Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior

Azzh Saad Alshehry; Safyan Mukhtar; Hena Saeed Khan; Rasool Shah

doi:10.1515/phys-2023-0121

Article Open Access

Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior

Azzh Saad Alshehry , Safyan Mukhtar , Hena Saeed Khan and Rasool Shah

Published/Copyright: November 15, 2023

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From the journal Open Physics Volume 21 Issue 1

Abstract

The human immunodeficiency virus, which attacks the immune system and especially targets CD4 cells that are crucial for immunological defense against infections, is the cause of the severe illness known as acquired immunodeficiency syndrome (AIDS). This condition has the potential to take a patient’s life. Understanding the dynamics of AIDS and evaluating potential methods of prevention and treatment have both significantly benefited from the use of mathematical modeling. This research article proposes a unique technique that solves a model system of differential equations representing diverse populations, such as susceptible populations, acute populations, asymptomatic populations, and symptomatic populations or populations with AIDS. The method uses an artificial neural network (ANN) to do this. A specific Caputo–Fabrizio derivative is included in the suggested method to validate the system’s stability via the use of Krassnoselskii’s and Banach’s fixed-point approach in combination with the exponential kernel. In order to solve the differential equations and get the required data, the Laplace Adomian Decomposition (LAD) technique is used. Training the ANN involves obtaining simulated data from LAD and doing it within the context of a supervised learning framework. The performance of the ANN is assessed by comparing its predicted solutions to the LAD solutions. This allows for the calculation of the average error for each of the system’s functions. This study presents a potentially useful computational tool for understanding the dynamics of AIDS and delivering important insights into the design of new prevention and treatment methods.

Keywords: fractional calculus; epidemic model; Atangana–Baleanu derivative; fixed-point theory; numerical method

1 Introduction

Mathematical modeling of biological, physical, and chemical phenomena and processes has been an ongoing concern for scientists in recent decades. We make the observation that mathematical models are effective resources for investigating a wide range of dynamic issues in the physical and life sciences. Bernoulli first proposed the idea in 1776. In contrast, McKendrick created the susceptible, infectious and recovered mode, a formal mathematical model of a three-compartment structure, in 1927. In subsequent years, mathematical modeling expanded to include infectious diseases. Various infectious diseases may now be better understood with the use of mathematical models of biological problems, and the appropriate approach for controlling or minimizing the spread of these diseases can be developed as a result. Many mathematical models have been developed to better comprehend the dynamics of infectious diseases and to guide the development of effective strategies for their containment and prevention [1–4]. These studies add significantly to a variety of scientific fields. In order to predict human immunodeficiency virus (HIV) dynamics and offer insights for more effective treatment approaches, Nazir et al. [5] used fractional-order derivatives. In order to shed insight on cultural views about mental health, Sabir and Umar [6] investigated the connection between medical theories of mental disease and stigma within the Orthodox Jewish community. Drosophila melanogaster is used as a model by Dionne and Schneider [7] to investigate infectious illnesses, providing information on the dynamics of disease transmission. Guo et al.’s [8] analyzis of multistage infectious disease models advances our knowledge of disease prevention and spread techniques. The function of the immune system is crucial to the body’s health because it must identify infections, differentiate them from the host’s healthy tissues, eliminate them, and keep track of them. Antigens are intruders or foreign substances to which the immune system reacts. Antibody proteins and cytotoxic or killer cells are often responsible for eliminating antigens. Both are white blood cell (lymphocyte) modifications that are pathogen-specific [9]. Undoubtedly, the wide variety of invading species makes it difficult for the immune system to recognize intruders. Furthermore, infections develop and adapt fast, making it difficult for the immune system to recognize and destroy them. However, viral strains may recombine to create more deadly strains, as shown in the 1918 influenza epidemic. The rapid evolution of infections such as the influenza virus necessitates the development of new countermeasures whenever they are encountered for the second time [10]. However, different defense systems have also developed, and complex creatures like humans now have defenses that can adapt to recognize certain infections over time. With adaptive immunity, the body is able to remember the particular pathogen it has previously encountered and respond more effectively the next time it comes into contact with it. Vaccines rely on an antibody’s capacity to recognize many similar antigens. Bernoulli began exploring mathematical models of infectious illnesses in 1766, and the field has since flourished [11,12].

New applications for ODENet were described in the work by Ata and Kiymaz, [13] who propose it as a replacement to ResNet blocks for tasks including time series modeling and prediction and normalization flux computation. Previous work by Lu et al. [14] and Haber and Ruthotto [15] investigated the possibility of creating new kinds of neural networks by relating the stepping methods used by ordinary differential equation (ODE) solvers to ResNet layers. To improve the ResNet design by drawing similarities with ODEs and partial differential equations (PDEs), Ruthotto and Haber [16] interpreted deep residual as nonlinear systems of PDEs and proposed a class of neural networks motivated by this interpretation. Raissi et al. [17,18] investigated data-driven methodologies for predicting dynamics guided by PDEs, and similar attempts have been undertaken to integrate machine learning techniques into conventional modeling of dynamical systems. They provide a method for learning PDEs that is based on common numerical ODE solvers and use a Gaussian prior on the dynamics and transitions between states. In contrast, the neural network-based method used by Raissi et al. [19] involves discretizing the dynamics using a multistep ODE solver technique. Some studies examine the use of ODE solver stepping approaches [20], while others investigate physics-informed networks that include knowledge about the dynamical system [21]. For forecasting unknown ODEs from data, Ayed et al. [22] presented a ResNet-based solution similar to ODENet that uses Euler’s forward algorithm as a fixed ODE solver. However, unlike ODENet, which supports various time intervals, they are limited to a single time-step due to the solver’s inflexibility.

The mathematical model of HIV is given as follows:

(1) d S d t = Λ − β 1 I 1 S N − β 2 I 2 S N − μ S , d I 1 d t = β 1 I 1 S N + β 2 I 2 S N − v I 2 − ( α + μ ) I 1 , d I 2 d t = α I 1 − ( v + γ + μ ) I 2 , d A d t = γ I 2 − ( μ + σ ) A ,

subject to the initial conditions

S ( 0 ) ≥ 0 , I 1 ( 0 ) ≥ 0 , I 2 ( 0 ) ≥ 0 , A ( 0 ) ≥ 0 ,

The following illustration is related to the given model: Here, S , I 1 , I 2 , and A shows the susceptible, acute, asymptomatic, and symptomatic or acquired immunodeficiency syndrome (AIDS) populations, respectively. Λ is the recruitment rate. β 1 and β 2 are the effective contact rates of S with I 1 and I 2 respectively, v is the treatment rate from asymptomatic to acute class, γ is the rate at which I 2 transfer to AIDS class. The natural death rate is μ , and the AIDS-associated death rate is σ .

To explain the dynamics of the spread of dengue fever, we use a Liouville-derivative framework of fractional order. In order to provide a more precise description, we followed the lead of Caputo [23–25] and maintained a constant dimension on both sides of the system. Therefore, the fractional Liouville–Caputo derivative of the AIDS system is given as:

(2) D t ϑ 0 L C S ( t ) = Λ − β 1 I 1 S N − β 2 I 2 S N − μ S , D t ϑ 0 L C I 1 ( t ) = β 1 I 1 S N + β 2 I 2 S N − v I 2 − ( α + μ ) I 1 , D t ϑ 0 L C I 2 ( t ) = α I 1 − ( v + γ + μ ) I 2 , D t ϑ 0 L C A ( t ) = γ I 2 − ( μ + σ ) A ,

subject to the initial conditions

S ( 0 ) ≥ 0 , I 1 ( 0 ) ≥ 0 , I 2 ( 0 ) ≥ 0 , A ( 0 ) ≥ 0 .

Recent years have seen increased use of fractional calculus, a powerful and well-suited technique for simulating practical situations [26]. Different branches of mathematics, engineering, biology, finance, economics, and the social sciences may all benefit from fractional calculus’ capacity to clarify complex dynamical systems with memory effects. Due to their widespread use in treating a wide variety of human health problems, several important fractional derivatives have been developed which serve to completely explain the memory effect. The fractional derivative in the Caputo meaning is one such novel formulation that has found widespread usage in simulating many kinds of application models. Examples are the models of an outbreak with two strains of the disease [27] and the blood ethanol concentration system [28]. Several mathematical models for addressing the issue of HIV infection through virus-to-cell transmission have been presented [29–34]. Synaptic transmission of the illness and its impact on the HIV pandemic were not recognized by scientists until the early 1990s [35–37]. For instance, Spouge et al. [38] offered two models for studying the dynamics of HIV infection: one describing the transfer of the virus from cell to cell and the other explaining the transmission of the virus from cell to cell. With the use of an elastic parameter regime, the research found that the two models exhibit the same asymptotic behavior. Wen and Lou [39] also analyzed a model and found that virus-to-cell infection in an HIV epidemic model is the most crucial and widespread scenario. However, as shown by Sigal et al. [40], cell-to-cell transmission is far more essential, may impact the immune system, and hence can cause the infection to fail to respond to therapy and perhaps contribute to viral resistance. Because of the significance of these models, many other researchers attempted to learn more about HIV therapy through modeling. Additional information is available in [41–43]. The provided set of references covers diverse areas of scientific research, encompassing various subjects such as epidemic modeling, chemotaxis, global stabilization, COVID-19 anxiety prevalence, transcranial stimulation for depression, and feature matching methods [44–46]. The references delve into the analysis of mathematical and biological models, mental health studies related to the COVID-19 epidemic, and innovative technological applications using convolutional neural networks [47–49]. The range of research includes both theoretical and practical aspects across different disciplines [50,51].

2 Basic definitions

Definition 1

The derivative of a function of order p that is not an integer is defined by Caputo [52] is as follows:

(3) D σ m C ψ ( ν , σ ) = 1 Γ ( m − p ) ∫ 0 σ ( σ − ϑ ) m − p − 1 ψ ( m ) ( ν , ϑ ) d ϑ , m − 1 < α ≤ m , t > 0 .

Definition 2

The expression for the non-integer Riemann integral is given by Derouich et al. [53] as follows:

(4) R σ p ψ ( ν , σ ) = 1 Γ ( p ) ∫ 0 σ ( σ − ϑ ) p − 1 ψ ( ν , ϑ ) d ϑ .

Definition 3

The Laplace transform (LT) of ψ ( ν , σ ) is presented by Syafruddin and Noorani [52] as follows:

(5) ψ ( ς , s ) = ℒ σ [ ψ ( ν , σ ) ] = ∫ 0 ∞ e − s σ ψ ( ν , τ ) d σ , s > p .

The following is the definition for the inverse LT:

(6) ψ ( ν , σ ) = ℒ σ − 1 [ ψ ( ν , s ) ] = ∫ b − i ∞ b + i ∞ e s σ ψ ( ν , s ) d s , b = Re ( s ) > b 0 .

Lemma 1

For n − 1 < k ≤ n , ζ > − 1 , ρ ≥ 0 , and ω ∈ R , we have:

D σ k σ ω = Γ ( k + 1 ) Γ ( ω − k + 1 ) σ ω − k
D σ k ρ = 0
D σ k R σ k ψ ( ν , σ ) = ψ ( ν , σ )
R σ k D σ k ψ ( ν , σ ) = ψ ( ν , σ ) − ∑ i = 0 n − 1 ∂ i ψ ( ν , 0 ) σ i i ! .

3 Propose model via artificial neural network (ANN)

The flow chart of the proposed AIDS model is shown in Figure 1. Each step contributes to the overall effect. The Laplace–Adomian decomposition (LAD) method is used to solve the problem and provide a training dataset for the ANN. With this dataset, the ANN may learn the skills it needs to make accurate predictions during training. Once the ANN has been trained using the provided data, the answer from the LAD approach is compared to the solution predicted by the ANN. By comparing outcomes, we can gauge the ANN’s ability to forecast the behavior of the AIDS differential equation model. More information on the proposed model and the procedures used to test it will be provided in subsequent parts of this investigation. We will go through the whole process from start to finish here, including everything from the LAD technique’s implementation to the ANN’s training to a side-by-side comparison of the two methods’ respective forecasts. Careful dissection of the model’s constituent parts will expose its advantages and disadvantages in handling AIDS dynamics.

Figure 1

Flow chart of the proposed model.

4 Basic definitions

In this research domain, we will define several important terms.

Definition 4

Let ϒ ∈ ℋ 1 ( a , b ) , b > a , ϱ ∈ ( 0 , 1 ) , then the given Caputo–Fabrizio fractional derivative (CFFD) is

(7) D t ϱ 0 C F ϒ ( t ) = K ( ϱ ) 1 − ϱ ∫ α t ϒ ′ ( Θ ) exp − t − Θ 1 − Θ d Θ ,

where K ( ϱ ) in Eq. (7) satisfies K ( 1 ) = K ( 0 ) = 1 . In addition, if ϒ is not in ℋ 1 ( a , b ) , then the equation becomes

D t ϱ 0 C F ϒ ( t ) = K ( ϱ ) 1 − ϱ ∫ α t ϒ ( t ) − ϒ ( Θ ) exp − t − Θ 1 − Θ d Θ .

Definition 5

Let ϱ ∈ ( 0 , 1 ] , and the integral of the function ϒ to the fractional order ϱ is

I t ϱ 0 C F ϒ ( t ) = ( 1 − ϱ ) K ( ϱ ) ϒ ( t ) + ϱ K ( ϱ ) ∫ 0 t ϒ ( Θ ) d Θ .

Lemma 2

The problem that occurs with CFFD is that

D t ϱ 0 C F ϒ ( t ) = z ( t ) , 0 < ϱ ≤ 1 , ϒ ( 0 ) = ϒ 0 , where ϒ is r e a l c o n s t a n t ,

and in other words, it is equivalent to the integral

ϒ ( t ) = ϒ 0 + 1 − ϱ K ( ϱ ) ϒ ( t ) + ϱ K ( ϱ ) ∫ 0 t ϒ ( Θ ) d Θ .

Definition 6

[52,53] The Caputo–Fabrizio fractional derivative of Laplace transform is D t ϱ 0 C F , ϱ ∈ ( 0 , 1 ] of M ( t ) is given as

L [ I t ϱ 0 C F M ( t ) ] = s L [ M ( t ) ] − M ( 0 ) s + ϱ ( 1 − s ) .

5 HIV model of fractional order: Existence and uniqueness results

We employ the theorems of Banach and Krassnoselskii to prove that there exists at least one solution to the model, i.e.,

(8) f 1 ( t , S , I 1 , I 2 , A ) = Λ − β 1 I 1 S N − β 2 I 2 S N − μ S , f 2 ( t , S , I 1 , I 2 , A ) = β 1 I 1 S N + β 2 I 2 S N − v I 2 − ( α + μ ) I 1 , f 3 ( t , S , I 1 , I 2 , A ) = α I 1 − ( v + γ + μ ) I 2 , f 4 ( t , S , I 1 , I 2 , A ) = γ I 2 − ( μ + σ ) A ,

where

S ( 0 ) = N 1 , I 1 ( 0 ) = N 2 , I 2 ( 0 ) = N 3 , A ( 0 ) = N 4 .

So our problem becomes

(9) D t ϑ 0 L C A ( t ) = f 1 ( t , S , I 1 , I 2 , A ) , D t ϑ 0 L C B ( t ) = f 2 ( t , S , I 1 , I 2 , A ) , D t ϑ 0 L C C ( t ) = f 3 ( t , S , I 1 , I 2 , A ) , D t ϑ 0 L C D ( t ) = f 4 ( t , S , I 1 , I 2 , A ) ,

where

S ( 0 ) = N 1 , I 1 ( 0 ) = N 2 , I 2 ( 0 ) = N 3 , A ( 0 ) = N 4 .

Here, we consider

ℏ ( t ) = S I 1 I 2 A , ℏ 0 = N 1 N 2 N 3 N 4 , F ( t , ℏ ( t ) ) = f 1 ( t , ℏ ( t ) ) f 2 ( t , ℏ ( t ) ) f 3 ( t , ℏ ( t ) ) f 4 ( t , ℏ ( t ) ) .

Therefore, system (9) may be written as follows:

(10) D t ϑ 0 L C ℏ t = F ( t , ℏ ( t ) ) , 0 < ϱ ≤ 1 , ℏ ( 0 ) = ℏ 0 .

The answer to Eq. (10) is provided by Lemma 1 if and only if the right side vanishes at 0, i.e.,

(11) ℏ ( t ) + ℏ 0 + X F ( t , ℏ ( t ) ) + X ̄ ∫ 0 t F ( ξ , ℏ ( Θ ) ) d Θ ,

where

X = 1 − ϱ κ ( ϱ ) and X ̄ = ϱ κ ( ϱ ) .

Let us now define the Banach space D = L [ 0 , T ] for further analyzis by defining the norm of D = L [ 0 , T ] on 0 < t ≤ T < ∞ .

∥ ℏ ∥ = t ε [ 0 , T ] sup { ∣ ℏ ( t ) ∣ : ℏ ∈ D }

Theorem

(Krassnoselski fixed-point theorem) Let D ⊂ X is a convex and closed subset and there two operators A and ℬ such that

A ℏ 1 + ℬ ℏ 2 ∈ D ;
ℬ is continuous and compact, while A is contraction;
∃ at least one fixed-point ℏ , such that A ℏ + ℬ ℏ = ℏ holds.

The following statement holds:

Let K F > 0 be a constant, then
∣ F ( t , ℏ ( t ) ) − F ( t , ℏ ¯ ( t ) ) ∣ ≤ K F ∣ ℏ − ℏ ¯ ∣ .
For the two constants C F > 0 and ℳ F > 0 , one has
∣ F ( t , ℏ ) ∣ ≤ C F ∣ ℏ ∣ + ℳ F ,

Theorem

Thanks to Theorem 4.1, the problem of Eq. (11) has at least one solution if G K F < 1 .

Proof

Let’s say we want to define the set D as the set that is both compact and closed. D = { ℏ ∈ X : ‖ ℏ ‖ ≤ r } . Let A and B are two operators, then:

(12) A ℏ ( t ) = ℏ 0 + G F ( t , ℏ ( t ) ) ℬ ℏ ( t ) = G ¯ ∫ 0 t F ( ξ , ℏ ( ξ ) ) d ξ .

For the contraction condition of A defined in Eq. (12), let ℏ , ℏ ¯ ∈ X , one has

(13) ‖ A ℏ − A ℏ ¯ ‖ = sup t ∈ [ 0 , T ] ∣ A ℏ ( t ) − A ℏ ¯ ( t ) ∣ = sup t ∈ [ 0 , T ] G ∣ F ( t , ℏ ( t ) ) − F ( t , ℏ ¯ ( t ) ) ∣ ≤ G K F [ ‖ ℏ − ℏ ¯ ‖ ] ,

thus A is contraction. For compactness of ℬ , consider

(14) ∣ ℬ ℏ ( t ) ∣ = G ¯ ∫ 0 t F ( Θ , ℏ ( Θ ) ) d Θ ≤ G ¯ ∫ 0 t ∣ F ( Θ , ℏ ( Θ ) ) ∣ d Θ .

Taking max of Eq. (14), we have

(15) ‖ ℬ ℏ ‖ ≤ G ¯ sup t ∈ [ 0 , T ] ∫ 0 t ∣ F ( Θ , ℏ ( Θ ) ) ∣ d Θ ≤ G ¯ sup t ∈ [ 0 , T ] ∫ 0 t [ C F ‖ ℏ ‖ + ℳ F ] d Θ ≤ G ¯ T ( C F r + ℳ F ) .

Then, ℬ is bounded in Eq. (15). Let the domain of t be t 1 < t 2 , we have

(16) ∣ ℬ ℏ ( t 2 ) − ℬ ℏ ( t 1 ) ∣ = G ¯ ∫ 0 t 2 F ( Θ , ℏ ) d Θ − G ¯ ∫ 0 t 1 F ( Θ , ℏ ) d Θ = G ¯ ∫ 0 t 2 F ( Θ , ℏ ) d Θ + G ¯ ∫ t 1 0 F ( Θ , ℏ ( Θ ) ) d Θ ≤ G ¯ ∫ t 1 t 2 ∣ F ( Θ , ℏ ) ∣ d Θ ≤ G ¯ ( C F r + ℳ F ) .

Upon t 2 → t 1 , the right side of Eq. (16) tends to zero. In addition, ℬ is uniformly bounded so

∣ ℬ ℏ ( t 2 ) − ℬ ℏ ( t 1 ) ∣ → 0 .

As a result, all of the assumptions of Theorem 5 apply, and the examined Model (10) has at least one solution, because ℬ is entirely continuous.□

Theorem

In light of ( ℋ 1 ) , if G ¯ ℱ ( 1 + T ) < 1 holds, then there is only one correct answer to the problem stated in Eq. (10). Therefore, there are more than one solution to the model (2).

Proof

Let P : X → X be an operator defined by

P ℏ ( t ) = ℏ 0 + G F ( t , ℏ ( t ) ) + G ¯ ∫ 0 t 1 F ( Θ , ℏ ( Θ ) ) d Θ .

Let ℏ , ℏ ¯ ∈ X , then

‖ P ( ℏ ) − P ( ℏ ¯ ) ‖ = sup t ∈ [ 0 , T ] ∣ P ( ℏ ) ( t ) − P ( ℏ ¯ ) ( t ) ∣ ≤ sup t ∈ [ 0 , T ] G ∣ F ( t , ℏ ( t ) ) − F ( t , ℏ ¯ ( t ) ) ∣ + G ¯ sup t ∈ [ 0 , T ] ∫ 0 t ∣ ( F ( Θ , ℏ ( Θ ) ) ) − F ( Θ , ( ℏ ¯ ( Θ ) ) ) ∣ d Θ ≤ G ¯ K F ‖ ℏ − ℏ ¯ ‖ + G K F T ‖ ℏ − ℏ ¯ ‖ ,

which suggests that

(17) ‖ P ( ℏ ( − P ( ℏ ¯ ) ) ) ‖ ≤ G ¯ K F ( 1 + T ) ‖ ℏ − ℏ ¯ ‖ .

Hence, problem (10) has at most one solution, which means that model (2) has one solution.□

6 Developing a generic algorithm to solve the model under consideration

Setting κ ( ϱ ) = 1 and applying LT yields the series-type solution to the issue. It is thus possible to construct the following algorithm:

(18) s ℒ [ S ( t ) ] − S ( 0 ) s + ϱ ( 1 − s ) = Λ − β 1 I 1 S N − β 2 I 2 S N − μ S s ℒ [ I 1 ( t ) ] − I 1 ( 0 ) s + ϱ ( 1 − s ) = β 1 I 1 S N + β 2 I 2 S N − v I 2 − ( α + μ ) I 1 s ℒ [ I 2 ( t ) ] − I 2 ( 0 ) s + ϱ ( 1 − s ) = [ α I 1 − ( v + γ + μ ) I 2 ] s ℒ [ A ( t ) ] − A ( 0 ) s + ϱ ( 1 − s ) = [ γ I 2 − ( μ + σ ) A ]

(19) ℒ [ S ( t ) ] = S ( 0 ) s + s + ϱ ( 1 − s ) s ℒ [ Λ − β 1 I 1 S N − β 2 I 2 S N − μ S ] ℒ [ I 1 ( t ) ] = I 1 ( 0 ) s + s + ϱ ( 1 − s ) s ℒ [ β 1 I 1 S N + β 2 I 2 S N − v I 2 − ( α + μ ) I 1 ] ℒ [ I 2 ( t ) ] = I 2 ( 0 ) s + s + ϱ ( 1 − s ) s ℒ [ α I 1 − ( v + γ + μ ) I 2 ] ℒ [ A ( t ) ] = A ( 0 ) s + s + ϱ ( 1 − s ) s ℒ [ γ I 2 − ( μ + σ ) A ] .

Using initial conditions of system (2)

(20) ℒ [ S ( t ) ] = N 1 s + s + ϱ ( 1 − s ) s ℒ [ Λ − β 1 I 1 S N − β 2 I 2 S N − μ S ] ℒ [ I 1 ( t ) ] = N 2 s + s + ϱ ( 1 − s ) s ℒ [ β 1 I 1 S N + β 2 I 2 S N − v I 2 − ( α + μ ) I 1 ] ℒ [ I 2 ( t ) ] = N 3 s + s + ϱ ( 1 − s ) s ℒ [ α I 1 − ( v + γ + μ ) I 2 ] ℒ [ A ( t ) ] = N 4 s + s + ϱ ( 1 − s ) s ℒ [ γ I 2 − ( μ + σ ) A ] .

Assume that the solution we calculate takes the form of an infinite series, as described below:

S ( t ) = ∑ n = 0 ∞ S n ( t ) , I 1 ( t ) = ∑ n = 0 ∞ ( I 1 ) n ( t ) ,

I 2 ( t ) = ∑ n = 0 ∞ ( I 2 ) n ( t ) , A ( t ) = ∑ n = 0 ∞ A n ( t ) .

The nonlinear terms I 1 S and I 2 S decompose in terms of Adomian polynomial as

I 1 ( t ) S ( t ) = ∑ n = 0 ∞ R n ( t ) ,

where

R n = 1 Ψ ( n + 1 ) d n d λ n ∑ k = 0 n λ k I 1 k ∑ k = 0 n λ k S k λ = o

n = 0 : R 0 = S 0 ( t ) ( I 1 ) 0 ( t ) n = 1 : R 1 = S 0 ( t ) ( I 1 ) 1 ( T ) + S 1 ( t ) ( I 1 ) 0 ( t ) n = 2 : R 2 = S 0 ( t ) ( I 1 ) 0 ( t ) + S 1 ( t ) ( I 1 ) 1 ( t ) + S 2 ( t ) ( I 1 ) 0 ( t ) n = 3 : R 3 = S 0 ( t ) ( I 1 ) 3 ( t ) + S 1 ( t ) ( I 1 ) 2 ( t ) + S 2 ( t ) ( I 1 ) 1 ( t ) + S 3 ( t ) ( I 1 ) 0 ( t ) n = 4 : R 4 = S 0 ( t ) ( I 1 ) 4 ( t ) + S 1 ( t ) ( I 1 ) 3 ( t ) + S 2 ( t ) ( I 1 ) 2 ( t ) + S 3 ( t ) ( I 1 ) 1 ( t ) + S 4 ( t ) ( I 1 ) 0 ( t ) ⋮ n = n : R n = S 0 ( t ) ( I 1 ) n ( t ) + S 1 ( t ) ( I 1 ) n − 1 ( t ) + ⋯ + S n − 1 ( t ) ( I 1 ) 1 ( t ) + S n ( t ) ( I 1 ) 0 ( t )

I 2 ( t ) S ( t ) = ∑ n = 0 ∞ M n ( t ) ,

where

M n = 1 Ψ ( n + 1 ) d n d λ n ∑ k = 0 n λ k I 2 k ∑ k = 0 n λ k S k λ = o

n = 0 : M 0 = S 0 ( t ) ( I 2 ) 0 ( t ) n = 1 : M 1 = S 0 ( t ) ( I 2 ) 1 ( T ) + S 1 ( t ) ( I 2 ) 0 ( t ) n = 2 : M 2 = S 0 ( t ) ( I 2 ) ( t ) 0 ( t ) + S 1 ( t ) ( I 2 ) 1 ( t ) + S 2 ( t ) ( I 2 ) 0 ( t ) n = 3 : M 3 = S 0 ( t ) ( I 2 ) 3 ( t ) + S 1 ( t ) ( I 2 ) 2 ( t ) + S 2 ( t ) ( I 2 ) 1 ( t ) + S 3 ( t ) ( I 2 ) 0 ( t ) n = 4 : M 4 = S 0 ( t ) ( I 2 ) 4 ( t ) + S 1 ( t ) ( I 2 ) 3 ( t ) + S 2 ( t ) ( I 2 ) 2 ( t ) + S 3 ( t ) ( I 2 ) 1 ( t ) + S 4 ( t ) ( I 2 ) 0 ( t ) ⋮ n = n : M n = S 0 ( t ) ( I 2 ) n ( t ) + S 1 ( t ) ( I 2 ) n − 1 ( t ) + ⋯ + S n − 1 ( t ) ( I 2 ) 1 ( t ) + S n ( t ) ( I 2 ) 0 ( t )

Considering these values, model develops

(21) ℒ ∑ k = 0 ∞ S k ( t ) = N 1 s + s + ϱ ( 1 − s ) s ℒ Λ − β 1 ∑ k = 0 ∞ R k N − β 2 ∑ k = 0 ∞ M k N − μ ∑ k = 0 ∞ S ℒ ∑ k = 0 ∞ I 1 k ( t ) = N 2 s + s + ϱ ( 1 − s ) s ℒ β 1 ∑ k = 0 ∞ R k N + β 2 ∑ k = 0 ∞ M k N − v ∑ k = 0 ∞ I 2 − ( α + μ ) ∑ k = 0 ∞ I 1 ℒ ∑ k = 0 ∞ I 2 k ( t ) = N 3 s + s + ϱ ( 1 − s ) s ℒ α ∑ k = 0 ∞ I 1 − ( v + γ + μ ) ∑ k = 0 ∞ I 2 ℒ ∑ k = 0 ∞ A k ( t ) = N 4 s + s + ϱ ( 1 − s ) s ℒ γ ∑ k = 0 ∞ I 2 − ( μ + σ ) ∑ k = 0 ∞ A

Comparing the terms of Eq. (21), the following complications arise:

Case 1. If we set n = 0 , then

(22) ℒ [ S 0 ( t ) ] = N 1 s + s + ϱ ( 1 − s ) s ℒ [ Λ ] ℒ [ ( I 1 ) 0 ( t ) ] = N 2 s ℒ [ ( I 2 ) 0 ( t ) ] = N 3 s ℒ [ A 0 ( t ) ] = N 4 s .

Taking inverse LT, we obtain

(23) S 0 ( t ) = N 1 + ( Λ ) [ 1 + ϱ ( t − 1 ) ] ( I 1 ) 0 ( t ) = N 2 ( I 2 ) 0 ( t ) = N 3 A 0 ( t ) = N 4 .

Case 2. If we set n = 1 , then

(24) ℒ [ s 1 ( t ) ] = s + ϱ ( 1 − s ) s ℒ − β 1 R 0 N − β 2 M 0 N − μ S 0 ℒ [ ( I 1 ) 1 ( t ) ] = s + ϱ ( 1 − s ) s ℒ β 1 R 0 N + β 2 M 0 N − v ( I 2 ) 0 − ( α + μ ) ( I 1 ) 0 ℒ [ ( I 2 ) 1 ( t ) ] = s + ϱ ( 1 − s ) s ℒ [ α ( I 1 ) 0 − ( v + γ + μ ) ( I 2 ) 0 ] ℒ [ A 1 ( t ) ] = s + ϱ ( 1 − s ) s ℒ [ γ ( I 2 ) 0 − ( μ + σ ) A 0 ] .

Taking inverse LT, we obtain

(25) S 1 ( t ) = − β 1 ( I 1 ) 0 S 0 N − β 2 ( I 2 ) 0 S 0 N − μ S 0 [ 1 + ϱ ( t − 1 ) ] ( I 1 ) 1 ( t ) = β 1 ( I 1 ) 0 S 0 N + β 2 ( I 2 ) 0 S 0 N − v ( I 2 ) 0 − ( α + μ ) ( I 1 ) 0 [ 1 + ϱ ( t − 1 ) ] ( I 2 ) 1 ( t ) = [ α ( I 1 ) 0 − ( v + γ + μ ) ( I 2 ) 0 ] [ 1 + ϱ ( t − 1 ) ] A 1 ( t ) = [ γ ( I 2 ) 0 − ( μ + σ ) A 0 ] [ 1 + ϱ ( t − 1 ) ] .

Case 3. If we set n = 2 , then

(26) ℒ [ S 2 ( t ) ] = s + ϱ ( 1 − s ) s ℒ − β 1 R 1 N − β 2 M 1 N − μ S 1 ℒ [ ( I 1 ) 2 ( t ) ] = s + ϱ ( 1 − s ) s ℒ β 1 R 1 N + β 2 M 1 N − v ( I 2 ) 1 − ( α + μ ) ( I 1 ) 1 ℒ [ ( I 2 ) 2 ( t ) ] = s + ϱ ( 1 − s ) s ℒ [ α ( I 1 ) 1 − ( v + γ + μ ) ( I 2 ) 1 ] ℒ [ A 2 ( t ) ] = s + ϱ ( 1 − s ) s ℒ [ γ ( I 2 ) 1 − ( μ + σ ) A 1 ] .

(27) ℒ [ S 2 ( t ) ] = s + ϱ ( 1 − s ) s ℒ − β 1 ( I 1 ) 1 S 1 N − β 2 ( I 2 ) 1 S 1 N − μ S 1 ℒ [ ( I 1 ) 2 ( t ) ] = s + ϱ ( 1 − s ) s ℒ β 1 ( I 1 ) 1 S 1 N + β 2 ( I 2 ) 1 S 1 N − v ( I 2 ) 1 − ( α + μ ) ( I 1 ) 1 ℒ [ ( I 2 ) 2 ( t ) ] = s + ϱ ( 1 − s ) s ℒ [ α ( I 1 ) 1 − ( v + γ + μ ) ( I 2 ) 1 ] ℒ [ A 2 ( t ) ] = s + ϱ ( 1 − s ) s ℒ [ γ ( I 2 ) 1 − ( μ + σ ) A 1 ] .

Taking inverse LT, we obtain

(28) S 2 ( t ) = − β 1 ( I 1 ) 0 S 0 N − β 2 ( I 2 ) 0 S 0 N − μ S 0 − β 1 β 1 ( I 1 ) 0 S 0 N + β 2 ( I 2 ) 0 S 0 N − v ( I 2 ) 0 − ( α + μ ) ( I 1 ) 0 N − β 2 ( α ( I 1 ) 0 − ( v + γ + μ ) ( I 2 ) 0 ) N − μ − β 1 ( I 1 ) 0 S 0 N − β 2 ( I 2 ) 0 S 0 N − μ S 0 1 + 2 ϱ ( t − 1 ) + ϱ 2 t 2 2 ! − 2 t + 1 ( I 1 ) 2 ( t ) = − β 1 ( I 1 ) 0 S 0 N − β 2 ( I 2 ) 0 S 0 N − μ S 0 β 1 β 1 ( I 1 ) 0 S 0 N + β 2 ( I 2 ) 0 S 0 N − v ( I 2 ) 0 − ( α + μ ) ( I 1 ) 0 N + β 2 ( α ( I 1 ) 0 − ( v + γ + μ ) ( I 2 ) 0 ) N v ( α ( I 1 ) 0 − ( v + γ + μ ) ( I 2 ) 0 ) − ( α + μ ) β 1 ( I 1 ) 0 S 0 N + β 2 ( I 2 ) 0 S 0 N − v ( I 2 ) 0 − ( α + μ ) ( I 1 ) 0 1 + 2 ϱ ( t − 1 ) + ϱ 2 t 2 2 ! − 2 t + 1 ( I 2 ) 2 ( t ) = α β 1 ( I 1 ) 0 S 0 N + β 2 ( I 2 ) 0 S 0 N − v ( I 2 ) 0 − ( α + μ ) ( I 1 ) 0 ) − ( v + γ + μ ) ( α ( I 1 ) 0 ] 1 + 2 ϱ ( t − 1 ) + ϱ 2 t 2 2 ! − 2 t + 1 A 2 ( t ) = [ γ ( α ( I 1 ) 0 − ( v + γ + μ ) ( I 2 ) 0 ) − ( μ + σ ) γ ( I 2 ) 0 ] 1 + 2 ϱ ( t − 1 ) + ϱ 2 t 2 2 ! − 2 t + 1

and so forth. To find more terms in the series solution, this method might be used. Consequently, we arrive at the solution as follows:

(29) S ( t ) = S 0 ( t ) + S 1 ( t ) + S 2 ( t ) + ⋯ I 1 ( t ) = ( I 1 ) 0 ( t ) + ( I 1 ) 1 ( t ) + ( I 1 ) 2 ( t ) + ⋯ I 2 ( t ) = ( I 2 ) 0 ( t ) + ( I 2 ) 1 ( t ) + ( I 2 ) 2 ( t ) + ⋯ A ( t ) = A 0 ( t ) + A 1 ( t ) + A 2 ( t ) + ⋯

7 ANN

A sort of machine learning model called an ANN regression model is used to predict continuous values based on input variables. It functions by altering and processing incoming data via many layers of networked nodes, also known as neurons. The ANN’s layers each apply a different mathematical operation to the data, enabling sophisticated transformations and feature extraction. As shown in Figure 2, an ANN generally consists of three primary layers: the input layer, hidden layers, and output layer. The input layer acts as the data’s primary entry point, where it is obtained and sent to the following levels. As their name implies, the hidden layers sit between the input and output layers and are in charge of carrying out computations and changing the data representation. Based on the information from the preceding levels that has been processed, the output layer finally generates the required result. The ANN regression model may successfully learn and generalize patterns in the input data by using this layered structure and proper mathematical operations, finally permitting accurate predictions of continuous values.

Figure 2

Basic architecture of ANN.

By accepting input characteristics or data and sending them to successive layers, the input layer plays a critical role in the neural network. It acts as the start of the information processing process. The hidden layers, which are found between the input and output layers, are the next thing we come across. The activation function, which is carried out by each neuron in a hidden layer, is a particular mathematical process. This process is in charge of adding nonlinearity to the neural network, which enables it to perceive and record complicated correlations in the data. The rectified linear unit (ReLU) function is used as the activation function in this study. ReLU is a popular activation function that, if the input is positive, outputs the input directly; if not, it returns zero. The neural network may create nonlinearity and effectively capture complex patterns and correlations within the data by using ReLU. If the input value is positive, ReLU returns the value; if not, it returns zero. It has been described as

f ( x ) = max ( 0 , x ) .

The output layer generates the regression model’s final output. In this case, the regression model simultaneously predicts several continuous values. In such a scenario, the output layer would consist of several neurons, and each of which would stand in for a distinct output variable. The following is a mathematical representation of the output layer function:

output i = w i * x + b i ,

where output i is the expected value for the i th output variable, x is the input from the previous layer, w i is the weight for the link between the previous layer and the i th output neuron, and b i is the bias term. The output layer contains five neurons, which give continuous predictions for each function. The parameters of ANN is shown in Table 1.

Table 1

HIV the estimated parameter values

Parameters and functions	Descriptions	Values	Resources
Λ	Recruitment rate	0.31	Assume
α	Rate at which I 1 transfer to I 2 class	0.2	Assume
v	Treatment rate from Asymptomatic to Acute	0.01	[5]
β 1	Effective contact rate of S with I 1	0.0000009	[5]
β 2	Effective contact rate of S with I 2	0.00000027	Assume
γ	Rate at which I 2 transfer to AIDS class	0.1	[52]
σ	Death rate related to AIDS	0.1	[53]
μ	Natural death rate	0.1	[52]
S 0	Initially susceptible population	300	Assume
( I 1 ) 0	Initially acute population	0.02	Assume
( I 2 ) 0	Initially asymptomatic population	0.04	[52]
A 0	Initially symptomatic	0.02	[53]

8 Results and discussion

The study of the suggested model for AIDS is the main topic of discussion in this portion of the article. The LAD solution is used to train the ANN. The performance of the ANN is analyzed by assessing a number of factors, such as its architecture, the caliber of the training data samples, the number of training epochs, convergence behavior, prediction accuracy, and the average error (AE) of the model’s predictions. In the study article, three distinct sets of epochs (250, 500, and 1,000) are used to demonstrate the performance of the ANN. The performance measures are monitored and evaluated in the article’s concluding section’s thorough performance analyzis. This study helps in the overall assessment of the performance and application of the ANN model by offering insightful information about its efficacy and dependability in forecasting AIDS dynamics.

8.1 AE

The AE of an ANN is an essential metric that provides a comprehensive evaluation of the network’s performance in predicting outcomes or estimating values. The AE represents the average discrepancy or difference between the ANN’s predicted outputs and the actual target values. It functions as a quantitative metric for assessing the precision and efficacy of the network’s predictions across multiple data points. By calculating AE, we obtain valuable insight into the ability of the ANN to generalize and make accurate predictions. Lower AE values show a higher level of precision and a closer match between predicted and actual values, whereas higher AE values indicate that the network’s predictions deviate more substantially from the ground truth. Consequently, monitoring and reducing the AE of an ANN is an essential responsibility for improving the network’s performance and ensuring accurate predictions across a variety of applications and domains. The AE is calculated as follows:

AE = mean ∣ y pred − y exact ∣ ) .

Figure 3 shows the AE for each function over different epochs.

Figure 3

AE for each function with respect to the epochs.

8.2 Loss function

A measure used to evaluate the degree of inaccuracy or dissimilarity between the network’s anticipated outputs and the actual target values included in the training dataset is known as the training loss of an ANN. It acts as a quantifiable metric that measures the network’s performance, especially on the training data, allowing for the assessment of how well the network is picking up on and adjusting to the new information. We may obtain useful information on the precision and effectiveness of the network’s predictions during training by calculating the training loss. The weights and biases of the neurons, as well as other model parameters, must be modified and optimized in order to reduce training loss and enhance network performance. As a result, by keeping an eye on and reducing the training loss, the network may progressively become better at identifying patterns, generalizing knowledge, and making precise predictions about the future. It is calculated for each by using the following equation:

Loss = 1 n * ∑ ( y true − y pred ) 2 .

The true target values are represented by y true , where n is the number of samples in the training dataset. The projected outputs of the neural network are represented by y pred . The loss function of the suggested model is shown in Figure 4 for various epochs. An increased epoch shows that the loss decreases and the model trains well.

Figure 4

Loss function of ANN.

8.3 Comparison of the prediction solution with the original solution

By comparing the prediction solution of an ANN with the solution used during training, one may further improve the performance assessment of the ANN. This method offers insightful information on how well the network captures the underlying dynamics and patterns of the provided data. In addition to adding visual interest, the graphical depiction of this comparison is a useful tool for understanding the model’s training procedure. Figures 5, 6, 7, 8 show thorough overview of the prediction solutions over 1,000 epochs for each particular function. This captivating trajectory enables us to see the neural network’s remarkable development as it harnesses the power of rising epochs to achieve more accuracy and reveal the mysteries concealed in the data.

$Figure 5 Comparison of y LAD {y}_{{\rm{LAD}}} and y AAN {y}_{{\rm{AAN}}} solution for S ( t ) S\left(t) over 1,000 epochs.$

Figure 5

Comparison of y LAD and y AAN solution for S ( t ) over 1,000 epochs.

$Figure 6 Comparison of y L A D {y}_{LAD} and y AAN {y}_{{\rm{AAN}}} solution for I 1 ( t ) {I}_{1}\left(t) over 1,000 epochs.$

Figure 6

Comparison of y L A D and y AAN solution for I 1 ( t ) over 1,000 epochs.

$Figure 7 Comparison of y L A D {y}_{LAD} and y AAN {y}_{{\rm{AAN}}} solution for I 2 ( t ) {I}_{2}\left(t) over 1,000 epochs.$

Figure 7

Comparison of y L A D and y AAN solution for I 2 ( t ) over 1,000 epochs.

$Figure 8 Comparison of y L A D {y}_{LAD} and y AAN {y}_{{\rm{AAN}}} solution for A ( t ) A\left(t) over 1,000 epochs.$

Figure 8

Comparison of y L A D and y AAN solution for A ( t ) over 1,000 epochs.

9 Conclusion

This study emphasizes the importance of employing mathematical modeling to improve our understanding of AIDS and investigate potential prevention and treatment strategies. The study introduces an innovative method for analyzing a model system comprised of differential equations that represent disparate AIDS-affected populations. This innovative method utilizes the strength of an ANN to efficiently solve the problem. To ensure the model’s stability, the researchers employ Krassnoselskii’s and Banach’s fixed-point approach, in addition to the exponential kernel and a specialized Caputo–Fabrizio derivative. Using the LAD method, the team is able to effectively solve the complex system of differential equations, thereby generating essential data for analyzis. The ANN is then trained using synthetic data derived from LAD in a supervised learning environment. The accuracy and performance of the ANN are meticulously evaluated by comparing its predicted solutions to those obtained from LAD, allowing for a comprehensive evaluation of AEs for each system operation. This study is a compelling illustration of the transformative potential of computer programs for enhancing our understanding of AIDS and providing invaluable insights for the development of innovative prevention and treatment strategies. By employing the synergistic power of mathematical modeling and artificial intelligence techniques, we can continue to make remarkable advances in our ongoing fight against AIDS, forging a healthier and sunnier future for all those afflicted by this debilitating illness.

Acknowledgments

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2023R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supportedby the Deanship of Scientific Research, the Vice Presidencyfor Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. 5004).

Funding information: Princess Nourah bint Abdulrahman University Researchers Supporting Project number(PNURSP2023R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, the Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (Grant No. 5004).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: Authors state no conflict of interest.

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Received: 2023-07-29

Revised: 2023-09-06

Accepted: 2023-09-28

Published Online: 2023-11-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2023-0121

Keywords for this article

fractional calculus; epidemic model; Atangana–Baleanu derivative; fixed-point theory; numerical method

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