Control of conjunctivitis virus with and without treatment measures: A bifurcation analysis

Muhammad Owais Kulachi; Aqeel Ahmad; Evren Hincal; Ali Hasan Ali; Muhammad Farman; Muhammad Taimoor

doi:10.1016/j.jksus.2024.103273

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Full Length Article

08 2024

:36;

103273

doi:

10.1016/j.jksus.2024.103273

Control of conjunctivitis virus with and without treatment measures: A bifurcation analysis

Muhammad Owais Kulachi^a, Aqeel Ahmad^a,b, Evren Hincal^b,c, Ali Hasan Ali^d,e,f,g,⁎, Muhammad Farman^h,i, Muhammad Taimoor^a

a Department of Mathematics, Ghazi University D G Khan 32200, Pakistan

b Mathematics Research Center, Near East University, Near East Boulevard, Nicosia North Cyprus, 99138, Cyprus

c Department of Mathematics, Near East University, Near East Boulevard, Nicosia North Cyprus, 99138, Cyprus

d Department of Mathematics, College of Education for Pure Sciences, University of Basrah, 61001, Basrah, Iraq

e Institute of Mathematics, University of Debrecen, Pf.400, H-4002, Debrecen, Hungary

f Technical Engineering College, Al-Ayen University, 64001, Dhi Qar, Iraq

g Department of Business Management, Al-imam University College, 34011, Balad, Iraq

h Faculty of Art and Science, University of Kyrenia, Kyrenia, TRNC, Mersin 10, Turkey

i Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

⁎Corresponding author. ali.hasan@science.unideb.hu (Ali Hasan Ali),

Received: 2024-1-30, Accepted: 2024-5-25,

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Abstract

This study aims to examine the early detection and treatment of conjunctivitis (eye infection) through vaccination and recovery measures, both with and without medication. We develop an immune system-boosting mathematical model and convert it to a fractionally ordered model using the ABC operator. Key properties, such as the uniqueness and boundedness of the model, are investigated using Banach space. To assess the stability of the newly developed $S E V I R_{u} R_{i}$ system and confirm the occurrence of flip bifurcation, we conduct both qualitative and quantitative analyses. We calculate the basic reproductive number, $R_{0}$ , using an advanced approach and analyze its impact across different sub-compartments. Sensitivity analysis is performed on each parameter to understand the rate of change sensitivity. The Atangana–Toufik method is employed to solve the system for various fractional values, providing a reliable bounded solution. Simulations are conducted to observe the real behavior and effects of the conjunctivitis virus, showing that individuals with a strong immune system can recover with or without medication. Finally, we determine the actual state of virus control post-early detection, accounting for treated and untreated individuals due to the robust immune system and precautionary measures.

Keywords

Mathematical modeling

Boundedness

Uniqueness

Reproductive number “R0”

Flip bifurcation

Atangana–Toufik

1 Introduction

In the 13th century, Fibonacci introduced the famous Fibonacci sequence to describe population growth, marking the beginning of mathematics in biology. Later, Daniel Bernoulli used mathematical concepts to explain microscopic organisms’ forms, and Johannes Reinke coined the term “bio math” in 1901. Essentially, bio math involves the theoretical study of mathematical models to understand the principles underlying biological systems’ structure and function.

The last few decades have seen significant advancements in the biological sciences, driven by technological progress. Mathematics has consistently contributed to society, bringing substantial advancements to the natural sciences. Similarly, biological research can be revolutionized through mathematical models, which help unravel life’s complexities. Modern computer algebra systems have simplified solving complex mathematical problems, enabling scientists to focus on understanding mathematical biology (Chou and Friedman, 2016; Yeargers et al., 2013).

Recent years have seen increased attention to the mathematical modeling of various biological, physical, and epibiological systems. This attention is due to mathematical models’ ability to incorporate complex elements. Scholars have particularly focused on mathematical biology in areas such as infectious disease modeling, human anatomy growth, and body fluid dynamics. Mathematical models have provided frameworks to understand biological processes, determine threshold parameters, clarify transmission dynamics, and suggest effective control strategies for infectious diseases (Murray, 2003; Kyere et al., 2018).

Conjunctivitis, or pink eye, is a highly contagious condition caused by inflammation of the conjunctiva, triggered by bacterial, viral, or allergic infections. Various forms of conjunctivitis, such as allergic conjunctivitis, occur seasonally due to exposure to allergens like pollen, dust mites, animal dander, and contact lenses (Center for Disease Control (CDC), n.d.; Fehily et al., 2015). Transmission occurs through contact with infected individuals or contaminated objects. Acute Hemorrhagic Conjunctivitis (ADC) is a specific viral type with an incubation period of 1 to 3 days, presenting symptoms such as tearing, irritation, photophobia, sore throat, lid swelling, and purulent discharges (Elliot, 1925).

Infectious conjunctivitis, including bacterial, viral, and ophthalmia neonatorum, involves infection of the conjunctiva and sclera. Symptoms include itching, soreness, tearing, pus discharge, and light sensitivity. Effective control measures include antibiotic eye drops, hygiene practices, isolation, and allowing the disease to run its course, typically lasting 2 to 3 weeks. Conjunctivitis is more prevalent in tropical regions (Malu, 2014; Kimberlin, 2018). In Fig. 1, the physical symptoms of eyes infected by the conjunctivitis virus are illustrated.

Fractional calculus is widely used in scientific disciplines, especially physics and engineering. Fractional order models are preferred over traditional integer order models for their ability to account for genetic and memory aspects of systems (Ahmad et al., 2024; Alsaud et al., 2024). Examples include fractional models for lung cancer and coronavirus, demonstrating the influence of fractional-order derivatives on disease transmission. The COVID-19 pandemic highlighted the need for such models due to social and economic disruptions (Tomar and Chadha, 2023; Chadha et al., 2023). The Generalized Damped Forced Korteweg–de Vries (GDFKdV) equation and the Damped Forced Korteweg–de Vries (DFKdV) equation have also been used to study nonlinear wave propagation and reaction dynamics (Tomar et al., 2023). Models of some diseases can be studied in certain environments such as fuzzy environment to employ fuzzy parameters accounts for the variability in parameter values among individuals within the population (Dayan et al., 2023).

Conjunctivitis often manifests during the rainy season when humid conditions favor the virus’s spread, particularly in tropical regions like Thailand Ghazali et al. (2003), Chansaenroj et al. (2015). Isolating infected individuals and granting sick leave for home isolation can accelerate recovery and reduce infectious interactions. The American Academy of Pediatrics recommends student isolation to prevent rapid transmission in schools (Chowell et al., 2006). Mathematical models of conjunctivitis have been developed to enhance understanding, with notable contributions from Suksawat and Naowarat (2014), Unyong and Naowarat (2014), Sangthongjeen et al. (2015), Alalhareth et al. (2023).

1.1

1.1 Basic definitions

Definition 1.1

Atangana–Baleanu’s fractional-order derivative in the Liouville–Caputo sense (ABC) is defined by Atangana and Baleanu (2016)

(1)

_{0}^{A B C} D_{t}^{ξ} {g (t)} = \frac{A B (ξ)}{n - ξ} \int_{ξ}^{t} \frac{d^{n}}{d w} g (w) E_{ξ} [- ξ \frac{{(t - w)}^{ξ}}{n - ξ}] d w, n - 1 < ξ < n,

where

E_{ξ}

represents the Mittag-Leffler function,

A B (ξ)

represents a normalization function, and

A B (0) = A B (1) = 1

Definition 1.2

The Laplace transform of Eq. (1) is given by: $[_{0}^{A B C} D_{t}^{ξ} g (t)] s = \frac{A B (ξ)}{1 - ξ} \frac{s^{ξ} L [g (t)] (s) - s^{ξ - 1} g (0)}{s^{ξ} + \frac{ξ}{1 - ξ}} .$

Definition 1.3

The Sumudu transform (ST) of Eq. (1) is given by: $S T {_{0}^{A B C} D_{t}^{ξ} g (t)} s = \frac{B (ξ)}{1 - ξ} (ξ Γ (ξ + 1) E_{ξ} (- \frac{1}{1 - ξ} ν^{ξ})) .$

Definition 1.4

The fractional integral of order $ξ$ for the Atangana–Baleanu function $g (t)$ is given by: $_{0}^{A B C} I_{t}^{ξ} [g (t)] = \frac{1 - (ξ)}{B - ξ} g (t) + \frac{(ξ)}{B (ξ) Γ (ξ)} \int_{ξ}^{t} g (s) {(t - s)}^{ξ - 1} d s .$

2 Formulation of SEVI $R_{u} R_{i}$ model

Conjunctivitis as a bacterial disease with pink eyes is discussed in Sangsawang et al. (2012), along with its medication. We formulate a mathematical model for the conjunctivitis virus by introducing recovered individuals without medication. This new model, denoted SEVI $R_{u} R_{i}$ , includes the following compartments: susceptible (S), exposed (E), vaccinated (V), infected (I), recovered without medication ( $R_{u}$ ), and recovered with treatment ( $R_{i}$ ). The key parameters in our model are:

$n$ : total population,
$β_{n}$ : birth rate,
$μ_{n}$ : death rate,
$β$ : transmission rate of Acute Hemorrhagic Conjunctivitis,
$ϕ$ : vaccination rate for uninfected individuals,
$κ$ : incubation rate for Acute Hemorrhagic Conjunctivitis,
$α$ : recovery rate with treatment,
$γ$ : recovery rate without medication,
$ρ$ : recovery rate without medication,
$σ$ : rate at which individuals become susceptible again.

The flowchart for the newly developed model SEVI $R_{u} R_{i}$ is given in Fig. 2.

The developed model, derived from the flowchart and the generalized constructed hypothesis mentioned above, is as follows:

Using the Atangana–Baleanu (ABC) fractional operator, the model becomes $_{0}^{A B C} D_{t}^{ξ} S (t) = n β_{n} + σ R_{u} - β I S - μ_{n} S,$ $_{0}^{A B C} D_{t}^{ξ} E (t) = β I S - ϕ E - μ_{n} E,$ $_{0}^{A B C} D_{t}^{ξ} V (t) = ϕ E - κ V - μ_{n} V,$

(2)

_{0}^{A B C} D_{t}^{ξ} I (t) = κ V - (α + γ + ρ + μ_{n}) I,

_{0}^{A B C} D_{t}^{ξ} R_{u} (t) = ρ I - (σ + μ_{n}) R_{u},

_{0}^{A B C} D_{t}^{ξ} R_{i} (t) = (α + γ) I - μ_{n} R_{i} .

Here, $_{0}^{A B C} D_{t}^{ξ}$ represents ABC derivatives, where $0 < ξ \leq 1$ . The initial conditions associated with this model are: $S (0) = S^{0}, E (0) = E^{0}, V (0) = V^{0}, I (0) = I^{0}, R_{u} (0) = R_{u}^{0}, R_{i} (0) = R_{i}^{0} .$ The parameters and their values involved in developed mathematical model are given in Table 1.

We employ the Sumudu transform on Eq. (2) and, after restructuring the system and applying the inverse Sumudu transform, we obtain: $S_{k + 1} (t) = S_{k} (0) + S T^{- 1} \{\begin{matrix} \frac{1 - ξ}{B (ξ) ξ Γ (ξ + 1) E_{ξ} (- \frac{1}{1 - ξ} w^{ξ})} \times S T [n β_{n} + σ R_{u} - β I S - μ_{n} S], \end{matrix}$ $E_{k + 1} (t) = E_{k} (0) + S T^{- 1} \{\begin{matrix} \frac{1 - ξ}{B (ξ) ξ Γ (ξ + 1) E_{ξ} (- \frac{1}{1 - ξ} w^{ξ})} \times S T [β I S - ϕ E - μ_{n} E], \end{matrix}$ $V_{k + 1} (t) = V_{k} (0) + S T^{- 1} \{\begin{matrix} \frac{1 - ξ}{B (ξ) ξ Γ (ξ + 1) E_{ξ} (- \frac{1}{1 - ξ} w^{ξ})} \times S T [ϕ E - κ V - μ_{n} V], \end{matrix}$

(3)

I_{k + 1} (t) = I_{k} (0) + S T^{- 1} \{\begin{matrix} \frac{1 - ξ}{B (ξ) ξ Γ (ξ + 1) E_{ξ} (- \frac{1}{1 - ξ} w^{ξ})} \times S T [κ V - (α + γ + ρ + μ_{n}) I], \end{matrix}

{R_{u}}_{k + 1} (t) = {R_{u}}_{k} (0) + S T^{- 1} \{\begin{matrix} \frac{1 - ξ}{B (ξ) ξ Γ (ξ + 1) E_{ξ} (- \frac{1}{1 - ξ} w^{ξ})} \times S T [ρ I - (σ + μ_{n}) R_{u}], \end{matrix}

{R_{i}}_{k + 1} (t) = {R_{i}}_{k} (0) + S T^{- 1} \{\begin{matrix} \frac{1 - ξ}{B (ξ) ξ Γ (ξ + 1) E_{ξ} (- \frac{1}{1 - ξ} w^{ξ})} \times S T [(α + γ) I - μ_{n} R_{i}] . \end{matrix}

The resultant solution of (3) is given as

S (t) = lim_{k \to \infty} S_{k} (t), E (t) = lim_{k \to \infty} E_{k} (t), V (t) = lim_{k \to \infty} V_{k} (t),

I (t) = lim_{k \to \infty} I_{k} (t), R_{u} (t) = lim_{k \to \infty} {R_{u}}_{k} (t), R_{i} (t) = lim_{k \to \infty} {R_{i}}_{k} (t) .

Table 1 Description of parameters.

Parameter	Representation	Estimated values	References
$n$	$Describes the whole population$	1	Sangsawang et al. (2012)
$β_{n}$	$Represents the birth rate$	0.00004215	Sangsawang et al. (2012)
$σ$	Rate at which individuals become susceptible	0.005	Assumed
$β$	$Transmission rate for Acute Hemorrhagic Conjunctivitis$	0.004	Sangsawang et al. (2012)
$μ_{n}$	$Represents the death rate$	0.00004215	Sangsawang et al. (2012)
$ϕ$	$Rate of vaccination for uninfected individuals$	0.3	Assumed
$κ$	Characterizes for Acute Hemorrhagic Conjunctivitis as incubation rate	0.04	Sangsawang et al. (2012)
$α$	Incidence rate for recovered with treatment measures	0.08	Sangsawang et al. (2012)
$γ$	Incidence rate for recovered individuals without medication	0.3	Sangsawang et al. (2012)
$ρ$	Recovery rate without medication	0.0000008	Assumed

Theorem 2.1

Consider a Banach space $(Y, ‖ \cdot ‖)$ and let $M$ be a self-map of $Y$ satisfying

(4)

‖ M y - M r ‖ \leq ξ ‖ y - r ‖, \forall y, r \in Y,

where

0 \leq ξ < 1

. Assume that

M

is Picard

M

-stable.

For Eq. (3), we obtain $\frac{1 - ξ}{B (ξ) ξ Γ (ξ + 1) E_{ξ} (- \frac{1}{1 - ξ} w^{ξ})} .$ The above expression shows the Lagrange multiplier.

Proof

Self-map $M$ is defined on Eq. (3). Using norm properties and the triangle inequality, we get $‖ M [s_{k} (t)] - M [S_{j} (t)] ‖ \leq ‖ s_{k} (t) - S_{j} (t) ‖ + S T^{- 1} {\frac{1 - ξ}{B (ξ) ξ Γ (ξ + 1) E_{ξ} (- \frac{1}{1 - ξ} w^{ξ})} \times S T [n β_{n} + σ (R_{u_{k}} (t) - R_{u_{j}} (t)) - β (I_{k} (t) - I_{j} (t)) (S_{k} (t) - S_{j} (t)) - μ_{n} (S_{k} (t) - S_{j} (t))]} .$ Similarly, we get analogous results for $‖ M [e_{k} (t)] - M [E_{j} (t)] ‖$ , $‖ M [v_{k} (t)] - M [V_{j} (t)] ‖$ , $‖ M [i_{k} (t)] - M [I_{j} (t)] ‖$ , $‖ M [r_{u_{k}} (t)] - M [R_{u_{j}} (t)] ‖$ , and $‖ M [r_{i_{k}} (t)] - M [R_{i_{j}} (t)] ‖$ .

$M$ fulfills the conditions of Theorem 2.1. Therefore, $M$ must be Picard $M$ -stable.

Theorem 2.2

The iteration approach is used to find a unique singular solution to Eq. (2).

Proof

Consider the Hilbert space $M = P^{2} ((v, u) \times (0, r))$ . In this context, $M$ denotes the set of measurable functions, $P$ represents the projection operator, and $((v, u) \times (0, r))$ is the Cartesian product of the intervals $(v, u)$ and $(0, r)$ . $h : (v, u) \times [0, T] \to R, \iint g h d g d h < \infty .$

Certain operators are taken into account: $ξ = (0, 0, 0, 0, 0, 0), ξ = \{\begin{matrix} n β_{n} + σ R_{u} - β I S - μ_{n} S, \\ β I S - ϕ E - μ_{n} E, \\ ϕ E - κ V - μ_{n} V, \\ κ V - (α + γ + ρ + μ_{n}) I, \\ ρ I - (σ + μ_{n}) R_{u}, \\ (α + γ) I - μ_{n} R_{i} . \end{matrix}$

We demonstrate that the inner product of $(T (S_{a 11} (t) - S_{a 12} (t), E_{b 21} (t) - E_{b 22} (t), V_{f 61} (t) - V_{f 62} (t), I_{c 31} (t)$ $- I_{c 32} (t), {R_{u}}_{d 41} (t) - {R_{u}}_{d 42} (t), {R_{i}}_{e 51} (t) - {R_{i}}_{e 52} (t)), (ν_{1}, ν_{2}, ν_{3}, ν_{4}, ν_{5})) .$ involving $(S_{a 11} (t) - S_{a 12} (t), E_{b 21} (t) - E_{b 22} (t), V_{f 61} (t) - V_{f 62} (t), I_{c 31} (t) - I_{c 32} (t), {R_{u}}_{d 41} (t) - {R_{u}}_{d 42} (t), {R_{i}}_{e 51} (t) - {R_{i}}_{e 52} (t))$ are the system’s unique solutions.

Considering the inner function and the norm, for large numbers $e_{1}, e_{2}, e_{3}, e_{4}, e_{5}$ , and $e_{6}$ , both solutions converge to the exact solution. Using topological notions, we derive five small positive parameters $(ɛ_{e_{1}}, ɛ_{e_{2}}, ɛ_{e_{3}}, ɛ_{e_{4}}, ɛ_{e_{5}}, ɛ_{e_{6}})$ such that $‖ ν_{1} ‖, ‖ ν_{2} ‖, ‖ ν_{3} ‖, ‖ ν_{4} ‖, ‖ ν_{5} ‖, ‖ ν_{6} ‖ \neq 0;$ $‖ S_{a 11} (t) - S_{a 12} (t) ‖, ‖ E_{b 21} (t) - E_{b 22} (t) ‖, ‖ V_{f 61} (t) - V_{f 62} (t) ‖, ‖ I_{c 31} (t) - I_{c 32} (t) ‖, ‖ {R_{u}}_{d 41} (t) - {R_{u}}_{d 42} (t) ‖, ‖ {R_{i}}_{e 51} (t) - {R_{i}}_{e 52} (t) ‖ \to 0;$ $S_{a 11} (t) = S_{a 12} (t), E_{b 21} (t) = E_{b 22} (t), V_{f 61} (t) = V_{f 62} (t),$ $I_{c 31} (t) = I_{c 32} (t), {R_{u}}_{d 41} (t) = {R_{u}}_{d 42} (t), {R_{i}}_{e 51} (t) = {R_{i}}_{e 52} (t) .$

This completes the proof of uniqueness.

3 Analysis of equilibrium points and reproductive number

In this section, we conduct a comprehensive analysis of equilibrium points. To determine these points, it is necessary to set the left-hand side of the system (2) to zero. The equilibrium point corresponding to the absence of disease in this model is: $D_{1} (S, E, V, I, R_{u}, R_{i}) = (\frac{n β_{n}}{μ_{n}}, 0, 0, 0, 0, 0) .$

Furthermore, the equilibrium point associated with the endemic state, after substituting the parameter values given in Table 1 and using Mathematica for simplification, is given by: $D_{2} (S^{*}, E^{*}, V^{*}, I^{*}, R_{u}^{*}, R_{i}^{*}) = (95.1242, 0.127117, 0.952377, 0.100239, 0.0000159042, 903.696) .$

3.1

3.1 Reproduction number and its analysis

The Jacobian matrices $F$ and $V$ are analyzed at the disease-free equilibrium point $D_{1}$ . In these matrices, the element at position $(i, j)$ of matrix $F$ denotes the rate at which a virus-infected individual in compartment $j$ spreads to compartment $i$ , while the element at $(i, j)$ in matrix $V$ indicates the progression of the disease within compartment $i$ . To compute the reproduction number, the spectral radius of the matrix $F V^{- 1}$ at the disease-free equilibrium point is evaluated. This matrix, known as the Next Generation Matrix, is given by: $J_{0} = (\begin{matrix} - μ_{n} & 0 & 0 & - \frac{β n β_{n}}{μ_{n}} & σ & 0 \\ 0 & - (μ_{n} + ϕ) & 0 & \frac{β n β_{n}}{μ_{n}} & 0 & 0 \\ 0 & ϕ & - (μ_{n} + κ) & 0 & 0 & 0 \\ 0 & 0 & κ & - (μ_{n} + α + γ + ρ) & 0 & 0 \\ 0 & 0 & 0 & ρ & - (μ_{n} + σ) & 0 \\ 0 & 0 & 0 & α + γ & 0 & - μ_{n} \end{matrix}) .$ $J_{0} = F - V .$

We use Eq. (1) to derive the vectors $F$ and $V$ in our model.

The resultant matrix $K$ is given by $K = F V^{- 1} .$

The characteristic equation is $| K - Λ I | = 0 .$

Solving this determinant, we obtain the eigenvalues $Λ$ . The principal eigenvalue from this characteristic equation is the reproduction number ( $R_{0}$ ), which is given by: $R_{0} = \frac{n β κ ϕ β_{n}}{μ_{n} (α + γ + ρ + μ_{n}) (κ + μ_{n}) (ϕ + μ_{n})} .$

3.2

3.2 Sensitivity analysis

Sensitivity analysis determines how different factors affect a model’s stability, especially with ambiguous data. It helps identify critical process factors. The sensitivity of $R_{0}$ is examined by computing the partial derivatives of the threshold with respect to relevant parameters.

The value of $R_{0}$ is sensitive to changes in parameters. Our analysis shows that $μ_{n}, α, ρ, γ$ , and $n, β_{n}, κ, β, ϕ$ show contraction and expansion, respectively. For effective infection control, prevention is advised over treatment. These indices help identify essential factors in defining the infection’s propagation capacity, as depicted in Fig. 3.

Reproductive number behavior for the newly developed system under different parameter effects.

4 Flip bifurcation analysis

From Ngoma et al. (2022), we observe that none of the eigenvalues equal 1 or -1, indicating that our model (2) may exhibit bifurcation if the constants are taken as: $(n, β_{n}, σ, β, μ_{n}, ϕ, κ, α, γ, ρ) .$ This is given by: $Ϝ |_{E (\frac{n β_{n}}{μ_{n}}, 0, 0, 0, 0, 0)} = \{(n, β_{n}, σ, β, μ_{n}, ϕ, κ, α, γ, ρ) : μ_{n} = 0\} .$ We need to prove that bifurcation does not exist at $E (\frac{n β_{n}}{μ_{n}}, 0, 0, 0, 0, 0),$ if the constants are taken as: $(n, β_{n}, σ, β, μ_{n}, ϕ, κ, α, γ, ρ) \in Ϝ |_{E (\frac{n β_{n}}{μ_{n}}, 0, 0, 0, 0, 0)} .$

Theorem 4.1

Our model (2) does not have flip bifurcation at $E (\frac{n β_{n}}{μ_{n}}, 0, 0, 0, 0, 0),$ if the constants are taken as: $(n, β_{n}, σ, β, μ_{n}, ϕ, κ, α, γ, ρ) \in Ϝ |_{E (\frac{n β_{n}}{μ_{n}}, 0, 0, 0, 0, 0)} .$

Proof

Our model in Eq. (2) is invariant with respect to $E, V, I, R_{u}$ and $R_{i} = 0$ . To verify the bifurcation existence, we take $E, V, I, R_{u}$ and $R_{i} = 0$ . As a result:

(5)

S (t) = h n β_{n} + (1 - h μ_{n}) S (t) .

Applying Eq. (5), we get:

(6)

f (S (t)) = h n β_{n} + (1 - h μ_{n}) S (t) .

Now, if

μ_{n} = 0

S (t) = \frac{n β_{n}}{μ_{n}}

. Applying Eq. (6), we get:

(7)

{\frac{\partial f (S (t))}{\partial S (t)} |}_{μ_{n} = 0, S (t) = \frac{n β_{n}}{μ_{n}}} = 1 .

Taking the partial derivative of

f (S (t))

with respect to

μ_{n}

, and applying the values for

S (t)

and

μ_{n}

, we get

- \frac{h n β_{n}}{μ_{n}} \neq 0

. Then, applying the second partial derivative on Eq. (7), we get:

(8)

\frac{\partial^{2} f (S (t))}{\partial S^{2} (t)} = 0 .

This shows that our model in Eq. (2) does not exhibit bifurcation if the constants are taken as follows:

(n, β_{n}, σ, β, μ_{n}, ϕ, κ, α, γ, ρ) \in Ϝ |_{E (\frac{n β_{n}}{μ_{n}}, 0, 0, 0, 0, 0)} .

Using a combination of immune system impacts and therapy, we investigate a newly designed conjunctivitis virus model. The population is affected in a complex, time-dependent way; the interaction between conjunctivitis and this system is continuous.

The linearization technique is used to achieve the stability and boundedness of the model equations in Fig. 4. From the bifurcation diagrams, we infer that the model’s behavior rises with control input and falls with extraction. The combination of medication and no medication creates a stable state for the conjunctivitis virus (pink eye) model. Fig. 4 supports our theoretical conclusions with time-stable graphs based on the parametric values.

5 Solutions by advanced numerical approach

We use an advanced numerical approach with a non-local, non-singular kernel for fractional differential equations to find reliable solutions. The system is given by:

(9)

\{\begin{matrix} _{0}^{A B C} D S (t) = f_{1} (t, S (t), G (t)), \\ _{0}^{A B C} D E (t) = f_{2} (t, S (t), G (t)), \\ _{0}^{A B C} D V (t) = f_{3} (t, S (t), G (t)), \\ _{0}^{A B C} D I (t) = f_{4} (t, S (t), G (t)), \\ _{0}^{A B C} D R_{u} (t) = f_{5} (t, S (t), G (t)), \\ _{0}^{A B C} D R_{i} (t) = f_{6} (t, S (t), G (t)), \\ where G (t) = (E (t), V (t), I (t), R_{u} (t), R_{i} (t)) . \\ S^{0} = S_{0}, E^{0} = E_{0}, V^{0} = V_{0}, I^{0} = I_{0}, R_{u}^{0} = {R_{u}}_{0}, R_{i}^{0} = {R_{i}}_{0} . \end{matrix}

These equations can be converted to fractional integral equations using fractional calculus:

S (t) - S (0) = \frac{(1 - ξ)}{A B C (ξ)} f_{1} (t, S (t), G (t)) + \frac{ξ}{A B C (ξ) Γ (ξ)} \int_{0}^{t} f_{1} (σ, S (σ), G (σ)) {(t - σ)}^{ξ - 1} d σ,

where

G (σ) = (E (σ), V (σ), I (σ), R_{u} (σ), R_{i} (σ))

Bifurcation analysis of continuous dynamics.

Using ABC operator at different fractional values.

Similarly, we get expressions for $E (t) - E (0)$ , $V (t) - V (0)$ , $I (t) - I (0)$ , $R_{u} (t) - R_{u} (0)$ , and $R_{i} (t) - R_{i} (0)$ .

Let $t = t_{n + 1}$ for $n = 0, 1, 2, \dots$ . The equations are reformulated as:

(10)

S (t_{n + 1}) - S (0) = \frac{(1 - ξ)}{A B C (ξ)} f_{1} (t_{n}, S (t_{n}), G (t_{n})) + \frac{ξ}{A B C (ξ) Γ (ξ)} \sum_{k = 0}^{n} \int_{t_{k}}^{t_{k + 1}} f_{1} (σ, S (σ), G (σ)) {(t_{n + 1} - σ)}^{ξ - 1} d σ,

where

G (t_{n}) = (E (t_{n}), V (t_{n}), I (t_{n}), R_{u} (t_{n}), R_{i} (t_{n}))

Similarly, we get expressions for $E (t_{n + 1}) - E (0)$ , $V (t_{n + 1}) - V (0)$ , $I (t_{n + 1}) - I (0)$ , $R_{u} (t_{n + 1}) - R_{u} (0)$ , and $R_{i} (t_{n + 1}) - R_{i} (0)$ .

Using two-step Lagrange polynomial interpolation within the interval $[t_{k}, t_{k + 1}]$ , we can simplify Eq. (10) to:

(11)

S_{n + 1} = S_{0} + \frac{(1 - ξ)}{A B C (ξ)} f_{1} (t_{n}, S (t_{n}), G (t_{n})) + \frac{ξ}{A B C (ξ) Γ (ξ)} \sum_{k = 0}^{n} (\frac{f_{1} (t_{k}, S_{k}, G_{k})}{h} \int_{t_{k}}^{t_{k + 1}} (σ - t_{k - 1}) {(t_{n + 1} - σ)}^{ξ - 1} d σ - \frac{f_{1} (t_{k - 1}, S_{k - 1}, G_{k - 1})}{h} \int_{t_{k}}^{t_{k + 1}} (σ - t_{k}) {(t_{n + 1} - σ)}^{ξ - 1} d σ),

where

G (t_{k}) = (E (t_{k}), V (t_{k}), I (t_{k}), R_{u} (t_{k}), R_{i} (t_{k}))

and

G (t_{k - 1}) = (E (t_{k - 1}), V (t_{k - 1}), I (t_{k - 1}), R_{u} (t_{k - 1}), R_{i} (t_{k - 1}))

Similarly, we get expressions for $E_{n + 1}$ , $V_{n + 1}$ , $I_{n + 1}$ , $R_{u_{n + 1}}$ , and $R_{i_{n + 1}}$ .

After integrating the terms, we replace them in Eq. (11) to get: $S_{n + 1} = S_{0} + \frac{(1 - ξ)}{A B C (ξ)} f_{1} (t_{n}, S (t_{n}), G (t_{n})) + \frac{ξ}{A B C (ξ)} \sum_{k = 0}^{n} (\frac{h^{ξ} f_{1} (S_{k}, G_{k})}{Γ (ξ + 2)} ({(n + 1 - k)}^{ξ} (n - k + 2 + ξ) - {(n - k)}^{ξ} (n - k + 2 + 2 ξ)) - \frac{h^{ξ} f_{1} (t_{k - 1}, S_{k - 1}, G_{k - 1})}{Γ (ξ + 2)} ({(n + 1 - k)}^{ξ + 1} - {(n - k)}^{ξ} (n - k + 1 + ξ))),$ Similarly, we get expressions for $E_{n + 1}$ , $V_{n + 1}$ , $I_{n + 1}$ , $R_{u_{n + 1}}$ , and $R_{i_{n + 1}}$ .

6 Simulation explanation

Theoretical results are obtained and examined using advanced approaches. Through simulations, the newly developed system is analyzed. The conjunctivitis model provides intriguing results using non-integer parametric values. Lowering fractional values yields reliable results for $S (t), E (t), V (t), I (t), R_{u} (t)$ , and $R_{i} (t)$ , as shown in Fig. 5.

Sub-figures (a) and (f) illustrate the dynamics of susceptible $S (t)$ and recovered with treatment $R_{i} (t)$ , respectively. Both compartments increase rapidly and then stabilize. Sub-figures (b), (c), (d), and (e) show the dynamics of exposed $E (t)$ , vaccinated $V (t)$ , infected $I (t)$ , and recovered without treatment $R_{u} (t)$ . Each compartment declines dramatically before stabilizing.

Sub-figures (c), (d), and (e) demonstrate a significant drop in infections due to combined vaccination and treatment strategies. Recovery with and without medication increases as fractional values decrease, as seen in the sub-figures (e) and (f).

These findings suggest future research directions for preventing the spread of conjunctivitis. The Atangana–Toufik method yields superior results for all sub-compartments at fractional derivatives compared to conventional derivatives. Reducing fractional values improves the accuracy and reliability of solutions across all compartments.

7 Conclusion

In this article, we formulated a fractional order conjunctivitis virus model, incorporating vaccination and recovery measures with and without medication. Using the Atangana–Toufik technique, we analyzed the model to produce reliable results. We recommend vaccinations and immune-boosting measures to prevent the spread of the virus, promoting a disease-free environment. The model examines the global impact of conjunctivitis with and without treatment and uses bifurcation analysis to ensure stability. Our findings confirm that the model does not exhibit flip bifurcation, and the unique and bounded solutions are validated using Banach space results. Calculating the reproductive number $R_{0}$ is crucial for understanding epidemic potential. Sensitivity analysis highlights the most significant factors affecting disease transmission. MATLAB simulations illustrate the dynamics of conjunctivitis control, demonstrating that combined measures vaccination and hand hygiene can control the virus without medication. These findings support future research in early detection and understanding the virus’s behavior and spread.

CRediT authorship contribution statement

Muhammad Owais Kulachi: Writing – original draft, Data curation, Conceptualization. Aqeel Ahmad: Writing – original draft, Validation, Supervision, Methodology. Evren Hincal: Writing – review & editing, Resources, Investigation, Funding acquisition. Ali Hasan Ali: Writing – review & editing, Visualization, Software, Project administration. Muhammad Farman: Writing – original draft, Investigation, Formal analysis, Conceptualization. Muhammad Taimoor: Writing – review & editing, Validation, Methodology, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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