Modeling the interactions of respiratory disease co-infection dynamics with fuzzy parameters via an ABC fractional-order approach

1 Introduction

Multi-infection respiratory diseases pose challenges owing to their complex transmission and epidemiological uncertainty. Mathematical modeling remains vital for analyzing, forecasting, and solving problems by connecting theories and real events. Mathematical models are crucial for predicting the behavior of viral pandemic diseases, as they help to comprehend the spread of disease and create plans to enhance public health. Because of interesting findings from such models in various fields, researchers continue to show significant interest in this area. Several models have been developed to analyze and explain the transmission of these diseases. A specific epidemiological model dealing with the process of multiple infections spreading in a host population is the co-infection model. It is useful in studying the interaction between the various pathogens, the dynamics of diseases in their coexistence, and the formulation of effective control policies. The necessity of mathematical modeling to describe co-infections of infectious diseases was highlighted by Inayaturohmat et al. [1]. Similarly, co-infection modeling, reproduction numbers, and an integrated approach to control were shown to be effective by Appiah et al. [2] and Tchoumi et al. [3].

Traditional mathematical models often fail to capture memory-dependent disease dynamics. Interestingly, fractional calculus is applied to describe the memory and inheritance of any physical phenomenon. This has led to its current widespread application across a variety of domains, including image processing, economics, mathematical biology, fluid dynamics, quantum physics, control and signal theory, and other fields beyond integer calculus. Kimeu [4] emphasized the importance of fractional calculus as a generalized method for the mathematical modeling of memory-based systems. Okundalaye et al. [5] and Shanta and Akbar [6] employed fractional-order susceptible–exposed–infectious–recovered (SEIR) models for the transmission of COVID-19 and Ebola, respectively, further proving the value of fractional operators in such investigations.

The Caputo derivative, one of the various definitions of fractional derivatives, has been widely used in epidemiology, engineering, and physics because it is compatible with classical initial conditions. Meenakshi et al. [7] proposed a Caputo fractional-order susceptible–exposed–infectious–recovered (SEIR) model for the co-infection of COVID-19 and influenza, which provides useful insights into fractional co-infection epidemic modeling and demonstrates the utility of fractional calculus in simulating disease dynamics. Although the Caputo derivative offers some benefits, its use may be limited by its singular kernel. It has recently been suggested that the Atangana–Baleanu–Caputo (ABC) derivative, which is distinguished by a non-singular Mittag–Leffler kernel, can more precisely capture both local and global memory effects. The fractional calculus method for modeling complex real-world processes using continuous-kernel operators was introduced by Atangana [8]. Atangana and Koca [9] developed the idea of the Atangana–Baleanu fractional derivative, which has a non-singular and non-local kernel. Wali et al. [10] and Thabet et al. [11] studied the stability and reproduction number for the transmission of coronavirus disease 2019 (COVID-19) using Atangana–Baleanu–Caputo fractional models. Alshehry et al. [12] further demonstrated the application of the Atangana–Baleanu fractional derivative.

To introduce time non-locality into mathematical modeling using a fractional differential operator with a non-singular kernel, Atangana and Baleanu developed the generalized Mittag-Leffler function as a novel kernel. The analyses by Uçar et al. [13] and Deressa and Duressa [14] suggest the application of Atangana–Baleanu fractional operators in modeling the transmission dynamics. The study by Meenakshi and Renuga [15] highlights how ABC fractional derivatives can be used to model simulated biological systems.

Using fractional calculus to create co-infection models is another area of interest for researchers. The parametric values for our model were adopted from the co-infection model of influenza and SARS-CoV-2 proposed by Liang et al. [16]. Raza et al. [17], Omame et al. [18], and Liang et al. [16] studied co-infection dynamics using fractional-order and age-structured mathematical models. Kanyiri et al. [19], Ojo et al. [20], and Gumel et al. [21] investigated influenza and COVID-19 transmission dynamics using mathematical models, emphasizing the role of vaccination, behavioral measures, and combined intervention strategies for effective disease control. In particular, influenza-related parameters were obtained from Kanyiri et al. [19]. Additionally, several other parametric values were obtained from Kanyiri et al. [19], Ojo et al. [20], and Gumel et al. [21].

This study builds upon the primary result of Mahato et al. [22], who demonstrated the ability of fuzzy numbers to handle uncertainty in epidemic processes by analyzing the fuzzy parameterization of a SARS-CoV-2 epidemic model. Arif et al. [23] introduced a susceptible–infectious–recovered (SIR) fuzzy model by combining fuzzy logic with the conventional SIR model to accurately describe uncertain disease transmission factors. Allehiany et al. [24] presented a fuzzy SIR model for the transmission of COVID-19 by incorporating uncertainty in infection, recovery, and mortality rate factors. Abdy et al. [25] proposed the idea of a fuzzy COVID-19 SIR model and took into consideration the variables of vaccination, treatment, health protocol, and virus load. A fuzzy SIR epidemic model with uncertain transmission, recovery, and mortality rate parameters was developed by Barros et al. [26]. In order to address the uncertainties occurring in the mathematical formulation, the parameters are taken as fuzzy parameters. The dynamics of co-infections were studied by Nainggolan and Ansori [27], Tamilalagan et al. [28], and Ren and Xue [29] employing mathematical models that included vaccination, fractional-order modeling techniques, and control measures. In addition to effective treatments to stop the spread of infectious diseases, their research emphasized the necessity of stability and sensitivity analyses.

The stability analysis was performed at the equilibrium points for the proposed fuzzy parametric fractional-order co-infection model, and the reproduction number was computed at the equilibrium point. The method used to find the analytical solution in this study was motivated by the traditional analysis given by Adomian [30], which was considered important in understanding the decomposition approaches in solving non-linear problems. The Laplace Adomian Decomposition Method (LADM) was used to obtain the approximate solutions [31–33]. Raina et al. [34] applied an adaptive neuro-fuzzy technique to the susceptible–exposed–infectious–recovered–susceptible (SEIRS) model of HIV infections, highlighting the relevance of fuzzy-based approaches in epidemic prediction. Modibbo [35] and Kamal et al. [36] demonstrated the efficacy of fractional and fuzzy optimization techniques.

It is evident from the comparison Table 1 that previous studies have examined co-infection models, fuzzy uncertainty, or fractional dynamics separately. However, not much work has been done on an integrated strategy for the co-infection of COVID-19 and influenza that uses fractional-order dynamics and fuzzy parameter uncertainty.

Table 1. Comparative summary of existing mathematical models in relation to this study.

Bold values indicate the modelling features of the existing studies that are considered and incorporated in the mathematical model proposed in the present study.

The novelty of this study lies in the fact that, although the dynamics of infectious diseases have been extensively studied in the literature using classical and fractional models, the addition of fuzzy parameters provides significant potential to capture uncertainty in the real world more precisely. There are many opportunities for advancement in this field of study owing to the relationships among these parameters that together shape the behavior of co-infection models. Using the ABC fractional derivative's non-singular Mittag-Leffler kernel better represents the memory effects observed during disease transmission. Its advantages remain unexplored in co-infection modeling, creating an obvious methodological gap that this study will attempt to fill. As a result, the goal of this study is to develop a fuzzy-parameter co-infection epidemiological model with ABC fractional derivatives to examine the co-dynamics and provide a fresh perspective on co-infection models.

This study is organized as follows: Section 2 gives a brief idea about the background and other preliminary concepts that are necessary for the study. In Section 3, the proposed model is formulated, incorporating fuzzy parameters along with graphical representations. Section 4 focuses on the stability analysis of the proposed model. Numerical simulations and graphical representations of the model variables are presented in Section 5. Section 6 presents the results and discussion of the proposed model. Section 7 concludes the study and highlights potential avenues for future studies.

2 Preliminaries

Definition 1 [8, 9]. Atangana–Baleanu–Caputo (ABC) fractional derivative

If y (t) ϵ H ¹(0, ζ) and 𝔮^ ϵ (0, 1] is the fractional operator

D𝔮^ABC(y(t))=N(𝔮^)(1 - 𝔮^)∫0ty′(x)E𝔮^[-𝔮^(t -x)𝔮^(1 - 𝔮^) ] dx(1)

E𝔮^[-𝔮^(t -x)𝔮^(1 - 𝔮^) ] is the Mittag-Leffler function, a generalization of the exponential function.

Proposition 1 [8, 9]. The Laplace transform of Equation 1 is,

Lap{D𝔮^ABC(y(t))}(s)=Lap{N(𝔮^)(1 - 𝔮^)∫0ty′(x)E𝔮^[-𝔮^(t - x)𝔮^(1 - 𝔮^) ]dx}=N(𝔮^)(1- 𝔮^)[ s𝔮^Lap{y(t)} - s𝔮^- 1y(0)s𝔮^+𝔮^1-𝔮^].

Definition 2 [23] A fuzzy subset A of the universe set X is given by ξ_A(x) : X → [0, 1] which is called a membership function, where ξ_A(x) indicates the degree of membership of A in the fuzzy set.

Definition 3 [24] A number A = (a₁, a₂, a₃, a₄) is called a trapezoidal fuzzy number if its membership function is defined as

ξA(x)={x - a1a2 - a1 a1<x≤a2, 1 a2<x ≤ a3,a4- xa4- a3 a3<x≤ a4,0 otherwise.

Definition 4 [7]. Disease-free equilibrium (DFE): If the disease eventually dies out, the solution of the system converges asymptotically to the disease-free equilibrium.

Definition 5 [7]. The reproduction number R0 is defined as the average number of new cases of an infection caused by one typical infected individual in a population consisting of susceptible individuals only.

Theorem 1. [7]. The disease-free equilibrium (DFE) is locally asymptotically stable if R0< 1 and is unstable if R0 > 1.

3 ABC fractional-order co-infection epidemic model for respiratory diseases with fuzzy parameters

3.1 Mathematical model

In this model, the total population is divided into 12 compartments as given in Equation 2:

N= 𝕊S𝕦s(t)+ 𝔼ℭ𝕠v(t) + 𝔼ℑ𝕟𝔣(t)+𝔼ℭ𝔗(t) + 𝕀ℭ𝕠v(t) +𝕀ℑ𝔣𝔼Cv(t)+𝕀ℑ𝕟𝔣(t) +𝕀ℭ𝔗(t)+𝕀Cv𝔼ℑ𝔣(t) +ℝℭ𝕠v(t) +ℝℑ𝕟𝔣(t) + ℝℭ𝔗(t)(2)

The model variables are listed in Table 2, and the flow diagram is shown in Figure 1.

Table 2. Brief description of the variables used in the model.

In epidemiology, the virus's presence or absence is crucial for differentiating between susceptible and infected individuals. By examining the infection in each person as a result of the viral load, we account for the model's variability. We presume that every infected individual does not contribute equally to the process of disease transmission, and each person's level of contagiousness varies according to the amount of virus present. Since the likelihood of transmission is unknown, even if it rises as infected individuals grow more contagious, we assume that the number of new infections is proportionate to the number of interactions between susceptible and infected individuals [24].

Each person's viral load can be taken into account when displaying the fuzzy parameters βC𝕠v(θ), βℑ𝕟𝔣(θ), δC𝕠v(θ), δ_ℑ𝕟𝔣(θ), δ_ℭ𝔗(θ), νC𝕠v(θ),ν_ℑ𝕟𝔣(θ), ν_ℭ𝔗(θ) are considered as functions of θ.

The equation governing the fuzzy parameter-based co-infection epidemiological model using ABC fractional-order is given by

D𝔮^ABC𝕊S𝕦s(t)=πS𝕦s-(μ𝕟 +λC𝕠v(θ) + λℑ𝕟𝔣(θ))𝕊S𝕦s(t),D𝔮^ABC𝔼ℭ𝕠v(t)= λC𝕠v(θ)(𝕊S𝕦s(t)) -(μ𝕟 + σC𝕠v+λℑ𝕟𝔣(θ))𝔼ℭ𝕠v(t),D𝔮^ABC𝔼ℑ𝕟𝔣(t)=λℑ𝕟𝔣(θ)(𝕊S𝕦s(t)) -(μ𝕟+σℑ𝕟𝔣 +λC𝕠v(θ))𝔼ℑ𝕟𝔣(t),D𝔮^ABC𝔼ℭ𝔗(t) =λC𝕠v(θ)𝔼ℑ𝕟𝔣(t)+ 𝔼ℭ𝕠v(t)λℑ𝕟𝔣(θ) -(μ𝕟+σℑ𝕟𝔣 +σC𝕠v)𝔼ℭ𝔗(t),D𝔮^ABC𝕀ℭ𝕠v(t)=σC𝕠v𝔼ℭ𝕠v(t) -(μ𝕟 + δC𝕠v(θ) + εℑ𝕟𝔣λℑ𝕟𝔣+ νC𝕠v(θ))𝕀ℭ𝕠v(t),D𝔮^ABC𝕀ℑ𝕟𝔣(t) = σℑ𝕟𝔣𝔼ℑ𝕟𝔣(t) -(μ𝕟 + δℑ𝕟𝔣(θ) +εC𝕠vλC𝕠v(θ) +νC𝕠v(θ))𝕀ℑ𝕟𝔣(t),D𝔮^ABC𝕀ℑ𝔣𝔼Cv(t) =σℑ𝕟𝔣𝔼ℭ𝔗(t) + εC𝕠vλC𝕠v(θ)𝕀ℑ𝕟𝔣(t) -(μ𝕟+δℑ𝕟𝔣(θ)+σC𝕠v)𝕀ℑ𝔣𝔼Cv(t),D𝔮^ABC𝕀Cv𝔼ℑ𝔣(t)=σC𝕠v𝔼ℭ𝔗(t)+ εℑ𝕟𝔣λℑ𝕟𝔣(θ)𝕀ℭ𝕠v(t) - (μ𝕟+δC𝕠v(θ)+σℑ𝕟𝔣)𝕀Cv𝔼ℑ𝔣(t),D𝔮^ABC𝕀ℭ𝔗(t) =σC𝕠v𝕀ℑ𝔣𝔼Cv(t)+σℑ𝕟𝔣𝕀Cv𝔼ℑ𝔣(t) -(μ𝕟+δℭ𝔗(θ)+νℭ𝔗(θ))𝕀ℭ𝔗(t),D𝔮^ABCℝℭ𝕠v(t) =νC𝕠v(θ) 𝕀ℭ𝕠v(t)-μ𝕟ℝℭ𝕠v(t),D𝔮^ABCℝℑ𝕟𝔣(t) =νℑ𝕟𝔣(θ)𝕀ℑ𝕟𝔣(t)-μ𝕟ℝℑ𝕟𝔣(t),D𝔮^ABCℝℭ𝔗(t)=νℭ𝔗(θ)𝕀ℭ𝔗(t) - μ𝕟ℝℭ𝔗(t).(3)

The initial conditions are given by

𝕊S𝕦s(0)>0,𝔼ℭ𝕠v(0) ≥ 0,𝔼ℑ𝕟𝔣(0)≥0, 𝔼ℭ𝔗(0) ≥0,𝕀ℭ𝕠v(0)≥0, 𝕀ℑ𝔣𝔼Cv(0) ≥ 0, 𝕀ℑ𝕟𝔣(0) ≥ 0, 𝕀ℭ𝔗(0) ≥ 0,𝕀Cv𝔼ℑ𝔣(0) ≥ 0,ℝℭ𝕠v(0) ≥0,ℝℑ𝕟𝔣(0) ≥ 0, ℝℭ𝔗(0) ≥0.

The parametric values used in the model to illustrate qualitative dynamics are obtained from previously published articles and are given in Table 3.

Table 3. Explanation of model parameters and crisp values.

Because of their specific characteristics, the illness transmission, recovery, and death rates of the affected individuals are regarded as fuzzy numbers. These fuzzy parameters are specified below and are indicated by β(θ), ν(θ), and δ (θ), respectively. The respective forces of infection λC𝕠v and λℑ𝕟𝔣 are given as λC𝕠v=βC𝕠v𝕀ℭ𝕠vN, λ_ℑ𝕟𝔣 = βℑ𝕟𝔣𝕀ℑ𝕟𝔣N. For the proposed model, due to its complexity, we use fuzzy values for the following parameters, namely, transmission, recovery, and death rates from the affected individuals of SARS-CoV-2, influenza and co-infection. These parameters are specified below and are indicated by βC𝕠v(θ), β_ℑ𝕟𝔣(θ) νC𝕠v(θ), ν_ℑ𝕟𝔣(θ), ν_ℭ𝔗(θ), δC𝕠v(θ), δ_ℑ𝕟𝔣(θ), δ_ℭ𝔗(θ).

For the transmission rate βC𝕠v(θ), β_ℑ𝕟𝔣(θ), recovery rate νC𝕠v(θ), ν_ℑ𝕟𝔣(θ), ν_ℭ𝔗(θ), death rate δC𝕠v(θ), δ_ℑ𝕟𝔣(θ), δ_ℭ𝔗(θ), caused by SARS-CoV-2, influenza, and Co-infection, we apply trapezoidal fuzzy membership functions in contrast to the study [24], where monotonic increasing functions at some maximum value were used. The trapezoidal function allows us to capture the growth and potential reduction of epidemiological parameters with varying virus load. From a biological perspective, this is valid for the reasons listed below.

The transmission rates βC𝕠v(θ) and βℑ𝕟𝔣(θ) are extremely low for low loads, but they increase as the load increases and reach their maximum at intermediate levels. However, at very high viral loads, infected people are more likely to be bedridden, hospitalized, or seriously ill, and have fewer social connections, which reduces transmission. Therefore, a biologically realistic decline at extreme viral loads is possible. βC𝕠v(θ) and β_ℑ𝕟𝔣(θ) are defined as in Barros et al. [26],

βC𝕠v(θ) ={x - θminθk- θl θmin<x≤θk,1 θk<x≤ θl,θmax- xθmax- θl θl <x≤ θmax ,0 θ ≤θmin or θ>θmax .

Similarly, β_ℑ𝕟𝔣(θ) is defined.

Rates of recovery are νC𝕠v(θ), ν_ℑ𝕟𝔣(θ), and ν_ℭ𝔗(θ). Individuals with mild infections are likely to recover. The likelihood of recovery falls and remains at a low level as the viral load increase. The trapezoidal fall-off at the tail indicates clinical severity and a poor response to treatment because recovery is rare at very high virus loads. Fuzzy parametric recovery rates νC𝕠v(θ), ν_ℑ𝕟𝔣(θ), and ν_ℭ𝔗(θ) follow trapezoidal membership as given in Definition 3.

Death rates are δC𝕠v(θ), δ_ℑ𝕟𝔣(θ), and δ_ℭ𝔗(θ). For mild infections, the risk of death is modest, increases gradually as viral loads rise, and then reaches a critical number. Because people may often pass away before they have had the opportunity to experience the full course of the virus, there may be a decrease in the highest viral loads in some cases, which lowers the recorded death rates in epidemiological measures.

Therefore, compared to purely monotonic membership functions, trapezoidal fuzzy functions offer a more flexible and physiologically realistic framework for the expression of variability in infection, recovery, and death dynamics. Table 4 gives the trapezoidal fuzzy parametric values.

Table 4. Explanation of model parameters, crisp, and fuzzy values.

3.2 Graphical representation of trapezoidal fuzzy parameters

The fuzzification and defuzzification for the fuzzy parameters, the transmission rate βC𝕠v(θ), β_ℑ𝕟𝔣(θ), death rate δC𝕠v(θ), δℑ𝕟𝔣(θ), and δ_ℭ𝔗(θ) and recovery rate νC𝕠v(θ), ν_ℑ𝕟𝔣(θ), ν_ℭ𝔗(θ) of the diseases are shown in Figures 2, 3.

4 Stability analysis

Stability analysis of the model is carried out by analyzing the analysis for the SARS-CoV-2-only model and the influenza-only model, and then it is generalized to the co-infection model. The reproduction number [7] for the SARS-CoV-2-only-model, influenza-only model, and the co-infection model is denoted by RC𝕠v, Rℑ𝕟𝔣,and R0, respectively, are the threshold that indicates which disease will dominate and are evaluated by employing the next-generation matrix method (NGM).

4.1 SARS-CoV-2-only

The equations governing SARS-CoV-2-only

D𝔮^ABCSSus(t)=πSus−(μn+λC𝕠v(θ))SSus(t)D𝔮^ABCEℭov(t)=λC𝕠v(θ)(SSus(t))−(μn+σC𝕠v)Eℭov(t),D𝔮^ABCIℭov(t)=σC𝕠vEℭov(t)−(μn+δC𝕠v(θ) +vC𝕠v(θ))Iℭov(t),D𝔮^ABCℝℭov(t)=vC𝕠v(θ)Iℭov(t)−μnℝℭov(t).(4)

Its initial conditions are given by 𝕊S𝕦s(0) > 0, 𝔼ℭ𝕠v(0) ≥ 0, 𝕀ℭ𝕠v(0)≥0, ℝℭ𝕠v(0) ≥ 0.

SARS-CoV-2 disease-free equilibrium (DFE) [7]

To identify the equilibrium point, consider each DABC𝕊𝔮^S𝕦s (t), DABC𝔼𝔮^ℭ𝕠v(t), DABC𝕀𝔮^ℭ𝕠v(t), DABCℝ𝔮^ℭ𝕠v(t) equal to 0 gives,

𝕊S𝕦s(t)=πS𝕦sμ𝕟,𝔼ℭ𝕠v(t)=0, 𝕀ℭ𝕠v(t)=0,ℝℭ𝕠v(t)=0.DFEPℭ𝕠v=(πS𝕦sμ𝕟,0,0, 0).

*Non-negativity* [7, 32]

Based on the system's dynamics, the system of equations R+4 = {x(t) ∈ R⁴, x ≥ 0}, x(t) = ( 𝕊S𝕦s(t), 𝔼ℭ𝕠v(t),𝕀ℭ𝕠v(t), ℝℭ𝕠v(t))T preserves positivity.

Theorem 2. [9] The SARS-CoV-2 sub-model initial value problem (Equation 4) has a unique solution that remains in R⁴, x ≥ 0.

*Reproduction number of SARS-CoV-2* [7]

Using NGM, the Jacobian of new infection terms 𝔛C𝕠v and transfer terms 𝔜C𝕠v are calculated and are given by

𝔛C𝕠v =[0βC𝕠v(θ)00] and𝔜C𝕠v=[σC𝕠v+μ𝕟0-σC𝕠vδC𝕠v(θ)+μ𝕟+νC𝕠v(θ)],

GC𝕠v= 𝔛C𝕠v𝔜C𝕠v-1

GC𝕠v=[βC𝕠v(θ)σC𝕠v(σC𝕠v + μ𝕟)(δC𝕠v(θ) + μ𝕟 + νC𝕠v(θ))βC𝕠v(θ)(δC𝕠v(θ) + μ𝕟 + νC𝕠v(θ))00],

The highest eigenvalue of GC𝕠v is RC𝕠v(θ) =βC𝕠v(θ)σC𝕠v(σC𝕠v+ μ𝕟)(δC𝕠v(θ) + μ𝕟 + νC𝕠v(θ)), the reproduction number of SARS-CoV-2, and is denoted as RC𝕠v(θ), which was calculated for each fuzzy parametric value (βC𝕠v(θ), δC𝕠v(θ), νC𝕠v(θ)) and is given by (1.779346, 1.779636, 1.780135, and 1.780350). Also, the reproduction number of RC𝕠v for the crisp parameters was 1.7799. Table 5 provides the comparison of the reproduction number of the COVID-19-only model for crisp and fuzzy parameters using the adopted percentage-based uncertainty (±5% for the inner bounds and ±10% for the outer bounds) around each crisp value.

Table 5. Comparison table of the reproduction number of COVID-19.

4.2 Influenza only

The equations governing influenza only

D𝔮^ABC𝕊S𝕦s(t)=πS𝕦s -(μ𝕟 +λℑ𝕟𝔣(θ)) 𝕊S𝕦s(t),D𝔮^ABC𝔼ℑ𝕟𝔣(t)=λℑ𝕟𝔣(θ)(𝕊S𝕦s(t)) -(μ𝕟 +σℑ𝕟𝔣)𝔼ℑ𝕟𝔣(t),D𝔮^ABC𝕀ℑ𝕟𝔣(t) = σℑ𝕟𝔣𝔼ℑ𝕟𝔣(t) -(μ𝕟 +δℑ𝕟𝔣(θ))𝕀ℑ𝕟𝔣(t),D𝔮^ABCℝℑ𝕟𝔣(t) = νℑ𝕟𝔣(θ)𝕀ℑ𝕟𝔣(t)-μ𝕟ℝℑ𝕟𝔣(t).(5)

The initial conditions are given by

𝕊S𝕦s(0)>0,𝔼ℑ𝕟𝔣(0)≥0,𝕀ℑ𝕟𝔣(0)≥0,ℝℑ𝕟𝔣(0)≥0.

*Influenza disease-free equilibrium(DFE)***

To identify the equilibrium point, consider each DABC𝕊𝔮^S𝕦s (t), DABC𝔼𝔮^ℑ𝕟𝔣 (t), DABC𝕀𝔮^ℑ𝕟𝔣 (t), and DABCℝ𝔮^ℑ𝕟𝔣 (t) are equal to zero.

*DFE* Pℑ𝕟𝔣 = (πS𝕦sμ𝕟, 0, 0, 0).

*Non-negativity* [7, 32]

Based on the system's dynamics of the system of Equations 5 R+4 = {x(t) ∈ R⁴, x ≥ 0}, x(t) = (𝕊S𝕦𝓈 (t), 𝔼ℑ𝕟𝔣(t), 𝕀ℑ𝕟𝔣(t), ℝℑ𝕟𝔣(t))T preserves positivity.

Theorem 3. [9] The influenza sub-model initial value problem (Equation 5) has a unique solution that remains in R⁴, x ≥ 0.

*Reproduction number of influenza* [7]

Using NGM, the Jacobian of new infection terms 𝔛_ℑ𝕟𝔣 and transfer terms 𝔜_{ℑ𝕟 𝔣} are calculated and are given by

𝔛ℑ𝕟𝔣 =[0βℑ𝕟𝔣(θ)00] and𝔜ℑ𝕟 𝔣 =[σℑ𝕟𝔣+μ𝕟0-σℑ𝕟𝔣δℑ𝕟𝔣(θ)+μ𝕟+νℑ𝕟𝔣(θ)],

Gℑ𝕟𝔣= 𝔛ℑ𝕟𝔣𝔜ℑ𝕟𝔣- 1,

Gℑ𝕟𝔣=[βℑ𝕟𝔣 (θ)σℑ𝕟𝔣(σℑ𝕟𝔣 + μ𝕟)(δℑ𝕟𝔣(θ) + μ𝕟 + νℑ𝕟𝔣(θ))&βℑ𝕟𝔣 (θ)(δℑ𝕟𝔣(θ) + μ𝕟 + νℑ𝕟𝔣(θ))0&0].

Highest eigenvalues of Gℑ𝕟𝔣 is Rℑ𝕟𝔣(θ)=βℑ𝕟𝔣 (θ)σℑ𝕟𝔣(σℑ𝕟𝔣 + μ𝕟)(δℑ𝕟𝔣(θ) + μ𝕟 +νℑ𝕟𝔣(θ))

is the reproduction number of the influenza sub-model and is denoted as Rℑ𝕟𝔣(θ).

Highest eigenvalues of Gℑ𝕟𝔣 is Rℑ𝕟𝔣(θ) for each (β_ℑ𝕟𝔣 (θ), δ_ℑ𝕟𝔣(θ), ν_ℑ𝕟𝔣(θ)) is given by

(0.004848, 0.004848, 0.004849, and 0.004850). Also, the reproduction number of Rℑ𝕟𝔣 for the crisp parameters was 0.004849. Table 6 provides the comparison of the reproduction number for crisp and fuzzy parameters of the influenza-only model using the adopted percentage-based uncertainty (±5% for the inner bounds and ±10% for the outer bounds) around each crisp value.

Table 6. Comparison table of the reproduction number of influenza.

4.3 Stability and comparative analysis of the reproduction number of the SARS-CoV-2 sub-model and influenza sub-model

4.3.1 Stability analysis

The stability study indicates whether the disease-free state is stable, and the comparison of reproduction numbers shows how easily the two diseases spread. When the reproduction number is less than one, the disease-free state is stable. The disease may continue to exist in the population if it is higher than 1 [7]. Figures 4, 5 show a quick comparison of the SARS-CoV-2 sub-model and influenza sub-model.

4.3.2 Uncertainty and statistical analysis

The fluctuation under parameter uncertainty was examined for the basic fuzzy-based reproduction number using statistical analysis. The mean, standard deviation, and confidence interval of the fuzzy-based reproduction number are presented for both respiratory diseases and expressed in Figure 6 to assess the stability findings degree of confidence, and robustness.

*SARS-CoV-2 sub-model and influenza sub-model*

4.4 Co-infection model

Equation 3 governs the co-infection model.

*Disease-free equilibrium(DFE)***

To identify the equilibrium point, consider each

D𝔮^ABC𝕊S𝕦s(t),D𝔮^ABC𝔼ℭ𝕠v(t),D𝔮^ABC𝔼ℭ𝔗(t),D𝔮^ABC𝕀ℭ𝕠v(t),D𝔮^ABCℝℭ𝕠v(t),D𝔮^ABC𝕀ℑ𝔣𝔼Cv(t),D𝔮^ABC 𝕀Cv𝔼ℑ𝔣(t),D𝔮^ABC 𝔼ℑ𝕟𝔣(t),D𝔮^ABC𝕀ℑ𝕟𝔣(t),D𝔮^ABC ℝℭ𝔗(t),ABCD𝔮^ℝℑ𝕟𝔣(t),D𝔮^ABC𝕀ℭ𝔗(t)

equal to 0.

*DFE* P₀ = (πS𝕦sμ𝕟, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0).

*Non-negativity* [7, 32]

Based on the system's dynamics for the system of Equation 5 R+12 = {x(t) ∈ R¹², x ≥ 0},

x(t)=x( 𝕊S𝕦s(t), 𝔼ℭ𝕠v(t),𝕀ℭ𝕠v(t), ℝℭ𝕠v(t),𝔼ℑ𝕟𝔣(t),𝕀ℑ𝕟𝔣(t), ℝℑ𝕟𝔣(t),𝔼ℭ𝔗(t),ℝℭ𝔗(t),𝕀ℑ𝔣𝔼Cv(t),𝕀Cv𝔼ℑ𝔣(t),𝕀ℭ𝔗(t))T.

Theorem 4. The co-infection model initial value problem (Equation 3) has a unique solution that remains in R¹², x ≥ 0 [9].

*Threshold number* [7]

R = max {RC𝕠v,Rℑ𝕟𝔣} gives the dominant disease threshold, which indicates the disease predominates at different fuzzy parametric values (θ) subject to co-infection interactions.

5 Numerical simulations using LADM and graphical analysis

LADM is a method that combines the Laplace technique with the Adomian decomposition method. In the LADM method, the model variables are expressed as an infinite series of the form v (t) = ∑n=1∞vn(t). The Laplace transform is applied to the system of derived fractional differential equations. This provides a recursive scheme for calculating the model variables. In this section, the procedure of applying LADM in order to obtain the solution of ABC fractional-order systems of equations is shown [31–34].

Lap { D𝔮^ABC𝕊S𝕦s(t)}=Lap{πS𝕦s -(μ𝕟 +λC𝕠v(θ) +λℑ𝕟𝔣(θ))𝕊S𝕦s(t)},Lap{D𝔮^ABC𝔼ℭ𝕠v(t)}=Lap{λC𝕠v(θ)(𝕊S𝕦s(t)) -(μ𝕟 + σC𝕠v+λℑ𝕟𝔣(θ))𝔼ℭ𝕠v(t)},Lap{D𝔮^ABC𝔼ℑ𝕟𝔣(t)}=Lap{λℑ𝕟𝔣(θ)(𝕊S𝕦s(t) -(μ𝕟+σℑ𝕟𝔣 +λC𝕠v(θ))𝔼ℑ𝕟𝔣(t)},Lap{ D𝔮^ABC𝔼ℭ𝔗(t)}=Lap{λC𝕠v(θ)𝔼ℑ𝕟𝔣(t) +𝔼ℭ𝕠v(t)λℑ𝕟𝔣(θ) -(μ𝕟+σℑ𝕟𝔣 +σC𝕠v)𝔼ℭ𝔗(t)},

Lap{ D𝔮^ABC𝕀ℭ𝕠v(t)}=Lap{ σC𝕠v𝔼ℭ𝕠v(t)-(μ𝕟+δC𝕠v(θ) + εℑ𝕟𝔣λℑ𝕟𝔣 +νC𝕠v(θ))𝕀ℭ𝕠v(t)},Lap{ D𝔮^ABC𝕀ℑ𝕟𝔣(t)}=Lap{σℑ𝕟𝔣𝔼ℑ𝕟𝔣(t) -(μ𝕟 +δℑ𝕟𝔣(θ) + εC𝕠vλC𝕠v(θ) +νC𝕠v(θ))𝕀ℑ𝕟𝔣(t)},Lap { D𝔮^ABC𝕀ℑ𝔣𝔼Cv(t)}=Lap{σℑ𝕟𝔣𝔼ℭ𝔗(t) + εC𝕠vλC𝕠v(θ)𝕀ℑ𝕟𝔣(t) -(μ𝕟+δℑ𝕟𝔣(θ)+σC𝕠v)𝕀ℑ𝔣𝔼Cv},Lap{ D𝔮^ABC𝕀Cv𝔼ℑ𝔣(t)}=Lap{σC𝕠v𝔼ℭ𝔗(t) + εℑ𝕟𝔣λℑ𝕟𝔣(θ)𝕀ℭ𝕠v(t) - (μ𝕟+δC𝕠v(θ)+σℑ𝕟𝔣)𝕀Cv𝔼ℑ𝔣(t)},Lap{ D𝔮^ABC𝕀ℭ𝔗(t)}=Lap{ σC𝕠v𝕀ℑ𝔣𝔼Cv(t)+σℑ𝕟𝔣𝕀Cv𝔼ℑ𝔣(t) -(μ𝕟+δℭ𝔗(θ)+ νℭ𝔗(θ)𝕀ℭ𝔗(t)},Lap{ D𝔮^ABCℝℭ𝕠v(t)}=Lap{νC𝕠v(θ)𝕀ℭ𝕠v(t) -μ𝕟ℝℭ𝕠v(t)},Lap{ D𝔮^ABCℝℑ𝕟𝔣(t)}=Lap{νℑ𝕟𝔣(θ)𝕀ℑ𝕟𝔣(t) -μ𝕟ℝℑ𝕟𝔣(t)},Lap{ D𝔮^ABC ℝℭ𝔗(t)}=Lap{νℭ𝔗(θ)𝕀ℭ𝔗(t) - μ𝕟ℝℭ𝔗(t)}.(6)

Upon solving the system using LADM, each variable is replaced by an infinite series. Apply the Laplace and inverse Laplace transforms to obtain the first few terms of the infinite series of the model variables, 𝕊S𝕦𝓈 (t),𝔼ℭ𝕠𝓿(t), 𝔼ℑ𝕟f(t),𝔼ℭT(t), 𝕀ℭ𝕠𝓿(t),𝕀ℑf𝔼C𝓿(t), 𝕀ℑ𝕟f(t),𝕀ℭT(t),𝕀C𝓿𝔼ℑf(t),ℝℭ𝕠𝓿(t),ℝℑ𝕟f(t),ℝℭT(t), which gives good approximations. Using the approximations and using Python (Python Software Foundation, Wilmington, DE, USA) the following graphs, Figures 7(i–v)–12(i–v) are obtained for each of the above variables with different fractional powers and at fuzzy parametric values. This allows the model to better capture a range of possible outcomes as a band rather than a single one. Also, the following graphs compare the model variables at their crisp values with fuzzy parameter values.

Similarly, applying LADM by considering the first few approximations of the variables 𝕀ℭT(t), 𝕀C𝓋Eℑf(t), ℝℭo𝓋(t), ℝℑnf(t), ℝℭT(t), the graphs of the respective variables can be obtained for different fractional powers 𝔮^ with crisp and fuzzy parametric values using Python.

6 Results and discussion

By accounting for memory effects and parameter uncertainty, the proposed ABC fractional-order fuzzy model successfully captures the dynamics of COVID-19 and influenza transmission. The suggested modeling framework is validated using numerical simulations with LADM, expressing the model variables as fuzzy bands. Overall, the fuzzy formulation preserves the biological properties of epidemiological parameters while providing a realistic representation of uncertainty.

In this study, the parameters, namely transmission, recovery, and death rates of the affected individuals, are regarded as fuzzy to address the uncertainties that occur in the real situation. We have used trapezoidal fuzzy membership functions to better represent the parameters βC𝕠v(θ), βℑ𝕟f(θ), νC𝕠v(θ), νℑ𝕟f(θ), νℭT(θ), δC𝕠v(θ), δℑ𝕟f(θ), δℭT(θ). The fuzzification and defuzzification of the chosen fuzzy parameters are represented graphically for a better understanding. Also, using fuzzy membership functions, the reproduction numbers are calculated for the SARS-CoV-2-only model and influenza-only model using NGM, and then the maximum of the reproduction numbers indicates which disease dominates in the co-dynamics. The proposed model's robustness can be justified by the consistency of the computed crisp and fuzzy basic reproduction numbers. This is proven by the closeness of the defuzzified number to the crisp value. In addition, the consistency of the two values implies that the selected uncertainty intervals do not significantly affect the basic reproduction number.

6.1 Numerical and graphical interpretations

Simulations for a range of fractional-order values 𝔮^ demonstrate the numerical solutions of the suggested fuzzy fractional multi-infection epidemiological model. Also, by applying LADM, the graphical analysis of the model variables at various fractional orders and at trapezoidal fuzzy parametric values is obtained using Python. For every state variable, the solutions derived using fuzzy parametric conditions result in restricted regions that form uncertainty bands. The lower and upper fuzzy solutions, which show the range of epidemic development owing to uncertainty in the transmission parameters, are linked to these uncertainty bands. The fractional operator's memory-dependent property is demonstrated by the observation that a decrease in the fractional order delays the onset of infection. Furthermore, the fuzzy band's spread shows how sensitive the epidemic's development is to changes in the parameters, offering a realistic prediction interval instead of a deterministic approach.

7 Conclusion

This study analyzes the system dynamics of a proposed co-infection model (SARS-CoV-2 and influenza) based on the inherent structure and assumptions using the ABC fractional order under fuzzy parameters. This study depicts the graphs of the fuzzy parameters (θ) and variables to model the uncertainties that exist in the system. The stability of the model is analyzed at the equilibrium points, and the reproduction number is calculated, which emphasizes the importance of the reproduction number in the spread of the disease. The SARS-CoV-2 model and the influenza model are analyzed separately and then generalized for the co-infection model.

The model is able to demonstrate the interaction between infections. The numerical simulation by LADM provides a graphical analysis of the model variables for different fractional powers, as well as for fuzzy parametric values based on the trapezoidal fuzzy idea. A more adaptable and physiologically realistic framework for expressing heterogeneity in infection, recovery, and death dynamics is provided using trapezoidal fuzzy functions. This method proves the capability of the model to describe the main system behavior and identify the significant factors that affect the system's development. In general, the above results provide a basis for dealing with more complex situations and further insight into the problem. However, this model takes into account only a few parameters as fuzzy, owing to its complexity. Future studies can consider more parameters as fuzzy, introduce vaccination policies, and also use real data to validate the model. This study contributes a foundation for future studies and adds to fuzzy-based co-infection dynamics.

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Footnotes

Edited by: Appanah Rao Appadu, University of the Western Cape, South Africa

Reviewed by: Umar Muhammad Modibbo, Modibbo Adama University, Nigeria

Mohammad Mahdi Ershadi, Amirkabir University of Technology, Iran