} \right ) $$

(25)

using the statistics and machine learning toolbox in MATLAB. This enables the assignment of identical coefficients of variation (CV) to each of the random parameters and $\sigma =5$ was used for the simulation. A time interval of $t\in [0,20]$ was used for a comparison with the original study [26]. Likewise, the initial conditions were used as $S_{0} = 50/42$, $E_{0} = 7$, $I_{0} = 6$, and $M_{0} = 4$.

The similarity between the deterministic results, random simulations, and numerical characteristics of the fractional-random system can be observed in both the solution curves shown in Fig. 4 and the extrema in Table 3. It should be noted that the curves for the fractional-random model obtained with ffEm move slightly away from the deterministic and random solution curves since $\alpha =0.98$.

Results show that in all models the changes in the number of nodes belonging to each compartment show similar patterns. The number of susceptible nodes sees a slight decrease in the beginning of the process, but after obtaining its minimum at around $t=1.0$ to $t=1.2$, it keeps increasing. The rest of the nodes follow a similar pattern where all of the numbers obtain their peak values in their initial state and keep decreasing until they reach their minimum values at $t=20$.

5.2 Numerical results for the fractional-random model

The numerical characteristics of the fractional-random model will be analyzed with two fractional schemes and the results will be compared with results from both deterministic and random cases.

5.2.1 Results from fde12

The fractional-random model (25) containing $\eta ^{*} \sim N(\eta ,\sigma _{1}^{2})$, $\beta ^{*} \sim N(\beta , \sigma _{2}^{2})$, $\lambda ^{*} \sim N(\lambda ,\sigma _{3}^{2})$ is analyzed with fde12.m, and the solution curves for various orders of derivation are shown in Fig. 5.

Figure 5

Solution curves of system (25) obtained by using fde12 with various α values

The extreme values for the variables are given in Table 4. Note that model (25) was simulated with $N=2^{15}$ repetitions for each α value ($\alpha =0.99,0.90,0.85,0.80$) and $\eta ^{*} \sim N(\eta ,(\frac{5\eta}{20})^{2})$, $\beta ^{*} \sim N(\beta ,(\frac{5\beta}{20})^{2})$, and $\lambda ^{*} \sim N(\lambda ,(\frac{5\lambda}{20})^{2})$ were used for the random parameters. The step-size $\Delta t = 0.1$, time interval $t \in [0,20]$, and the initial values remain the same through the study.

Table 4 Comparison of extreme values for the compartments in (25) obtained with fde12 using $\sigma = 5$ and $\alpha =0.99,\:0.90,\:0.85,\:0.80$

Note that as α gets smaller, the timing of the minima for S increases and the timing of the maxima for E decreases. The overall behavior of the curves for all expected values $E(S(t))$, $E(E(t))$, $E(I(t))$, and $E(M(t))$ are similar for all orders of derivation, with the gradual downward shift in the increase rate of S being the most clear difference as α gets smaller. Still, the minimum and maximum values seen above are similarly obtained for all α values. At $t=10$, it is seen that $E(S(20)) = 1.14857$ for $\alpha = 0.99$, $E(S(20)) = 0.931423$ for $\alpha = 0.90$, $E(S(20)) = 0.823179$ for $\alpha = 0.85$, and $E(S(20)) = 0.728237$ for $\alpha = 0.85$.

Apart from the expected values, which substitute the approximate solutions in the deterministic model, the fractional-random model enables the analysis of the distribution of results around the expected values. Standard deviations of the results can be similarly obtained from analyzing the outcomes of $N=2^{15}$ simulations. Figure 6 shows the $\sim 99\%$ confidence intervals for the expectations of the susceptible nodes, obtained by using three standard deviations for the upper and lower ends, i.e., $[E(S(t))-3\times std(S(t)),E(S(t))+3\times std(S(t))]$. The rest of the confidence intervals can be shown similarly but are omitted for an abstract presentation.

Figure 6

Confidence intervals for the expectations in (25) obtained by using fde12 with various α

The extreme values obtained for the $\sim 99\%$ confidence intervals of the expectations of the compartments are given in Table 5.

Table 5 Extreme values for the confidence intervals of the expectations of compartments in (25) obtained with fde12 using $\sigma = 5$ and $\alpha =0.99,\:0.90,\:0.85,\:0.80$

The upper limits for the confidence intervals are shown with blue dash-dotted curves and the lower limits with green dotted curves, whereas the solid black lines refer to the expected value for each case. It is seen that the gap between two ends increases temporarily for $\alpha = 0.99$, whereas this increase is permanent for the other orders of derivation.

In particular, the expected value for $\alpha =0.85$ at $t=20$ is found as 0.823179, whereas three standard deviation confidence interval of the expectation indicates that $S(t)$ has an approximately 99% probability of being anywhere between $(0.67114,0.975219)$. This interval is found as $(0.808937,1.05391)$ for $\alpha = 0.90$ and $(0.55952,0.896955)$ for $\alpha =0.80$. This is an important result for the randomness of the system. For instance, the simulation with $\alpha =0.85$, $\sigma =5$, $N=2^{15}$ indicates that the deterministic system, which neglects the random nature of the system, could be approximately 18% misleading. This is an important marker which underlines the advantages of using a random system. The random system provides expectations, which substitute deterministic solution curves, in addition to providing the ability to analyze the variability in the results.

Results show that as the number of susceptible nodes is increasing with a growing variability on their numbers, α gets smaller and $t \rightarrow 20$. The variability in the number of exposed, infected, and mutated nodes is at a lower level, which will be discussed in the following sections.

5.2.2 Results from ffEm

Note that the expected values of the compartments obtained with forward fractional Euler method (ffEm) is almost identical to the results obtained with fde12. Hence, the graphs of the expectations will be omitted for a concise presentation. The values for the extrema will be used to interpret the differences instead.

The minimum and maximum points for the expected values of the solutions of the fractional-random model are given in Table 6 with their timing. The results have been obtained for $\sigma =5$, $N=2^{15}$, and $\Delta t = 0.1$ for this section.

Table 6 Comparison of extreme values for the compartments in (25) obtained with ffEm using $\sigma = 5$ and $\alpha =0.99,\:0.90,\:0.85,\:0.80$

Although there is a general correspondence between the results from fde12 and ffEm, some values—such as the maxima of $E(t)$ for various α—differ from the results in the previous section. fde12.m method is based on the predictor-corrector algorithm of Adams–Bashforth–Moulton and, compared to the Euler method, is expected to provide higher accuracy. This fact should be taken into consideration when analyzing the difference between the extrema for both methods.

The variability in the random results can also be investigated for ffEm. The minimum and maximum values of the expected values $E(S(t))$, $E(E(t))$, $E(I(t))$, and $E(M(t))$ within three standard deviations are given in Table 7 with their timing.

Table 7 Extreme values for the confidence intervals of the expectations of compartments in (25) obtained with ffEm using $\sigma = 5$ and $\alpha =0.99,\:0.90,\:0.85,\:0.80$

Once again, for $S(t)$, the expected value of the compartment with $\alpha = 0.85$ is obtained as 0.825713, whereas this simulations suggest that $S(t)$ has an approximately 99% probability of being within the interval $(0.67526,0.976166)$. This interval is found as $(0.812006,1.05469)$ for $\alpha = 0.90$ and $(0.561995,0.900275)$ for $\alpha =0.80$. Similarly, the simulation with $\alpha =0.85$, $\sigma =5$, $N=2^{15}$ indicates around 18% potentially misleading results from the deterministic results for arbitrary realizations the event.

Results for the extreme values and confidence intervals show that the changes in the number of nodes in each compartment can be monitored similarly using each of the fractional methods.

5.2.3 Relative standard deviation comparison

The coefficients of variations (CV), or the relative standard deviations, are one of the most practical statistics to investigate the variability of the results obtained with fde12 and ffEm. Once again, the solution curves for both methods are identical, and hence only the curves for fde12 method are given (Fig. 7).

Figure 7

Confidence intervals of the compartments in (25) obtained by using fde12 with various α

The maximum values of the coefficients of variation obtained with both methods for $\alpha =0.99$, $\alpha = 0.90$, $\alpha = 0.85$, and $\alpha =0.80$, $\sigma =5$, $N=2^{15}$, and $\Delta t =0.1$ are given in Table 8. Note that all of the minimum values for the coefficients of variation are 0% since no randomness is defined for the initial conditions, resulting in zero initial deviation.

Table 8 Maximum values for the coefficients of variation for the compartments in (25) obtained with ffEm and ffEm using $\sigma = 5$ and $\alpha =0.99,\:0.90,\:0.85,\:0.80$

The CV are seen to show a steady decrease as the order of derivation decreases, with the exception of $S(t)$ for $\alpha \in [0.8,0.9]$. Both the overall trend and extrema timing for $E(t)$, $I(t)$, and $M(t)$ are almost identical for both methods and α values. It is seen that the results for $S(t)$ show a significantly larger amount or variability compared to the other compartments. The equations for S and E both contain two random components, whereas I and M contain only one random variable. However, relative standard deviation analysis shows that there is around three times higher variability in the results for $S(t)$, meaning that the deterministic results for this compartment are less reliable.

This situation can be further observed by increasing the standard deviation value σ in the simulations. The extremum values for the compartments (Table 9) and the maximum values of the variation coefficients (Table 10) are given above for $\sigma =5,\:10$, $\alpha =0.95$, and $N=2^{15},\:2^{16},\:2^{17}$ using fde12 and ffEm methods for the fractional-random model and the simulation of fractional model.

Table 9 Extreme values for the expectations of compartments in (a) the random model with $\sigma =5$, $N=2^{15}$; (b) the fractional-random model obtained with fde12 with $\alpha =0.95$, $\sigma =5$, $N=2^{16}$; (c) the fractional-random model obtained with fde12 with $\alpha =0.95$, $\sigma =10$, $N=2^{17}$; and (d) the fractional-random model obtained with ffEm with $\alpha =0.95$, $\sigma =10$, $N=2^{17}$

Table 10 Maximum values for the coefficients of variation for the compartments in (a) the random model with $\sigma =5$, $N=2^{15}$; (b) the fractional-random model obtained with fde12 with $\alpha =0.95$, $\sigma =5$, $N=2^{16}$; (c) the fractional-random model obtained with fde12 with $\alpha =0.95$, $\sigma =10$, $N=2^{17}$; and (d) the fractional-random model obtained with ffEm with $\alpha =0.95$, $\sigma =10$, $N=2^{17}$

The results show that for the coefficients of variation, as the standard deviation factor σ increases, the coefficient of variation for $S(t)$ increases significantly more than the other compartments. The results given in Table 8 contain the maximal CV for $\sigma =5$ and the maximum values in Table 10 shows that although the standard deviation of the parameters are doubled, the coefficient of variation increases between 3 to 5 times. This is another indicator of the random nature of the change in susceptible node size, which is neglected in the deterministic model. Hence, an overall summary can be given for the results of the CV by underlining the fact that the number of susceptible nodes is more sensitive to changes in spread dynamics represented by the parameters.

6 Discussion

This study proposes an alternative perspective to uncertainty analysis in fractional models using the example of a model of malware spread. The variability of the results in the model containing random parameters is measured by using various statistics from simulation sets. The numerical methods, total number of repetitions for the simulation, orders of derivation in the fractional models, and standard deviations of the random components are altered to perform an extensive analysis of the randomness.

The majority of the studies using tools of fractional calculus neglect the randomness of the events being modeled. The minority of these studies which do consider uncertainty focus on the use of stochastic noise and stay within the framework of stochastic calculus. In this study, the alternative approach to modeling uncertainty in systems consisting of differential equations—the use of random variables—is deployed. The study uses Caputo derivatives for the fractional model construction and fde12 and forward fractional Euler methods (ffEm) for the numerical analysis of the system. Since the main focus is the analysis of randomness, the methods and definition of fractional derivation is kept at a minimum level; however, the fractional-random system can easily be expanded for a more detailed analysis of the fractional dynamics.

As an alternative to studies such as [40, 41] by Omar et al. and Ali et al., where stochastic noise is utilized in fractional equation systems to model uncertainty in real-life phenomena, this model utilizes the use of random variables. The use of stochastic variables is generally preferred in studies focusing on the stability and optimal control of models. The use of random variables enables the analysis of numerous numerical characteristics of random solutions such as expected values, standard deviations, skewness, kurtosis, central moments, etc. This study presents results for the expected values, relative standard deviations, and confidence intervals of the expected values for a concise presentation; however, the results of the simulations can be expanded greatly to offer the interpretation of the aforementioned statistics. This study shows that while the fractional-random model offers the expected values results (that are alternatives to the solution curves of the ordinary-deterministic model), the randomness of the virus transmission presented with the relative standard deviations is at a significant level. Using $\alpha =0.99$, $\sigma =5$, $N=2^{15}$, and $\Delta t =0.1$ reveals a maximal relative standard deviation of $\sim 29\%$ for $S(t)$ with forward fractional Euler method (ffEm). The variability can also be seen in the confidence intervals where the expected value of $S(t)$ is found to be 0.825713 with $\alpha =0.85$, $\sigma =5$, $N=2^{15}$, and $\Delta t =0.1$, whereas this value is seen to be within the interval $(0.67526,0.976166)$ with approximately 99% probability. An overall conclusion for the variability results states that the use of FRDEs enables an effortless analysis of uncertainty compared to SDEs.

The literature on sensitivity analysis of mathematical models consisting of fractional/ordinary differential equations systems, such as [42, 43], is generally carried out together with stability and optimal control investigations. In this study, random characteristics of the parameters are interpreted in relation to the normalized forward sensitivity index (NFSI) of these parameters. The standard deviation of the basic reproduction number ($R_{0}$) containing the random parameter of a parameter p and the NFSI of p is shown to follow the same pattern, as seen in Fig. 3. Both the standard deviation of the randomized $R_{0}$ and the NFSI measure the changes in transmission as a result of the changes in the parameters, and this study presents a novel use of NFSI to highlight this resemblance. The ratios of NFSI and standard deviations for the random parameters $b^{*}$, $\lambda ^{*}$, and $\beta ^{*}$, i.e., $\Upsilon _{b}^{R_{0}}/std(R_{0}^{*})_{b}$, $\Upsilon _{\lambda}^{R_{0}}/std(R_{0}^{*})_{ \lambda}$, $\Upsilon _{\beta}^{R_{0}}/std(R_{0}^{*})_{\beta}$, are all found to be around 0.004 showing the similarity of using these measure to analyze the variability in transmission dynamics resulting from the changes in these parameters.

In contrast to stochastic or fractional systems for the modeling of computer viruses such as [20, 23, 25], this study utilizes the randomization of the parameters of an ordinary-deterministic system given by Ren et al. [26]. Numerical simulations of the fractional-random system enables the analysis the effects of changes in model dynamics on the results. Key components for the simulation of the fractional-random system such as order of derivation, standard deviation of random parameters, number of repetitions used for simulations and the numerical method used, are altered to present an overall numerical investigation, in contrast to the popular tendency in the literature to study other aspects of the models such as the effects of using various definitions of fractional derivation or stochastic noise [44, 45]. Results show that the use of fde12 method, which is based on the predictor-corrector approach, produces results that are closer to their deterministic counterparts. Increasing the standard deviation of parameters increases the length of all confidence intervals, as expected; however, this increase is significantly greater for susceptible nodes. Decreasing the order of derivation within $\alpha \in [0.99,0.8]$ affects the change in the number of susceptible nodes more than the other compartments, where the decrease in the slope of the solution curve after its minimal point around $t=1.3$ is clearly smaller as $\alpha \rightarrow 0.8$.

These results can be extended for other values for the aforementioned simulation dynamics, and it can be said that a fractional-random analysis provides a more generalized point of view compared to the ordinary-deterministic model. The variability for the number of susceptible nodes is further shown using simulation results for increased standard deviations $\sigma = 10$ and repetitions $N = 2^{17}$ using both methods. In this study, the focus was to address the possible variability of the results occurring as a result of the randomness in transmission dynamics. Most of the literature on fraction models, such as [20, 46], presents solution curves for various orders of derivation for numerical investigation. However, this study presents an extensive random simulation section to analyze the uncertainty in the spread.

7 Conclusion

In this study, an existing ordinary-deterministic model of malware spread in networks under mutation is analyzed with fractional derivatives and random parameters. The fractional-random equation system provides a more generalized modeling approach where the fractional derivatives enable the analysis of memory effects and the random parameters enable the analysis of the variability in the results. In this regard, the parameters concerning new infections and mutations in the network are transformed into normally distributed random variables. The resulting model is analyzed by using the predictor-corrector method based fde12 algorithm and the fractional forward Euler method. The numerical solutions are repeated $N=2^{15}$ to $N=2^{17}$ times with various orders of derivation such as $\alpha \in \{0.85,\:0.90,\:0.95,\:0.98,\:0.99\}$ and with $\sigma =5$ or $\sigma =10$ for the random parameters to obtain numerical characteristics of the spread.

The fractional-random model provides expected values of the compartments, which act as alternatives to the solutions of a deterministic model. However, in addition, the fractional-random model also provides results to monitor the variability of the spread dynamics. The random parameters $\eta ^{\ast}$, $\beta ^{\ast}$, and $\lambda ^{\ast}$ are defined to have matching coefficients of variation each, but the simulations show that the results for the spread dynamics have significant CV, especially for the number of susceptible nodes, meaning that the variability of the results should not be neglected. The ordinary deterministic model suggests that after $t=20$ units of time all the nodes in the network would be susceptible, and while the fractional-random model provides the same expectation, variability analysis shows a potential 18% deviation. Sensitivity analysis of the basic reproduction number to changes in the parameters is handled in relation to the deviation of the random parameters. Results show that both analyses provide compatible results for the variability in spread dynamics caused by randomness in parameters.

Many other numerical characteristics for the fractional-random model could be analyzed; however, a summarizing study is aimed to provide the reader a fundamental approach to using random variables within the fractional framework. Numerical investigation shows that the number of susceptible nodes is more affected by changes in transmission dynamics. This approach could be generalized to many more modeling problems and provide useful results for the investigation of real-life systems. Comparison of all model outputs emphasizes that the use of FRDEs provides a more generalized analysis and can be used to obtain extensive numerical characteristics to investigate the random nature of the systems being studied.

The similarity between the results for the approximate expected values of the fractional-random model and the numerical solution of the ordinary-deterministic model is a key indicator for the future implementations of this methodology. The 18% deviation in the results of the randomized model suggests that a similar randomization approach could be used to investigate the variability of solutions in various models and systems from different areas. The variability of spread dynamics such as the number of infections for an infectious disease, the alterations in the amount of pollution spreading in a system of lakes, and the deviations in a financial model can be modeled more effectively by the use of a random model. The alignment of the results for the normalized forward sensitivity indices and the standard deviations could provide an alternative approach to the investigation of uncertainty in systems that are susceptible to effects of external factors. The use of other fractional derivative definitions, other numerical and approximate-analytical methods, and different probability distributions for the random parameters is also an idea for future investigations. The use of stochastic processes to model the uncertainty or the investigation of other numerical characteristics in the simulation are other means for a more extensive random analysis. Considering that most of the studies on fractional calculus are carried out on a deterministic level, the findings and methods in this study could provide the basis for future studies that focus on the mentioned investigations.