EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6364 ISSN 1307-5543 – ejpam.com Published by New York Business Global Modeling of Eye Infection Transmitting by Conjunctivitis Adenovirus: Deterministic and Stochastic Approach Rahim ud Din1,∗, Naveed Ahmad2, Sadique Ahmad3, Puntani Pongsumpun1 1 Department of Mathematics, School of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok, Thailand 2 EIAS Data Science and BlockChain Lab, CCIS, Prince Sultan University, Riyadh 11586, Saudi Arabia 3 Department of Computer Science CCIS, Prince Sultan University, Riyadh 11586, Saudi Arabia Abstract. This study develops a mathematical model to investigate the early diagnosis and treat- ment of conjunctivitis caused by adenovirus, incorporating both deterministic and stochastic ap- proaches to capture disease dynamics. The model’s fundamental properties, including boundedness and uniqueness, are analyzed to ensure reliability, and equilibrium points are established for the deterministic framework. The basic reproduction number is derived and subjected to sensitivity analysis to evaluate how key parameters influence the spread of infection. Numerical simulations, conducted using a nonstandard finite difference (NSFD) scheme for the deterministic model and stochastic methods for the probabilistic model, reveal that individuals with strong immune sys- tems can recover without medical intervention, highlighting the critical role of immune strength in controlling disease transmission. These findings contribute to a deeper understanding of con- junctivitis dynamics and provide valuable insights for designing effective control strategies based on early diagnosis and treatment. 2020 Mathematics Subject Classifications: 92B05, 92D30, 93A30 Key Words and Phrases: Eye infection, adenovirus, sensitivity analysis, SITR model, NSFD scheme, stochastic model 1. Introduction Over the past few decades, the field of biological research has expanded considerably, and this growth is expected to continue with ongoing technological advancements. Mathe- maticians have played a key role in driving progress, contributing significant developments ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6364 Email addresses: 67056116@kmitl.ac.th (R. ud Din), Nahmed@psu.edu.sa (N. Ahmad), saahmad@psu.edu.sa (S. Ahmad), puntani.po@kmitl.ac.th (P. Pongsumpun) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) R. ud Din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6364 2 of 23 that provide lasting benefits to society [1]. Mathematics has not only advanced the nat- ural sciences but also holds great potential to transform biological research. Through mathematical modeling, biology gains valuable tools to interpret and analyze its inherent complexity. Moreover, recent progress in computer algebra systems has made it easier to address complex mathematical problems [2], enabling researchers to concentrate more on the study of mathematical biology rather than on technical problem-solving. In recent years, there has been growing interest in the mathematical modeling of biological, physical, and epidemiological phenomena. This surge is largely due to the ability of math- ematical models to account for numerous influencing factors. Mathematical biology, in particular, has attracted significant attention from researchers in areas such as body fluid dynamics, human growth modeling, infectious disease modeling, and related fields [3–5]. The widespread application of mathematical models has deepened our understanding of the complex nature of biological processes by shedding light on fundamental concepts. In the context of infectious diseases, mathematical modeling plays a vital role in identifying threshold parameters, clarifying transmission patterns, and formulating effective manage- ment strategies [6]. Moreover, by offering practical insights into treatment, mathematical models have proven valuable in both the prevention and control of infectious diseases, serving as a powerful tool in the fight against viral outbreaks [7]. Conjunctivitis, commonly known as “pink eye” or “bloodshot eyes,” is an infection of the conjunctive the thin tissue covering the white part of the eye and the inner surface of the eyelid. It is a highly contagious condition that may be caused by bacteria, viruses, or allergic reactions, all of which can trigger inflammation. Conjunctivitis occurs in sev- eral forms; for instance, allergic conjunctivitis is often caused by seasonal exposure to antigens or irritants such as house dust mites, pollen, animal dander, and contact lenses [8, 9]. Transmission typically occurs when a susceptible individual comes into direct con- tact with an infected person or with contaminated objects, such as foreign particles that enter the eyes, or through exposure to fluids discharged from the conjunctiva or the upper respiratory tract of an infected individual. Furthermore, studies suggest a direct associ- ation between parental chlamydial or gonococcal infections and the risk of transmission to infants. This work focuses on a viral form of the disease known as acute hemorrhagic conjunctivitis (AHC). The incubation period of AHC ranges from one to three days, after which symptoms such as photophobia, sore throat, tearing, eye discomfort, and swelling may develop, often accompanied by ocular discharge [10]. Effective strategies to prevent the spread of conjunctivitis include the use of antibiotic eye drops, maintaining proper hygiene, isolating infected individuals, and allowing the viral infection to resolve natu- rally, which typically occurs within two to three weeks. The disease is more prevalent in tropical regions [11, 12] and is commonly observed during the rainy season, when humid conditions favor its transmission [13]. High occurrence rates have been reported in tropical areas such as Thailand [14, 15]. Isolation of affected individuals is strongly recommended to limit the spread of infection [16]. Beyond reducing transmission, home isolation also facilitates recovery while minimizing both the number and duration of infectious contacts [17]. Notable contributions to this field include the studies reported in [16, 18, 19]. Furthermore, conjunctivitis most commonly emerges during the rainy season, when high R. ud Din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6364 3 of 23 Figure 1: Different kinds of eye infection image. humidity creates favorable conditions for viral transmission [20]. The disease is frequently reported in tropical regions such as Thailand [21, 22]. Isolating affected individuals is strongly recommended to curb its spread. Allowing patients to take leave from work and remain in home isolation not only supports faster recovery but also reduces both the fre- quency and intensity of potential infectious contacts [23]. In Figure 2, we present the different types of eye infections. We develop a mathematical model to study the dynamics of eye infections by dividing the total population into four compartments: S (susceptible), I (infected), T (under treat- ment), and R (recovered). The model incorporates a saturated incidence rate to describe the transmission of infection. For the deterministic framework, local stability is analyzed, and the basic reproduction number is derived using the next-generation matrix approach. Numerical simulations of the deterministic model are carried out using the NSFD scheme. For the stochastic model, properties such as exactness, uniqueness, the basic reproduction number, stability, and numerical simulations are discussed in subsequent sections. 1.1. Model Formulation In this section of our work, we develop model for the Conjunctivitis virus. The whole population are divide into four compartments, in which S stand for susceptible people, I R. ud Din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6364 4 of 23 S RTI Figure 2: Representation of the model schematically (1). Parameters Physical Meaning S Susceptible population I Infected population T Under treatment population R Recovered population b Birth rate K Transmission rate β Saturation constant α Recovery without medication µ medication rate d0 Natural death rate r Recovery rate of under treatment populations Table 1: The model (1) describes and specifies the system’s parameters. for infected people, T for under treatment, and R represent recovered populations. dS dt = b− KSI 1 + βI − d0S, dI dt = KSI 1 + βI − (α+ µ+ d0)I, dT dt = µI − (r + d0)T , dR dt = rT + αI − d0R. (1) A flowchart for the above system (1) are represented in figure (2), which shows how the community dynamics of different compartments have evolved over time. The following table, 1.1, contains the parameters utilized in the algorithm (1) and their actual meaning. Components in the model (1) and how they vary from one another are also given in . R. ud Din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6364 5 of 23 2. Uniqueness and Existence We look for the existence and uniqueness of the system (1). In order to attained this, the system (1) is described by S ′ = h1 (t,S, I, T ,R) , I ′ = h2 (t,S, I, T ,R) , T ′ = h3 (t,S, I, T ,R) , R′ = h4 (t,S, I, T ,R) . (2) Norm is define as ||χ||∞ = sup t∈x . (3) where x ∈ [0, t]. We assumed that for every t, S, I, T , and R are limited in [0, t] and that for each t that belongs to [0, t], there exist k1, k2, k3,, and k4, such that ||S||∞ < k1, ||I||∞ < k2, ||T ||∞ < k3, ||R||∞ < k4. Now, we have to prove (2) is bonded |h1 (t,S, I, T ,R) | =|b− KSI 1 + βI − d0S| ≤b+ K|S||I| 1 + β|I| + d0|S| ≤b+ K 1 + βk2 k1k2 + d0k1 < ∞. (4) where t ∈ [0, t] = M . Additionally, using the same process, we have |h2 (t,S, I, T ,R) | =| KSI 1 + βI − (α+ µ+ d0)I| ≤ K 1 + βk2 k1k2 + (α+ µ+ d0)k2 < ∞. (5) |h3 (t,S, I, T ,R) | =|µI − (r + d0)T | ≤µk2 + (r + d0)k3 < ∞. (6) |h4 (t,S, I, T ,R) | =|rT + αI − d0R| ≤rk3 + αk2 + (d0)k4 < ∞. (7) R. ud Din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6364 6 of 23 Thus S, I, T and R are bounded and there exist L1, L2, L3 and L4 such that sup M |h1 (t,S, I, T ,R) | < L1, sup M |h2 (t,S, I, T ,R) | < L2, sup M |h3 (t,S, I, T ,R) | < L3, sup M |h4 (t,S, I, T ,R) | < L4. Next, we have to prove |h1 (t,S1, I, T ,R)− h1 (t,S2, I, T ,R) | =|b− KS1I 1 + βI − d0S1 − b− KS2I 1 + βI − d0S2| <( KI 1 + βI + d0)|S1 − S2| 0, therefore (RS 1 − 1) < 0, implies RS 1 < 1. Therefore, if RS 1 < 1, then the DFE point E0 is locally asymptotically stable. 9. Numerically analysis of the stochastic model Using the Euler-Maruyama approach, the trajectories or approximated solutions of a stochastic system problem are found using the following equation: χti+1 = χti + α(ti, χti) + β(ti, χti)∆Ai. (24) The case where i = 0, 1, ..., n − 1. Comprehending the computation of ∆Ai is necessary for the computational implementation of the process. instances in which i = 0, 1, ..., n−1. The computer implementation of the procedure requires an understanding of how to com- pute of ∆Ai. Suppose that η is a random variable with an average η ∼ M(0, 1). A normal distribution is thus indicated by √ ∆tη1 with zero mean and variance ∆t, that is,√ ∆tη1 ∼ M(0,∆t). The system of dynamic differential equations (15) must be appropri- ately separated for our suggested model in order to apply the Euler-Maruyama approach method similarly (24). This can be done by Sti+1 = Sti + [ b− KSI 1+βI − d0S ] ∆t+ √ ∆tη1, Iti+1 = Iti + [ KSI 1+βI − (α+ µ+ d0)I ] ∆t+ √ ∆tη2, Tti+1 = Tti + [µI − (r + d0)T ]∆t+ √ ∆tη3, Rti+1 = Rti + [rT + αI − d0R]∆t+ √ ∆tη4. (25) We consider the initial data as given by (S(0), I(0), T (0),R(0)) = (800, 700, 500, 200) and use the parameters values given in Table 3 to simulate the results of stochastic model in figure 8. In addition, we compare the deterministic and stochastic model in figures 9-12 respectively of different classes. In the model, the susceptible population gradually decreases as individuals become infected. The number of infected cases initially rises for a R. ud Din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6364 19 of 23 Time (days) 0 20 40 60 80 100 P o p u la ti o n 0 500 1000 1500 Susceptible class Infected class Treatment class Recovered class Figure 8: Papulation dynamics of stochastic model (15). Time (days) 0 5 10 15 20 25 S u s c ip ta b le c la s s 0 200 400 600 800 1000 Stochastic Deteministic Figure 9: Comparison between deterministic and stochastic numerical interpretation of susceptible class under stochastic type model. period of time but then begins to decline as treatment is introduced. During this process, some individuals recover naturally without treatment, while others recover after receiving treatment. As a result, the recovered population increases steadily and grows rapidly over time. Furthermore, a comparison between the numerical results of the stochastic and deterministic models shows that both approaches are in close agreement, confirming the reliability of the simulations. 10. Conclusion In recent years, eye infectious diseases have attracted increasing attention in research, highlighting the need for new mathematical frameworks to study their dynamics from multiple perspectives. In this work, we developed and analyzed a mathematical model of eye infections using both deterministic and stochastic approaches. Key analyses were conducted, including the computation of disease-free and endemic equilibria, as well as the derivation of the basic reproduction numbers for both models. Theoretical results on existence and stability were established using tools from nonlinear analysis. For the R. ud Din et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6364 20 of 23 Time (days) 0 5 10 15 20 25 In fe c te d c la s s 500 1000 1500 2000 2500 3000 Stochastic Deteministic Figure 10: Comparison between deterministic and stochastic numerical interpretation of infected class under stochastic type model. Time (days) 0 5 10 15 20 25 T re a tm e n t c la s s 0 500 1000 1500 2000 Stochastic Deteministic Figure 11: Comparison between deterministic and stochastic numerical interpretation of treatment class under stochastic type model. deterministic model, a nonstandard finite difference (NSFD) scheme was employed, while for the stochastic model, a robust numerical algorithm was implemented. Numerical sim- ulations and graphical illustrations were provided, and a comparison between the two modeling approaches demonstrated close agreement, thereby validating the accuracy and reliability of the proposed methodology. 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