Fractional insights into Zika virus transmission: Exploring preventive measures from a dynamical perspective

Rashid Jan; Normy Norfiza Abdul Razak; Salah Boulaaras; Ziad Ur Rehman

doi:10.1515/nleng-2022-0352

Article Open Access

Fractional insights into Zika virus transmission: Exploring preventive measures from a dynamical perspective

Rashid Jan , Normy Norfiza Abdul Razak , Salah Boulaaras and Ziad Ur Rehman

Published/Copyright: December 16, 2023

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From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

Mathematical models for infectious diseases can help researchers, public health officials, and policymakers to predict the course of an outbreak. We formulate an epidemic model for the transmission dynamics of Zika infection with carriers to understand the intricate progression route of the infection. In our study, we focused on the visualization of the transmission patterns of the Zika with asymptomatic carriers, using fractional calculus. For the validity of the model, we have shown that the solutions of the system are positive and bounded. Moreover, we conduct a qualitative analysis and examine the dynamical behavior of Zika dynamics. The existence and uniqueness of the solution of the system have been proved through analytic skills. We establish the necessary conditions to ensure the stability of the recommended system based on the Ulam–Hyers stability concept (UHS). Our research emphasizes the most critical factors, specifically the mosquito biting rate and the existence of asymptomatic carriers, in increasing the complexity of virus control efforts. Furthermore, we predict that the asymptomatic fraction has the ability to spread the infection to non-infected regions. Furthermore, treatment due to medication, the fractional parameter or memory index, and vaccination can serve as effective control measures in combating this viral infection.

Keywords: fractional calculus; Zika virus infection; mathematical model; vaccination and treatment; stability analysis; dynamical behavior

1 Introduction

Zika infection is a contagious illness induced by the Zika virus, primarily propagate with the bite of Aedes mosquitoes that are infected, particularly Aedes aegypti [1]. The primary method of Zika virus transmission is through mosquito bites, predominantly from Aedes species. However, the transmission of the virus can also occur through blood transfusions, sexual contact, and from a pregnant woman to her unborn child during pregnancy [1]. One of the most alarming aspects of Zika virus infection is its potential to induce birth defects in infants born to infected mothers. When a pregnant woman contracts the virus, it can be transmitted to the developing fetus, leading to severe neurological birth defects such as microcephaly. Additional abnormalities in brain development, eye malformations, hearing impairment, and stunted growth have also been linked to Zika infection during pregnancy [2]. The majority of individuals who contract the Zika virus either remain asymptomatic or experience mild symptoms that typically last for a few days to a week. Common symptoms encompass fever, rash, joint and muscle discomfort, headache, conjunctivitis, and fatigue [3]. These symptoms frequently resemble those of other diseases transmitted by mosquitoes, like chikungunya and dengue. The presence of Zika virus infection has been associated with an increased risk of Guillain-Barr syndrome (GBS), a rare neurological condition. This syndrome can lead to paralysis, muscle weakness, and, in severe instances, respiratory problems while the exact mechanism through which the Zika virus triggers GBS is not yet fully comprehended, it is believed to involve an autoimmune response [4]. Implementing preventive measures like using mosquito repellents, staying in screened areas, wearing protective clothing, and eliminating stagnant water sources where mosquitoes can breed are recommended interventions.

The applications of mathematics are vast and diverse, touching almost every aspect of our lives. Degenerate random variables and degenerate gamma functions are mathematical concepts that have been studied in recent research [5–8]. These concepts have been used in various applications, such as predicting the distribution of extreme values and studying mathematical models. Mathematical models are widely utilized to visualize the spread of infectious diseases and to provide better policies for the management of diseases [9,10]. Several epidemic models have been structured to study the progression route of Zika infection. The potential spread of the Zika virus through sexual contact was initially identified by the researchers [11]. Gao et al. [12] formulated a mathematical model that considered both sexual transmission and mosquito-borne, using epidemic data from different countries. Their study concluded that sexual transmission played a relatively minor role in Zika virus transmission. However, it was found that sexual transmission heightened the infection risk and increased the epidemic size, potentially prolonging the outbreak. He et al. conducted simulation-based study on the reported data of Zika virus infections in the State of Bahia in Brazil, Colombia, and French Polynesia [13]. By examining and contrasting the outcomes of these simulations, they acquired an enhanced understanding of the anticipated advancement of the Zika virus, strengthen our capacity to manage the spread of the epidemic, and avert potential transmissions. Baca-Carrasco et al. [14] presented three mathematical models that considered population migration, sexual transmission, and vector transmission. The collective findings from these models indicated that the incidence of endemic disease subsequent to the Zika outbreak was relatively low, with sexual transmission playing a significant role in the magnitude of the outbreak.

Agusto et al. [15] developed an epidemic model for the Zika virus that specifically considered scenarios without any disease-related deaths. After that, they extended their model with disease-induced mortality and perform stability analysis. Imran et al. [16] formulated an inclusive model that considered the vertical transmission of the Zika in both vectors and hosts. They performed an extensive examination of the dynamical behavior of the model. Dénes et al. [17] devised an intricate transmission model that distinguished between females and males regarding the method of sexual transmission. They recognized the substantial effect of sexual transmission on the dissemination of the infection. Ibrahim and Denes [18] formulated a model to examine the transmission of the Zika, considering sexually transmitted infections and asymptomatic carriers. In addition, the researchers investigated the effect of weather patterns on parameters associated with mosquitoes. In a separate study, Yuan et al. [19] introduced three modes of Zika virus transmission, including transmission between mosquitoes and humans, transmission through sexual contact, and transmission within mosquitoes through vertical transmission. The researchers evaluated the influence of each transmission pathway on the basic reproduction number. By means of computational simulations, they substantiated that sexual and vertical transmission manifested distinct influences on the early and prolonged kinetics of Zika propagation. Moreover, a preceding inquiry conducted by Busenberg and Cooke [20] delved into the modeling and dynamic scrutiny of diverse vertically transmitted illnesses. This study focuses on the analysis of the transmission dynamics of Zika infection, considering the impact of vaccination, treatment, and the memory index. In addition, our aim is to explore the role of asymptomatic carriers in both controlling and potentially amplifying the infection’s dissemination.

Fractional calculus provides a powerful mathematical tool for understanding and solving real-world problems across diverse fields [21–23]. This is due to its ability to capture the complex dynamics of natural problem that can not be accurately represented by classical one [24]. In the context of vector-borne diseases, memory plays a vital role and is associated with both the host and mosquitoes [25,26]. Fractional derivatives offer an effective means of addressing the impact of memory in these biological processes. A wide range of practical problems in biology, economics, mathematics, physics, control systems, and other scientific disciplines are successfully represented through effective modeling [27–30]. In the realm of diseases transmitted by vectors, the memory index is a crucial factor that can be effectively represented using fractional frameworks. Thus, we employ a fractional framework to characterize the dynamics of the Zika, with the objective of highlighting the significance of memory in mitigating the propagation of this viral infection.

The structure of the research work is outlined as follows: In Section 2, a comprehensive summary of the fundamental principles and results of fractional calculus is presented. Section 3 introduces an epidemic model that integrates vaccination, asymptomatic carriers, and treatment to provide a more realistic representation of Zika transmission. The analysis of the recommended system of infection is conducted in Section 4, while Section 5 establishes the conditions required for UHS. Finally, the concluding section summarizes the overall findings of the study and provides concluding remarks.

2 Fundamental theory and results

Here, we will outline the fundamental concepts and results of the fractional theory, which will serve as the basis for analyzing the proposed model. The key advantage of fractional calculus lies in its ability to incorporate the notion of memory, which plays an important part in understanding the transmission dynamics of vector-borne diseases. More specifically, the researchers directed their focus towards fractional systems due to their extensive range of applications. The basic principles are presented as follows:

Definition 2.1

[31] Consider a function ρ ( t ) with the condition ρ ( t ) ∈ L 1 ( [ m 1 , h ] , R ) and let ℵ be the fractional order, then

(1) I m 1 + m 1 ℵ ρ ( t ) = 1 Γ ( ℵ ) ∫ 0 t ( t − r ) ℵ − 1 ρ ( r ) d r

denotes the fractional integral and 0 < ℵ ≤ 1 and Γ is gamma function that serves as a bridge between the discrete world of integers and the continuous realm of fractional calculus, providing a mathematical framework to explore and understand phenomena that exhibit fractional-order behavior.

Definition 2.2

[31] Assume a function ρ ( t ) in a manner that ρ ( t ) ∈ G n [ m 1 , h ] , then

(2) D 0 + ℵ L C ρ ( t ) = 1 Γ ( n − ℵ ) ∫ 0 t ( t − r ) n − ℵ − 1 h n ( r ) d r ,

where D 0 + ℵ L C ρ ( t ) represents the Liouville-Caputo derivative.

Lemma 2.1

[31] Let ρ ( h ) be a function and consider the system

(3) D 0 + ℵ L C ρ ( t ) = v ( t ) , t ∈ [ 0 , φ ] , n − 1 < ℵ < n , ρ ( 0 ) = v 0

in which v ( t ) ∈ G ( [ 0 , φ ] ) , then

(4) ρ ( t ) = ∑ i = 0 n − 1 d i t i , d i ∈ R a n d f o r i = 0 , 1 , … , n − 1 .

Definition 2.3

[32] For the LC derivative, the Laplace transform is given by

(5) £ [ D 0 + ℵ L C ρ ( t ) ] = r ℵ ρ ( r ) − ∑ k = 0 n − 1 r ℵ − k − 1 ρ k ( 0 ) ,

with n − 1 < ℵ < n . Moreover, taking norm on Z as follows:

(6) ‖ ρ ‖ = max t ∈ [ 0 , φ ] { ∣ ρ ∣ , for all ρ ∈ Z } .

Theorem 2.1

[33] Let us consider Z as a Banach space in which D : Z → Z is both compact and continuous, then D has a fixed-point if

(7) ℐ = { ρ ∈ Z : ρ = η D ρ , η ∈ ( 0 , 1 ) }

is bounded.

3 Formulation of Zika model

In developing the model, we categorized the human population, represented by N h , into several distinct groups: susceptible individuals ( ℋ S ), individuals in the exposed stage ( ℋ E ), infected individuals ( ℋ I ), carriers ( ℋ C ), and individuals who have recovered ( ℋ R ). Similarly, the female mosquito population, denoted as N v , was categorized into susceptible ( V S ), exposed ( V E ), and infectious ( V I ) mosquitoes. We made the assumption that the natural death and birth rates remained constant for both populations, denoted as ζ h for the human population and ζ v for the mosquito population. In this formulation, the rates at which individuals transition from the susceptible class ( ℋ S ) to ℋ E are denoted by ( b β 1 N h ( ℋ I + ℋ C ) ) and ( b β 2 N h V I ). In addition, the rate of transition from the susceptible class ( V S ) to V E is represented by ( b β 3 N h ( ℋ I + ℋ C ) ). The flow of individuals from the exposed classes, ℋ E and V E , is indicated by η h and η v respectively, within the context of the transmission process.

In this formulation, we made the assumption that a fraction θ represents asymptotic carriers, and γ represents the recovery rate. The vaccination and treatment rates are denoted by v and φ respectively, while the biting rate of the vector is symbolized by b . Furthermore, the transmission probabilities are assumed to be β 1 , β 2 , and β 3 . With these considerations, the dynamics of the Zika virus infection can be described as follows:

(8) D t ℵ 0 L C ℋ S = Λ h − β 1 b N h ( ℋ I + ℋ C ) ℋ S − β 2 b N h ℋ S V I − v ℋ S − ζ h ℋ S , D t ℵ 0 L C ℋ E = β 1 b N h ( ℋ I + ℋ C ) ℋ S + β 2 b N h ℋ S V I − η h ℋ E − ζ h ℋ E , D t ℵ 0 L C ℋ I = ( 1 − θ ) η h ℋ E − ( ζ h + φ + γ ) ℋ I , D t ℵ 0 L C ℋ C = θ η h ℋ E − ( ζ h + γ ) ℋ C , D t ℵ 0 L C ℋ R = v ℋ S + γ ( ℋ I + ℋ C ) + φ ℋ I − ζ h V h , D t ℵ 0 L C V S = Λ v − β 3 b N h ( ℋ I + ℋ C ) V S − ( ζ v + c ) V S , D t ℵ 0 L C V E = β 3 b N h ( ℋ I + ℋ C ) V S − ( ζ v + c + η v ) V E , D t ℵ 0 L C V I = η v V E − ( ζ v + c ) V I ,

with

ℋ S ( 0 ) ≥ 0 , ℋ E ( 0 ) ≥ 0 , ℋ I ( 0 ) ≥ 0 , ℋ C ( 0 ) ≥ 0 , ℋ R ( 0 ) ≥ 0 ,

and

V S ( 0 ) ≥ 0 , V E ( 0 ) ≥ 0 , V I ( 0 ) ≥ 0 .

In addition, the vector size is given by

N v = V S + V E + V I ,

and the host size is given as follows:

N h = ℋ S + ℋ E + ℋ I + ℋ C + ℋ R .

In the context mentioned earlier, we use the notation 0 L C D t ℵ to represent the Liouville–Caputo operator, where the memory index is denoted as ℵ . By incorporating fractional systems, we can obtain more accurate and reliable outcomes, thanks to the non-local nature of biological processes. Furthermore, fractional systems exhibit a characteristic of heredity, which allows us to gather information about their previous and present states for future examination. Caputo’s derivative is widely acknowledged for its improved reliability and versatility in analytical applications. As a result, we utilized fractional theory to represent the dynamics of the Zika virus.

Theorem 3.1

The solutions ( ℋ S , ℋ E , ℋ I , ℋ C , ℋ R , V S , V E , V I ) of the fractional system (8) representing the Zika virus exhibit both positivity and boundedness.

Proof

To demonstrate the desired outcome, we proceed by following these steps:

(9) D t ℵ 0 L C ℋ S ∣ ℋ S = 0 = Λ h ≥ 0 , D t ℵ 0 L C ℋ E ∣ ℋ E = 0 = β 1 b N h ( ℋ I + ℋ C ) ℋ S + β 2 b N h ℋ S V I ≥ 0 , D t ℵ 0 L C ℋ I ∣ ℋ I = 0 = ( 1 − θ ) η h ℋ E ≥ 0 , D t ℵ 0 L C ℋ C ∣ ℋ C = 0 = θ η h ℋ E ≥ 0 , D t ℵ 0 L C ℋ R ∣ ℋ R = 0 = v ℋ S + γ ( ℋ I + ℋ C ) + φ ℋ I ≥ 0 , D t ℵ 0 L C V S ∣ V S = 0 = Λ v ≥ 0 , D t ℵ 0 L C V E ∣ V E = 0 = β 3 b N h ( ℋ I + ℋ C ) V S ≥ 0 , D t ℵ 0 L C V I ∣ V I = 0 = η v V E ≥ 0 .

Hence, the solutions of our fractional system (8) are ensured to be positive. To establish the boundedness of the solutions, we begin by summing all compartments of the host population, as follows:

(10) D t ℵ 0 L C ( ℋ S + ℋ E + ℋ I + ℋ C + ℋ R ) ≤ ℳ − ζ h ( ℋ S + ℋ E + ℋ I + ℋ C + ℋ R ) ,

where ℳ = Λ h . By evaluating the aforementioned inequality, we obtain that

( ( ℋ S + ℋ E + ℋ I + ℋ C + ℋ R ) ) ≤ ℋ S ( 0 ) + ℋ E ( 0 ) + ℋ I ( 0 ) + ℋ C ( 0 ) + ℋ R ( 0 ) − ℳ ζ h × E ℵ ( − ζ h t ) + ℳ ζ h .

By utilizing the Mittag-Leffler function (for instance, see the study by Kilbas et al. [31]), we obtain that

( ℋ S + ℋ E + ℋ I + ℋ C + ℋ R ) ≤ ℳ ζ h ≅ ℳ 1 .

Likewise, for the mosquitoes portion of the system (8), we derive the inequality V S + V E + V I ≤ ℳ 2 , where ℳ 2 = N ζ v . As a result, the solutions of the Zika virus system (8) are bounded and positive.

The disease-free steady state of our suggested system (8) of Zika virus infection, which is denoted by ℰ 0 ( ℋ S , ℋ E , ℋ I , ℋ C , ℋ R , V S , V E , V I ) is stated as follows:

Λ h ν + ζ h , 0 , 0 , 0 , ν Λ h ζ h ( ν + ζ h ) , Λ v ζ h + c , 0 , 0 .

In the upcoming section, we will investigate the solution of the recommended fractional model of Zika virus infection.

4 Theory of existence

In this section of the study, we will analyze the qualitative properties of the recommended system (8) of the Zika virus infection by employing the theory of existence. To accomplish this, we will proceed through the following steps:

(11) Z 1 ( t , ℋ S , ℋ E , ℋ I , ℋ C , V h , V S , V E , V I ) = Λ h − β 1 b N h ( ℋ I + ℋ C ) ℋ S − β 2 b N h ℋ S V I − v ℋ S − ζ h ℋ S , Z 2 ( t , ℋ S , ℋ E , ℋ I , ℋ C , V h , V S , V E , V I ) = β 1 b N h ( ℋ I + ℋ C ) ℋ S + β 2 b N h ℋ S V I − η h ℋ E − ζ h ℋ E , Z 3 ( t , ℋ S , ℋ E , ℋ I , ℋ C , V h , V S , V E , V I ) = ( 1 − θ ) η h ℋ E − ( ζ h + φ + γ ) ℋ I , Z 4 ( t , ℋ S , ℋ E , ℋ I , ℋ C , V h , V S , V E , V I ) = θ η h ℋ E − ( ζ h + γ ) ℋ C , Z 5 ( t , ℋ S , ℋ E , ℋ I , ℋ C , V h , V S , V E , V I ) = v ℋ S + γ ( ℋ I + ℋ C ) + φ ℋ I − ζ h ℋ R , Z 6 ( t , ℋ S , ℋ E , ℋ I , ℋ C , V h , V S , V E , V I ) = Λ v − β 3 b N h ( ℋ I + ℋ C ) V S − ( ζ v + c ) V S , Z 7 ( t , ℋ S , ℋ E , ℋ I , ℋ C , V h , V S , V E , V I ) = β 3 b N h ( ℋ I + ℋ C ) V S − ( ζ v + c + η v ) V E , Z 8 ( t , ℋ S , ℋ E , ℋ I , ℋ C , V h , V S , V E , V I ) = η v V E − ( ζ v + c ) V I .

We can also write the system (11) of Zika virus as follows:

(12) D 0 + ℵ L C Z ( t ) = J ( t , Z ( t ) ) , t ∈ [ 0 , φ ] , Z ( 0 ) = Z 0 , 0 < ℵ ≤ 1 ,

where

(13) Z ( t ) = ℋ S ( t ) , ℋ E ( t ) , ℋ I ( t ) , ℋ C ( t ) , ℋ R ( t ) , V S ( t ) , V E ( t ) , V I ( t ) . Z 0 ( t ) = ℋ S 0 , ℋ E 0 , ℋ I 0 , ℋ C 0 , ℋ R 0 , V S 0 , V E 0 , V I 0 . J ( t , Z ( t ) ) = Z 1 ( t , ℋ S , ℋ E , ℋ I , ℋ C , ℋ R , V S , V E , V I ) Z 2 ( t , ℋ S , ℋ E , ℋ I , ℋ C , ℋ R , V S , V E , V I ) Z 3 ( t , ℋ S , ℋ E , ℋ I , ℋ C , ℋ R , V S , V E , V I ) Z 4 ( t , ℋ S , ℋ E , ℋ I , ℋ C , ℋ R , V S , V E , V I ) Z 5 ( t , ℋ S , ℋ E , ℋ I , ℋ C , ℋ R , V S , V E , V I ) Z 6 ( t , ℋ S , ℋ E , ℋ I , ℋ C , ℋ R , V S , V E , V I ) Z 7 ( t , ℋ S , ℋ E , ℋ I , ℋ C , ℋ R , V S , V E , V I ) Z 8 ( t , ℋ S , ℋ E , ℋ I , ℋ C , ℋ R , V S , V E , V I ) .

By using the previously mentioned Lemma (2.1), we can express the system (12) in an equivalent integral form as follows:

(14) Z ( t ) = Z 0 ( t ) + 1 Γ ( ℵ ) ∫ 0 t ( t − r ) ℵ − 1 J ( r , Z ( r ) ) d r .

To analyze our proposed model, we apply the Lipschitz criteria outlined as follows: (C1) For ı ∈ [ 0 , 1 ) , ∃ X J , Y J in such a way that

(15) ∣ J ( t , Z ( t ) ) ∣ ≤ X Z ∣ Z ∣ ı + Y J .

(C2) We have M J > 0 , and all Z , Z ¯ ∈ Z such that

(16) ∣ J ( t , Z ) − J ( t , Z ¯ ) ∣ ≤ M J [ ∣ Z − Z ¯ ∣ ] .

Assuming a map ℬ on Z defined by

(17) ℬ Z ( t ) = Z 0 ( t ) + 1 Γ ( ℵ ) ∫ 0 t ( t − r ) ℵ − 1 J ( r , Z ( r ) ) d r .

If conditions C 1 and C 2 are satisfied, then there exists at least one solution of (12). To analyze the solution of our Zika virus system, we proceed in the following manner:

Theorem 4.1

The suggested system (8) of Zika virus possesses at least one solution when the assumptions C1 and C2 are met.

Proof

To demonstrate the intended outcome, we will make use of Schaefer’s fixed point theorem. The delineation of this theorem will be presented in four sequential stages, outlined as follows:

P1: In the first phase, we will establish the continuity of the operator ℬ . We assume that Z i for i = 1 , 2 , … , 9 is continuous, which shows that J ( t , Z ( t ) ) is also continuous. In the subsequent steps, considering Z j and Z ∈ Z such that Z j → Z , we observe that ℬ Z j → ℬ Z . In addition, let us assume

(18) ‖ ℬ Z j − ℬ Z ‖ = max t ∈ [ 0 , φ ] 1 Γ ( ℵ ) ∫ 0 t ( t − r ) ℵ − 1 Q j ( r , Z j ( r ) ) d r − 1 Γ ( ℵ ) ∫ 0 t ( t − r ) ℵ − 1 J ( r , Z ( r ) ) d s ≤ max t ∈ [ 0 , φ ] ∫ 0 t ( t − r ) ℵ − 1 Γ ( ℵ ) ∣ J j ( r , Z j ( r ) ) − J ( r , Z ( r ) ) ∣ d r ≤ φ ℵ M J Γ ( ℵ + 1 ) ‖ Z j − Z ‖ → 0 as j → ∞ .

The continuity of the expression ℬ Z j → ℬ Z is guaranteed due to the continuity of J , which in turn ensures the continuity of ℬ .

P2: In the second step, we will establish the boundedness of ℬ . Consider Z ∈ Z . The following properties hold true for the operator ℬ :

(19) ‖ ℬ Z ‖ = max t ∈ [ 0 , φ ] Z o ( t ) + 1 Γ ( ℵ ) ∫ 0 t ( t − r ) ℵ − 1 J ( r , Z ( r ) ) d r ≤ ∣ Z 0 ∣ max t ∈ [ 0 , φ ] 1 Γ ( ℵ ) ∫ 0 t ∣ ( t − r ) ℵ − 1 ‖ J ( r , Z ( r ) ) ∣ d r ≤ ∣ Z 0 ∣ + φ ℵ Γ ( ℵ + 1 ) { A Z ‖ Z ‖ ı + V J } .

Next, we will demonstrate the boundedness of ℬ ( V ) for a bounded subset V of Z . Let Z ∈ V , and since P is bounded, it follows that there exists A ≥ 0 such that

(20) ‖ Z ‖ ≤ A , ∀ Z ∈ V .

Consequently, for any Z ∈ V , we obtain the following result: p

(21) ‖ ℬ W ‖ ≤ ∣ Z 0 ∣ + φ ℵ Γ ( ℵ + 1 ) [ A J ‖ Z ‖ ı + V J ] ≤ ∣ Z 0 ∣ + φ ℵ Γ ( ℵ + 1 ) [ A J A ı + V J ] .

Thus, the operator ℬ ( V ) is bounded.

P3: For the equi-continuity, take u 1 , u 2 ∈ [ 0 , φ ] with u 1 ≥ u 2 , then we have

(22) ∣ ℬ Z ( u 1 ) − ℬ Z ( u 1 ) ∣ = 1 Γ ( ℵ ) ∫ 0 u 1 ∣ ( u 1 − r ) ℵ − 1 ‖ J ( r , Z ( r ) ) ∣ d r − 1 Γ ( ℵ ) ∫ 0 u 2 ∣ ( u 2 − r ) ℵ − 1 ∣ ∣ J ( r , Z ( r ) ) ∣ d r ≤ 1 Γ ( ℵ ) ∫ 0 u 1 ∣ ( u 1 − r ) ℵ − 1 ∣ − 1 Γ ( ℵ ) ∫ 0 u 2 ∣ ( u 2 − r ) ℵ − 1 ∣ ∣ J ( r , Z ( r ) ) ∣ d r ≤ φ ℵ Γ ( ℵ + 1 ) [ A J ‖ Z ‖ ı + V J ] [ u 1 ℵ − u 2 ℵ ] → 0 as u 1 → u 2 .

This guarantees the relative compactness of ℬ ( V ) by means of the Arzela-Ascoli theorem.

P4: In the fourth step, we will consider the following equation:

(23) ℐ = { Z ∈ Z : Z = χ ℬ Z , χ ∈ ( 0 , 1 ) } .

To demonstrate the boundedness of set ℐ , we consider Z ∈ ℐ and observe that the following condition holds for every t ∈ [ 0 , φ ] :

(24) ‖ Z ‖ = χ ‖ ℬ Z ‖ ≤ χ ∣ Z 0 ∣ φ ℵ Γ ( ℵ + 1 ) [ A J ‖ Z ‖ ı + V J ] .

This confirms the boundedness of set ℐ . By applying Schaefer’s theorem, we conclude that the operator ℬ possesses a fixed point. Therefore, our proposed system (12) for Zika virus possess at least one solution.

Remark 4.1

If the condition C 1 is satisfied for ı = 1 , then Theorem 4.1 can be demonstrated when φ ℵ A Z Γ ( ℵ + 1 ) < 1 .

Theorem 4.2

The proposed system (12) of the infection has a unique solution if φ ℵ A Z Γ ( ℵ + 1 ) < 1 satisfies.

Proof

To prove the statement, we utilize Banach’s contraction theorem with the assumption that Z and Z ¯ ∈ Z as follows:

(25) ‖ ℬ Z − ℬ Z ¯ ‖ ≤ max t ∈ [ 0 , φ ] 1 Γ ( ℵ ) ∫ 0 t ∣ ( t − r ) ℵ − 1 ‖ J ( r , Z ( r ) ) − J ( r , Z ¯ ( r ) ) ∣ d r ≤ φ ℵ A J Γ ( ℵ + 1 ) ‖ Z − Z ¯ ‖ .

As a result, a unique fixed point of ℬ is guaranteed, leading to the existence of a unique solution of model (12) of Zika virus infection.

5 Stability analysis

Here, we will demonstrate the UHS of our model of Zika virus infection. The UHS concept was first introduced by Ulam in 1940 and later expanded upon by Hyers [34,35]. Researchers from various disciplines have since employed the notion of UHS in their studies [36,37]. The fundamental theory can be summarized as follows:

Let G : Z → Z , where

(26) G Y = Y for Y ∈ Z .

Definition 5.1

Eq. (26) satisfies the UHS property if for ℜ > 0 and every solution Z ∈ Z , we have

(27) ‖ Y − G Y ‖ ≤ ℜ for t ∈ [ 0 , φ ] .

Moreover, there is a unique solution Z ¯ of the previously mentioned Eq. (26) such that 0 < C ı , and the following condition is satisfied

(28) ‖ Y ¯ − Y ‖ ≤ C ı ℜ , t ∈ [ 0 , φ ] .

Definition 5.2

Let Z and Z ¯ are solutions of Eq. (26). The system (26) is considered to be generalized UHS if

(29) ‖ Y ¯ − Y ‖ ≤ J ( ℜ ) ,

where the image of 0 is 0 and J ∈ G ( R , R ) .

Remark 5.1

If the solution Z ¯ ∈ Z holds true (28) and ∀ t ∈ [ 0 , φ ] , the below stated satisfies

∣ ℑ ( t ) ∣ ≤ ℜ , in which ℑ ∈ G ( [ 0 , φ ] ; R )
G Y ¯ ( V ) = Y ¯ + ℑ ( V ) .

Then, with slight modifications, system (12) can be transformed into:

(30) D 0 + ℵ C Y ( t ) = Z ( t , Y ( t ) ) + ℑ ( t ) , Y ( 0 ) = Y 0 .

Lemma 5.1

System (30) also fulfills

(31) ∣ Y ( t ) − V Y ( t ) ∣ ≤ a ℜ , in w h i c h a = φ ℵ Γ ( ℵ + 1 ) .

By employing Lemma 2.1 and taking into account Remark 5.1, we can easily prove this result.

Theorem 5.1

If the condition φ ℵ L Z Γ ( ℵ + 1 ) < 1 satisfied, we can establish that the solution of (12) exhibits UHS and generalized UHS based on Lemma 5.1.

Proof

To prove this, we consider the solutions Y ∈ X and Y ¯ ∈ X of (12), thus

(32) ∣ Y ( t ) − Y ¯ ( t ) ∣ = ∣ Y ( t ) − Y ¯ ( t ) ∣ ≤ ∣ Y ( t ) − V Y ¯ ( t ) ∣ ≤ ∣ Y ( t ) − V Y ¯ ( t ) ∣ ≤ a ℜ + φ ξ L X Γ ( ξ + 1 ) ∣ Y ( t ) − Y ¯ ( t ) ∣ ≤ a ℜ 1 − φ ξ L X Γ ( ξ + 1 ) .

As a result, it can be concluded that the fractional system (12) of Zika virus exhibits UHS and generalized UHS.

Definition 5.3

For any solution Y ∈ Z , system (26) is considered to be Ulam–Hyers–Rassias stable (UHRS) if the following condition is satisfied:

(33) ‖ Y − G Y ‖ ≤ F ( t ) ℜ , for t ∈ [ 0 , φ ] ,

where F ∈ G [ [ 0 , φ ] , R ] and ℜ > 0 . If C ı > 0 , then there exits a solution Y ¯ of (26), which holds the following

(34) ‖ Y ¯ − Y ‖ ≤ C ı F ( t ) ℜ ,

for all t in [ 0 , φ ] .

Definition 5.4

Assume Z ¯ be the unique solution, and suppose Z is another solution of (26) such that

(35) ‖ Y ¯ − Y ‖ ≤ G ı , г F ( t ) ℜ ,

in which t ∈ [ 0 , φ ] , F ∈ D [ [ 0 , φ ] , R ] in a manner G ı , г and ℜ > 0 . This shows that the solution of (26) is generalized UHRS.

Remark 5.2

Choose Y ¯ ∈ X , this solution will satisfying (28) if ∀ t ∈ [ 0 , φ ] , we have

∣ ℑ ( t ) ∣ ≤ ℜ F ( t ) , where ℑ ( t ) ∈ C ( [ 0 , φ ] ; R )
G Y ¯ ( t ) = Y ¯ + ℑ ( t ) .

Lemma 5.2

The perturbed system (5.1) fulfills the following condition:

(36) ∣ Y ( t ) − V Y ( V ) ∣ ≤ a F ( t ) ℜ , i n w h i c h a = φ ℵ Γ ( ℵ + 1 ) .

By employing Lemma 2.1 and taking into account Remark 5.2, we can easily prove the result with the help of analytic skills.

Theorem 5.2

If the condition φ ℵ L X Γ ( ℵ + 1 ) < 1 is satisfied, then the solution of (12) exhibits UHRS and generalized UHRS, as per Lemma 5.2.

Proof

Let us assume that Y ¯ ∈ Z be a unique solution and any alternative solution Y ∈ Z of (12). It can be observed that the following holds true:

(37) ∣ Y ( t ) − Y ¯ ( t ) ∣ = ∣ Y ( t ) − Y ¯ ( t ) ∣ ≤ ∣ Y ( t ) − V Y ¯ ( t ) ∣ ≤ ∣ Y ( t ) − V Y ¯ ( t ) ∣ ≤ a F ( t ) ℜ + φ ℵ L Z Γ ( ℵ + 1 ) ∣ Y ( t ) − Y ¯ ( t ) ∣ ≤ a F ( t ) ℜ 1 − φ ℵ L Z Γ ( ℵ + 1 ) .

As a result of this, the solution of (12) exhibits UHRS and generalized UHRS.

6 Numerical scheme for the model

In this section, we introduce the numerical scheme for the solution of our fractional model of Zika virus infection as defined under the Caputo framework (8). The technique employed here to address the fractional nonlinear system is detailed in prior works, specifically in the study by [38]. To present the numerical scheme, we proceed as follows:

(38) D t α 0 C z ( t ) = f ( t , z ( t ) ) ,

Eq. (38) implies through definition

(39) z ( t ) − z ( 0 ) = 1 Γ ( α ) ∫ 0 t f ( χ , z ( χ ) ) ( t − χ ) α − 1 d χ ,

then, at t = t n + 1 , n = 0 , 1 , … , we obtain the following equation:

(40) z ( t n + 1 ) − z ( 0 ) = 1 Γ ( α ) ∫ 0 t n + 1 ( t n + 1 − t ) α − 1 f ( t , z ( t ) ) d t

and

(41) z ( t n ) − z ( 0 ) = 1 Γ ( α ) ∫ 0 t n ( t n − t ) α − 1 f ( t , z ( t ) ) d t .

From (41) and (40), we have

(42) z ( t n + 1 ) = z ( t n ) + 1 Γ ( α ) ∫ 0 t n + 1 ( t n + 1 − t ) α − 1 f ( t , z ( t ) ) d t ︸ A α , 1 − 1 Γ ( α ) ∫ 0 t n ( t n − t ) α − 1 f ( t , z ( t ) ) d t ︸ A α , 2 ,

where

(43) A α , 1 = 1 Γ ( α ) ∫ 0 t n + 1 ( t n + 1 − t ) α − 1 f ( t , z ( t ) ) d t

and

(44) A α , 2 = 1 Γ ( α ) ∫ 0 t n ( t n − t ) α − 1 f ( t , z ( t ) ) d t .

The Lagrange approximation for f ( t , z ( t ) ) is given by

(45) P ( t ) ≃ t − t n − 1 t n − t n − 1 f ( t n , z n ) + t − t n t n − 1 − t n f ( t n − 1 , z n − 1 ) = f ( t n , z n ) h ( t − t n − 1 ) − f ( t n − 1 , z n − 1 ) h ( t − t n ) .

The use of the aforementioned expression leads to,

(46) A α , 1 = f ( t n , z n ) h Γ ( α ) ∫ 0 t n + 1 ( t n + 1 − t ) α − 1 ( t − t n − 1 ) d t − f ( t n − 1 , z n − 1 ) h Γ ( α ) ∫ 0 t n + 1 ( t n + 1 − t ) α − 1 ( t − t n ) d t .

We have after further simplification,

(47) A α , 1 = f ( t n , z n ) h Γ ( α ) 2 h α t n + 1 α − t n + 1 α + 1 α + 1 − f ( t n − 1 , z n − 1 ) h Γ ( α ) h α t n + 1 α − 1 α + 1 t n + 1 α + 1 .

Similarly,

(48) A α , 2 = 1 Γ ( α ) ∫ 0 t n ( t n − t ) α − 1 × f ( t n , z n ) h ( t − t n − 1 ) − f ( t n − 1 , z n − 1 ) h ( t − t n ) d t .

Further simplifying, we obtain

(49) A α , 2 = f ( t n , z n ) h Γ ( α ) h α t n α − t n α + 1 α + 1 + f ( t n − 1 , z n − 1 ) h Γ ( α ) 1 α + 1 t n α + 1 .

We can derive the ultimate approximate solution for the fractional nonlinear system by substituting Eqs (48) and Eq. (49) into Eq. (42). This solution is expressed as follows:

(50) z ( t n + 1 ) = z ( t n ) + f ( t n , z n ) h Γ ( α ) 2 h t n + 1 α α − t n + 1 α + 1 α + 1 + h α t n α − t n + 1 α + 1 α + 1 + f ( t n − 1 , z n − 1 ) h Γ ( α ) − h α t n + 1 α + t n + 1 α + 1 α + 1 + t n α + 1 α + 1 .

Here, we present numerical findings that illustrate the pathways of the system follows as input parameters experience fluctuations. The objective of this section involves analyzing the impact of alterations in these input parameters on system’s behavior. Through exploring diverse scenarios, our aim is to gain a comprehensive understanding of how the system responds to varying conditions. By scrutinizing these numerical outcomes, our intention is to emphasize system’s resilience and susceptibility to different input variations. This section plays a pivotal role in showcasing system’s performance across various conditions, thereby deepening comprehension of its reactions and furnishing valuable insights for applications in engineering or scientific contexts. It is important to note that for numerical purposes, we have introduced assumptions concerning the initial values of state variables and system parameters.

We performed several simulations to conceptualize the proposed dynamics of viral infection with variation of different parameters. Plotting the trajectories of infected individuals of both the classes over time can provide a clear view of how the epidemic progresses. In Figures 1 and 2, we have shown the impact of fractional parameter on the infected classes of the model. In Figure 1, a competitive analysis of integer and non-integer derivative is also shown where we assumed the value of fractional parameter to be 0.85, 0.90, 0.95, and 1.00, while in Figure 2, we assumed 0.50, 0.60, 0.70, and 0.80. It is clear from these findings that decreasing the index of memory or fractional parameter can decrease the infection level of Zika virus in the society. In this scenario, memory emerges as a valuable ally in the continuous fight against vector-borne diseases, paving the way for more efficient and enduring control strategies.

$Figure 1 Time series analysis of the (a) carrier hosts, (b) infected hosts and (c) infected vectors of model (8) with different values of fractional order ℵ \aleph , i.e., ℵ = 0.85 \aleph =0.85 , 0.90, 0.95, and 1.00.$

Figure 1

Time series analysis of the (a) carrier hosts, (b) infected hosts and (c) infected vectors of model (8) with different values of fractional order ℵ , i.e., ℵ = 0.85 , 0.90, 0.95, and 1.00.

$Figure 2 Time series analysis of the (a) carrier hosts, (b) infected hosts and (c) infected vectors of model (8) with different values of fractional order ℵ \aleph , i.e., ℵ = 0.50 \aleph =0.50 , 0.60, 0.70, and 0.80.$

Figure 2

Time series analysis of the (a) carrier hosts, (b) infected hosts and (c) infected vectors of model (8) with different values of fractional order ℵ , i.e., ℵ = 0.50 , 0.60, 0.70, and 0.80.

In Figures 3 and 4, the role of transmission probability and biting rate of mosquitoes has been visualized. In this scenario, the value of transmission probability is assumed to be 0.55, 0.65, 0.75, and 0.85 in Figure 3, while the rate of biting is considered to be 0.40, 0.46, 0.52, and 0.58. We noticed that these input factors are critical and can increase the risk of the infection in the society. The impact of vaccination fraction has been illustrated in Figure 5. It can be seen that vaccination has a positive impact on the system and can be used as a control measure for the infection of Zika virus. In the last simulation presented in Figure 6, we visualized the effect of the input parameter c on the infected individuals of vectors and hosts. From the Figure, we noticed that this preventive measure can also effectively control the infection in the society.

$Figure 3 Graphical view analysis of the (a) carriers hosts, (b) infected hosts and (c) infected vectors of system (8) with different values of the transmission rate β 1 {\beta }_{1} , i.e., β 1 = 0.55 {\beta }_{1}=0.55 , 0.65, 0.75, and 0.85.$

Figure 3

Graphical view analysis of the (a) carriers hosts, (b) infected hosts and (c) infected vectors of system (8) with different values of the transmission rate β 1 , i.e., β 1 = 0.55 , 0.65, 0.75, and 0.85.

Figure 4

Graphical view analysis of the (a) carrier hosts, (b) infected hosts and (c) infected vectors of the system (8) with different values of the biting rate b , i.e., b = 0.40 , 0.46, 0.52, and 0.58.

$Figure 5 Representation of the solution pathways of the (a) carrier hosts, (b) infected hosts and (c) infected vectors of the system (8) with various assumptions of the vaccination fraction v v , i.e., v = 0.20 v=0.20 , 0.24, 0.28, and 0.32.$

Figure 5

Representation of the solution pathways of the (a) carrier hosts, (b) infected hosts and (c) infected vectors of the system (8) with various assumptions of the vaccination fraction v , i.e., v = 0.20 , 0.24, 0.28, and 0.32.

Figure 6

Representation of the solution pathways of the (a) carrier hosts, (b) infected hosts and (c) infected vectors of system (8) with different values of input parameter c , i.e., c = 0.155 , 0.205, 0.255, and 0.305.

In this work, we visually represent the interactions between variables, the progression of the epidemic over time, and the effects of parameter changes. Our findings predicted that the index of memory, vaccination rate, and the input factor c can be utilized as preventive measures for the infection of Zika virus, while the transmission probability and biting rate are critical and can increase the risk of infection.

7 Concluding remarks

The Zika virus infection presents a substantial threats to worldwide health and safety, carrying the potential for life-threatening consequences. Addressing this challenge requires effective strategies from policymakers, researchers, and public health officials. In our study, we structured a novel epidemic model for the transmission of Zika virus infection that incorporates vaccination, treatment, and asymptomatic carriers using fractional calculus. The positivity and boundedness of the solutions of the proposed model are established through mathematical skills. Through qualitative analysis and the establishment of the existence and uniqueness of solutions, we delved into the dynamical behavior of Zika dynamics. We have established UHS results for the solution of our model. The predictive aspect of our study anticipates that the fraction of asymptomatic carriers has the potential to extend the reach of infection to non-infected regions, emphasizing the need for vigilant monitoring and control measures. Moreover, we identify treatment through medication, the fractional parameter or memory index, and vaccination as effective strategies in the combat against this viral infection.

Acknowledgment

Researchers would like to thank the Deanship of Scientific Research, Qassim University for funding publication of this project.

Author contributions: All the authors contributed significantly to this research work. Rashid Jan conceptualized, formulated, analyzed the work, and wrote the first draft. Normy Norfiza Abdul Razak supervised, analyzed, validated, and revised this research work. Salah Boulaaras supervised, visualized, investigated, and wrote the first draft. Ziad Ur Rehman contributed to the formulation, investigation and find out the numerical results of the work. In the end, all the authors thoroughly checked and approved the final output.
Conflict of interest: No conflict of interest regarding this work.
Data availability statement: Not applicable here.

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Received: 2023-09-03

Revised: 2023-10-20

Accepted: 2023-11-03

Published Online: 2023-12-16

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0352

Keywords for this article

fractional calculus; Zika virus infection; mathematical model; vaccination and treatment; stability analysis; dynamical behavior

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