EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6517 ISSN 1307-5543 – ejpam.com Published by New York Business Global Numerical and Statistical Sensitivity Analysis of Deterministic and Stochastic SVEITR Model for Measles Infection Madhia Mohiudin1,∗, Sana Ullah2, Shema Khan3, Naveed Ahmad4, Sadique Ahmad4, Mohammed A. Elaffendi4 1 Department of Mathematics, COMSATS University Islamabad, Islamabad Campus, Park Road, Tarlai Kalan, Islamabad 45550, Pakistan 2 Department of Mathematics, University of Malakand, Chakdara Dir(L), 18000, Khyber Pakhtunkhwa, Pakistan 3 Department of Mathematics, Government Post Graduate Jahanzeb College, Swat, KPK, Pakistan 4 IAS Data Science and BlockChain Laboratory, College of Computer and Information Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia Abstract. Measles is a global health concern due to its high contagiousness and rapid transmis- sion, especially in low-vaccination areas. This article uses an SVEITR model to explain measles transmission and control dynamics. The model is stable at two equilibria and can be verified using the Non-Standard Finite Difference (NSFD) technique. The study offers valuable advice to pub- lic health professionals and legislators working to prevent and manage measles, highlighting the importance of international cooperation and vaccination in preventing the disease’s return. 2020 Mathematics Subject Classifications: 92D30, 34D20, 60H10, 65L12 Key Words and Phrases: Statistical sensitivity analysis, Basic reproduction number, Global stability, Lyapunov function, Stochastic SVEITR model 1. Introduction Measles is an extremely transmissible virus that has remained a major worldwide health problem in spite of medical science improvements. Measles, which is characterized by a characteristic rash and influenza-like symptoms, is a serious concern, especially in places with poor vaccination rates and limited access to healthcare [1]. Respiratory droplets are ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6517 Email addresses: madhia.mohiudin@gmail.com (M. Mohiudin), safi1987maths@gmail.com (S. Ullah), shemakhan805@gmail.com (S. Khan), nahmed@psu.edu.sa (N. Ahmad), saahmad@psu.edu.sa (S. Ahmad), affendi@psu.edu.sa (M. A. Elaffendi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 2 of 40 the main way that the Measles virus (MeV), which causes the viral infection, is spread. Its high contagiousness is highlighted by the fact that an infected person might spread the illness to others before symptom manifest. Measles, which is spread by inhaling air polluted by asymptomatic individuals or by coming into contact with contagious respira- tory or swallowing secretions (things like coughing or sneezing), is one of the most widely transmitted contagious diseases in the world. For as long as two hours, the virus can be spread through the atmosphere or on contaminated surfaces. Because of this, measles is extremely contagious, and an infected individual can spread the disease to nine among ten of their close, unvaccinated acquaintances. An infected person can spread it between four days before the rash appears and four days after it does. Outbreaks of measles can cause serious complications and even death, particularly in young, undernourished children. In nations nearing measles eradication, imported cases continue to be a significant source of infection. Measles may spread quickly, which makes it a powerful outbreak agent, partic- ularly in areas with high population density [2, 3]. The first signs of a measles infection appear after an incubation time frame of roughly 10 to 14 days. Red, watery eyes, a runny nose, coughing, a severe temperature, and a body rash are common symptoms. The trait Measles rash usually starts on the neck and face and spreads down the body 7-18 days later contact [4]. It often fades after 5-6 days. After infecting the respiratory system, the measles virus progresses throughout the body. Although it can infect anyone, chil- dren are more frequently infected with the measles virus. Complications can occur, even though most people recover in a few weeks, particularly in susceptible groups such small children, pregnant women, and those with compromised immune systems. Women who contract measles during pregnancy run the risk of harming themselves and giving birth to a preterm, underweight child. Measles can cause serious complications, such as brain in- flammation, pneumonia, blindness, severe diarrhea and associated dehydration, infections of the ears, severe respiratory issues, and even death [5]. Complications are expected to affect children under five and adults over thirty. They are more common in malnourished youngsters, especially those whose don’t get enough vitamin A or which have compromised immune systems because of HIV or other diseases. Since measles itself weakens immunity and might make the body lose how to fight off illnesses, children are especially vulnerable. Since vaccinations can largely prevent measles-related deaths, immunization campaigns are essential to halting the disease’s spread. Since measles itself weakens immunity and might make the body lose how to fight off illnesses, children are especially vulnerable. Immunization campaigns are crucial to stop- ping the spread of measles since immunizations can significantly reduce the number of deaths caused by the illness. A safe and efficient method of establishing immunity for the virus is the measles vaccination, which is frequently given as a component of the rubella-mumps-Measles (RMM) vaccine [6, 7]. In many regions of the world, extensive vaccination campaigns have resulted in a notable decline in cases of measles and fatal- ities. To stop outbreaks and keep disease under control, high immunization rates must be maintained. When communities achieve high vaccination coverage, particularly with a two-dose regimen, they can create herd immunity, which provides additional security to those who cannot receive vaccinations for medical reasons [8, 9]. Many nations and M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 3 of 40 international health organizations aim to eradicate measles. It will take consistent work to achieve eradication, including making sure vaccines are accessible, bolstering health- care systems, and combating vaccine reluctance and disinformation. Given how quickly measles can spread across borders, international cooperation and coordination are crucial. Despite advancements, difficulties still exist. Measles control efforts may be hampered by vaccine hesitancy, which occurs when people postpone or refuse immunization even though they have access to it [10, 11]. Measles outbreaks have been made possible by pockets of unvaccinated people who have been influenced by false information on vaccines and their safety. Due to limited outbreaks and a return of cases in certain locations, measles has received fresh attention in recent years. These occurrences highlight the significance of aggressive immunization campaigns and ongoing attention [12]. Every child ought to have a measles vaccination. The vaccination is affordable, safe, and effective. To guarantee immunity, children must get two doses of the vaccination. In nations where measles is prevalent, the first dosage is often administered at nine months of age; in other nations, it is given between 12 and 15 months. Later in childhood, often between 15 and 18 months, a second dosage should be administered. The measles vaccine is administered either by itself or frequently in conjunction with the varicella, mumps, and rubella vaccines. Reducing the number of measles deaths worldwide requires both routine vaccination and major immunization programs in nations with high case rates. For less than 1USD per child, the measles vaccine remains in use for almost 60 years. In an emer- gency, the measles vaccine also serves to prevent the spread of outbreaks. Refugees are more vulnerable to measles epidemics, so they should get vaccinated as quickly as feasi- ble [13]. Combining vaccines gives a boost of immunity of rubella, a particularly common vaccine-preventable disease that may infect unborn children, while also allowing for pooled delivery and administration costs. About 83% of children worldwide received one dose of the measles vaccine before the time of their first birth, and 74% of infants received both doses in 2022 [14]. Since not all children gain immunity after the first dose, two boosters of the vaccination are advised to guarantee immunity and stop outbreaks. In 2022, almost 22 million babies did not receive at least one dose of the measles vaccination as part of their regular immunization schedule. Measles does not have a specific therapy. The goals of care should be to reduce symptoms, ensure the patient is comfortable, and avoid com- plications. Fluids lost due to vomiting or diarrhea can be replaced by drinking adequate water and receiving dehydration treatments. Consuming a nutritious diet is also crucial. Antibiotics are sometimes prescribed by doctors to treat ears and eye infection as well as pneumonia. Vitamin A supplements should be administered in two doses, separated by 24 hours, to all children and adults who have measles. This raises the vitamin’s levels that are low, even in youngsters who are fed a healthy diet. It can aid in preventing blindness and eye damage. Supplementing with vitamin A may help lower the incidence of measles deaths. A key approach for examining the dynamics of disease transmission within communities is mathematical modeling of pandemics, which makes use of computational simulations and mathematical equations [7, 15]. These models enable public health professionals make well-informed decisions by illustrating the interactions between vulnerable, recovered, and M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 4 of 40 infected individuals. This gives them insights into how illnesses propagate. These models forecast how an epidemic will develop by taking into account a number of variables, in- cluding population size, transmission rates, and response tactics. They are able to predict when peak cases will occur, the possible number of infections, and the effects of vari- ous control strategies. Sensitivity analysis finds important parameters that affect results and aids in evaluating the model’s robustness. Furthermore, mathematical modeling has played a critical role in guiding the implementation of measures like lockdowns, testing, and vaccination campaigns in response to epidemic outbreaks. Models are still essential resources for comprehending epidemics and creating successful treatments to safeguard public health, despite their limitations brought on by uncertainties and simplifications [16]. Analyzing the effects of vaccine hesitancy, taking into account geographical con- siderations for limited pandemics, and dealing with the connection between measles and other illnesses within large epidemiological networks have been the main focuses of recent advances in mathematical simulation of measles [17]. A mathematical approach called optimal control theory, when applied to epidemic models, seeks to determine the best ways to stop the spread of infectious illnesses inside the commu- nity [17–19]. The theory aims to ascertain the best way to distribute resources, including vaccination campaigns, quarantine policies, medication transportation, and various other public health initiatives, by using mathematical models that illustrate the dynamics of dis- ease transmission. This entails developing a mathematical function of objective variables that measures the expenses related to the effects of the disease and the use of preventative interventions. Then, over time, the control variables-which stand in for the interventions- are modified to minimize the desired function while respecting realistic restrictions like financial and resource constraints. By taking into account both epidemiological dynamics and pragmatic factors, optimum theory of control [20] offers decision-makers a methodical and mathematical system for proactively managing epidemics. Furthermore, sensitivity evaluation and measurement of uncertainty are being used in optimal control studies to deal with the inherent risk in disease models [21, 22]. This makes it easier to comprehend how resilient control techniques are to different situations and parameter uncertainty. Optimal control measures today frequently incorporate thorough cost-benefit analysis in addition to minimizing illness burden. This means assessing how different solutions will affect the economy in terms of medical expenses, lost output, and intervention costs. By taking into account both financial and health outcomes, this method helps policymakers make well-informed decisions [23–25]. Measles continues to pose a concern to world health because of its high contagiousness and quick dissemination, particularly in places having low immunization rates. The high frequency of cross-border transmission emphasizes how urgently worldwide vaccination and concerted efforts to stop epidemics are needed. The ultimate goal is to contribute to the possible eradication of measles by creating efficient disease management systems. The goal of this study is to improve knowledge about measles control and transmission. Thus, a thorough investigation of viral transmission dynamics, with an emphasis on compre- hending, controlling, and improving control methods, serves as the driving force behind the study. The goal of the study is to make a significant contribution to the fields of pub- M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 5 of 40 lic health policy and infectious disease modeling. The project is to support international health initiatives and policies targeted at avoiding and managing measles outbreaks by evaluating the efficacy of various control tactics, such as continuous measures and optimal interventions. Enhancing our knowledge of measles behavior and offering evidence-based suggestions for better disease management globally are the ultimate goals. A research team recently created an SEIR Measles simulation [26], which was released in 2019. Their goals were to conduct numerical simulations to examine the evolution of Measles disease in the future and to assess the model’s existence, uniqueness, and local stability in a disease-free state only. But there are several problems with their strategy. For instance, they employed a selectively converging RK4 numerical scheme to confirm their analytical conclusions, car- ried out an insufficient sensitivity analysis, and failed to demonstrate the models global and local asymptotic processes at its equilibrium states. Measles can be eliminated from the human population, but their study doesn’t show how or by what means. It is essential to comprehend how interventions can affect the dynamics of measles in order to influence public health policies. It would be beneficial to encourage the researchers to investigate other well-thought-out intervention options, such vaccination campaigns or restrictions on quarantine and therapy, and to assess how well they work to control or eradicate measles from the human population. In order to examine the long-term dynamics of measles dis- ease, we developed an expanded SVEITR model in the current study by including therapy and vaccination compartments [26]. Those that are infected are referred to treatment, where they are segregated or given the necessary medical attention. In the meanwhile, those with strong immune systems can heal themselves without medical intervention. In situations where there is still a possibility of exposure for vaccinated persons due to an incomplete vaccine, susceptible individuals are vaccinated. We extended the model [26] and showed its existence and uniqueness, key properties, and both global and local hy- perbolic stability at its equilibrium states in order to make the problem well-posed. To evaluate the effect of the various parameters on the spread of the disease, sensitivity anal- ysis is also carried out. In order to compare the deterministic and stochastic numerical simulations, we also transform the deterministic model to the stochastic model. The com- parison demonstrates the relationship between the dynamics and how they change over time. There are ten sections in the manuscript. We developed a dynamic deterministic measles model in Section 2 and evaluated a number of its qualitative biological charac- teristics. We have demonstrated that there is a unique, positive, and bounded solution in order to establish epidemiologic significance, making the problem clearly specified in Sect. 3. The model’s identification with both disease-free and epidemic equilibrium states was crucial to understanding the long-term behavior of the illness. The numerical value of the threshold factor is calculated using the next-generation approach in section 4. The model’s local and global stable at its equilibria was established by a thorough stability study, with the threshold number subjected to the required circumstances. Nonetheless, Lyapunov theory was used to demonstrate global stability in section 5. The model’s sen- sitivity analysis is shown in section 6, and the deterministic model’s numerical analysis is presented in section 7. Section 8 presents the deterministic model in its stochastic form, while Section 9 presents its numerical simulation. The conclusion of the entire book is M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 6 of 40 presented in the last section. 2. Formulation of the Model Numerous papers have detailed a number of mathematical models to examine the dy- namics behind the spread of measles [27–31]. These models are useful in a variety of epidemiological fields and are essential for aiding public health planners and policymakers in their decision-making processes. The underlying evolution of infectious diseases like measles, transmission patterns, and the possible effects of public health measures are all better understood by mathematical modeling [32, 33]. In order to assess interventions, plan studies, and contain epidemic outbreaks, modeling has become more and more im- portant. In addition to helping with the testing and development of important theories, epidemiological models [34, 35] offer a conceptual grasp of how diseases propagate. These models aid in the creation of hypotheses, the validation of experimental designs, the inter- pretation of results, the derivation of diagnoses from observable symptoms and indications, the support of decision-making, and the validation of findings [36]. In essence, modeling is a crucial statistical tool in the continuous fight against infectious illnesses, including measles. In order to examine the variations in measles transmission, we have developed a novel real- world SEITR model in this study. Assessing the effectiveness of vaccination and treatment plans in preventing the spread of measles, both before and after optimization stages, is our main goal. It is important to emphasize that the foundation of our concept is the idea that vaccination tactics are not infallible. This indicates that it takes into account the chance that individuals may be re-susceptible throughout the vaccine period, which could result interesting dynamic. Additionally, it takes into consideration the chance that some people may end up healthy and unaffected before treatment. Our research enables us to investigate the effects of treatment and immunization plans in real-world scenarios, when it might not be possible to achieve 100% vaccine coverage. We believe that our research will greatly advance our knowledge of practical strategies for halting the spread of measles. All things considered, our research tackles a significant public health issue and seeks to offer insightful information about the mechanisms of measles spread and the effi- cacy of intervention techniques in practical settings. There are six different compartments for the entire population, denoted by M(t). A healthy individual who runs the danger of getting sick after coming into contact with an infected person is said to be susceptible S(t) represents the first class. The second class, known as the vaccinated class or V(t), is made up of healthy people who have been vaccinated to lessen their vulnerability to the illness. The next compartment, known as the exposed class and represented by E(t), is made up of exposed people. The exposed persons are those who are infected and, while they may or may not exhibit symptoms, are unable to transmit others. People who are at present infectious and able to infect others are included in the fourth class, denoted by I(t), sometimes known as the infected class. Those receiving medical treatment are included in the fifth class, represented by T(t), which is referred to as the treatment class. Lastly, the sixth group, designated R(t), also known as the recovered class, is made up M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 7 of 40 of people who have been cured of the illness by therapy or vaccination and are unable to contract it again. The total population is taken as M(t) = S(t) + V(t) + E(t) + I(t) + T(t) + R(t). New people joining the susceptible class is happening at b rate. A common approach to depict the frequency of infection spread from susceptible to people who are exposed due to interaction with infected individuals is to use the strength of disease, which is represented by the symbol k. In essence, it is a measurement of the likelihood of transmission for each interaction between an infectious person and a susceptible person. The frequency with which susceptible people are immunized to develop disease immunity is indicated by the metric c2. The µ0 represents the intrinsic rate of death in the susceptible group, while c1 are re-susceptible rate. Thus, the overall pace of transformation of those who are most susceptible can be expressed as dS(t) dt = b+ c1V− kSI M − (c2 + µ0)S. Vaccinated people may become susceptible because of their compromised immunity or other reasons, and they will then move to the exposed compartment E(t) at the amount represented by c1. Recruitment of new individuals from the susceptible compartment S(t) to the vaccinated class occurs at a rate of c2. µ0 is a representation of the vaccination compartment natural mortality rate. The following is the net rate of change for those who have received vaccinations, dV(t) dt = c2S− (c1 + µ0)V. k represent the susceptible compartment contribute additional members to the compart- ment of exposed individuals. α shows the pace at which members of this exposed com- partment migrate to the infected compartment. r1 is the recovering rate for people in the exposed compartment. Additionally, the rate µ0 indicates a departure from the exposed compartment because of natural death. The usual differential equation describes the total change in the number of people in the exposed group as, dE(t) dt = kSI M − (α+ r1 + µ0)E. At a rate of α, new members of the infectious class are recruited by the exposed compart- ment. At a rate of θ, those in the infectious group who are severely afflicted are sent for treatment. The natural death rate is shown by µ0, while the infection death rate denoted by d1. Consequently, the net rate of change of people within the infectious class is taken into consideration using the following equation, dI(t) dt = αE− (θ + d1 + µ0)I. Infectious individuals enter the treatment compartment at θ, and go to the recovered com- partment at r2. The rate of natural death causes an leave from the therapy compartment M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 8 of 40 at µo, while the death rate during treatment is by d2. Consequently, the following formula represents the net variation in the treatment compartment, dT(t) dt = θI− (r2 + d2 + µ0)T. Accordingly, r1 and r2 represent the transmission rates from the treatment and exposed compartment to the recovered compartment. Due to the natural death rate µ0, some people leave the recovered class. The formula that determines the total rate of change for people in the recovered compartment over time is hence dR(t) dt = r2T+ r1E− µ0R. To demonstrate the dynamics of measles, we develop a thorough mathematical model by integrating all of the previously discussed differential equations. dS(t) dt = b+ c1V− kSI M − (c2 + µ0)S, dV(t) dt = c2S− (c1 + µ0)V, dE(t) dt = kSI M − (α+ r1 + µ0)E, dI(t) dt = αE− (θ + d1 + µ0)I, dT(t) dt = θI− (r2 + d2 + µ0)T, dR(t) dt = r2T+ r1E− µ0R. (1) Subject to the initial condition: S(0) ≥ 0,V(0) ≥ 0,E(0) ≥ 0, I(0) ≥ 0,T(0) ≥ 0,R(0) ≥ 0. The model is displayed in the form of flow chart as follows: S V E I T R b kSI M c2 c1 α r1 θ r2 µ0 µ0 µ0 µ0 + d1 µ0 + d2 µ0 M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 9 of 40 3. Mathematical Analysis This section covers the model (1) characteristics, including the solution’s boundedness, positivity, and well-posedness t ≥ 0. Let y : [0 → ∞) =⇒ R6 be provide by y = (S+ V+ E+ I+ T+ R)T a vector common in R6 comprising six state parameters. Let g(y) = (g1(y) + g2(y) + g3(y) + g4(y) + g5(y) + g6(y)) T be provided by g : R6 =⇒ R6, a function with a vector value defined by R6 where g1(y) = b+ c1V− kSI M − µ0S− c2S, g2(y) = c2S− c1V− µ0V, g3(y) = kSI M − αE− r1E− µ0E, g4(y) = αE− θ)I− (d1 + µ0)I, g5(y) = θI− r2T− (d2 + µ0)T, g6(y) = r2T+ r1E− µ0R. (2) In compact form, the system (1)− (2) can be expressed as: dx dt = G(y), x(0) = x0 (3) Here x0 = (S(0),P(0), I(0),T(0),R(0))T Theorem 1. The function G is continuous Lipschitly in y. Proof. As six state variable, (S,V,E, I,T and R) are continuous differentiable equations of t and hence are the elements, g1(y), g2(y), g3(y), g4(y), g5(y) and g6(y) of the function G. Let assume, L(y, z, b) = x+ b(z − y) : b ∈ [0, 1], y, z ∈ R6. A set L(y, z, b) is a section of a line in R6 which meets the point y to z as b varies from 0 to 1, and is a compact subset of R6. Therefore, we can select a point c ∈ L(y, z, b) so that the subsequent equalities hold by using the Mean Value Theorem, ∥G(z)−G(y)∥∞ = ∥G,(c; z − y)∥∞ (4) Here G(c; z − y) is a directional derivatives of the function G at c in the direction of z-y. So, we have ∥G,(c; z − y)∥∞ = 6∑ i=1 ∥Jgi(z).(z − y)ei∥∞ ≤ 6∑ i=1 ∥Jgi(z)∥∞∥z − y|∞. The abounded linear operator Jgi(c) and the ith coordinate unit column ei in R6 + are used here. Given the limited nature of the partial derivatives of the function gi, i = 1, 2, 3, 4, 5, 6, there exists an integer K1 such that, 6∑ i=1 ∥Jgi(z)∥∞ ≤ K1, M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 10 of 40 for c ∈ L(y, z, b) ⊆ R6 +. Consequently, we have ∥G(z)−G(y)∥∞ ≤ K1∥z − y|∞, and so its prove that G is Lipschitz continuous. Theorem 2. The system (4) has a unique solution. Proof. According to Theorem 1, a system G presented in (4) is Lipschitz continuous in y. Thus, the system (4) has a unique solution according to the Picard theorem of existence and uniqueness. It is considered practical if the system solution (1), or equivalently (4), is bounded and nonnegative. Since the values of a realistic solution may be determined by gathering data, it is physically dependable. Theorem 3. In R6 +, the nonlinear system (1) contains a global solution (S,V,E, I,T, R) for any nonnegative initial condition. All solutions that begin with that starting data will therefore continue to be advantageous for any t ≥ 0. Proof. Suppose t̄ = sup{t ≥ 0 : S(t) ≥ 0,P(t) ≥ 0, I(t) ≥ 0,T(t) ≥ 0,R(t) ≥ 0}, Clearly t̄ ≥ 0. The first equation of the system leads to dS(t) dt = b+ c1V− kSI M − µ0S− c2S, (5) from (5), we have dS(t) dt + ( kI M + c2 + µ0)S = b+ c1V, (6) now, put b+ c1V = b1 in (6), implies that dS(t) dt = ( kI M + c2 + µ0)S = b1. (7) take left hand of (7), as kI M + c2 + µ0S = g(t), so we have dS(t) dt + g(t)S = b1, By integrating factor, I.F = e ∫ t 0 g(t)dt = e ∫ t 0 g(t) dt, d dt S(t)e ∫ t 0 g(t) dt = b1e ∫ t 0 g(t) dt, integrating from t = t̄ to t = 0, and multiply e− ∫ t̄ 0 f(t) dt, we get S(t̄)e− ∫ t̄ 0 f(t) dt = S(0)e− ∫ t̄ 0 f(t) dt + b1e − ∫ t̄ 0 f(t) dt ∫ t̄ 0 ∣∣∣e∫ y 0 g(t) dt ∣∣∣ dy. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 11 of 40 Since S(0) ≥ 0, the sum of the positive terms is positive, hence S(t) is positive. Similarly V(t), E(t), I(t), T(t), and R(t) for any set of nonnegative initial conditions used in R6 + can be shown to hold for any t ≥ 0. Moreover, the conclusion remains x = 0 for all t ≥ 0 if x0 = 0. R6 + is therefore a positively invariant region. Which follow the conclusion. Theorem 4. The theorems (3.2) and (3.3) provide bounded solutions to the systems (1). Proof. The system (1) has positive solution using initial condition, S(0) ≥ 0,V(0) ≥ 0,E(0) ≥ 0, I(0) ≥ 0,T(0) ≥ 0,R(0) ≥ 0. To prove the bondness of the solution, we used the total population M(t) as, M(t) = S(t) + V(t) + E(t),+I(t) + T(t) + R(t) Differentiating M(t) with respect to t and adding the six equation from system (1), we obtained that dM(t) dt = b− µ0M(t)− d1I(t)− d2T(t). This implies that dM(t) dt ≤ b− µ0M(t). By integrating, we get∣∣∣∣ln( b− µ0M(t) b− µ0M(0) )∣∣∣∣ ≥ −µ0t, Simplify, M(t) ≤ b µ0 + (M(0)− b µ0 )e−µ0t. If we take t =⇒ ∞, then the factor e−µ0t = 0, So M(t) ≤ b µ0 . Hence all solution of the model (1) are bounded in the region ϕ. ϕ = { (S(t),V(t),E(t), I(t),T(t),R(t)) ∈ R6 + ∣∣ S(t) + V(t) + E(t) + I(t) + T(t) + R(t) ≤ b µ0 , S(t),V(t),E(t), I(t),T(t),R(t) ≥ 0 } As M(t) ≤ b µ0 , when time t approaches to infinity. Consequently, ϕ is a positively invariant closed set. Thus, the proposed measle model (1) is well-posed and biologically meaningful since all of its solutions are positive and bounded. Disease Free Equilibrium Points: In this section, we find the measle free equilibrium points for the system (1). The SVEITR model (1) has a unique measles free equilibrium point that is given as, X 0 = ( S0,V0E0, I0,T0,R0 ) = ( S0,V0, 0, 0, 0, 0 ) = ( b(c1 + µ0) (c2 + µ0)(c1 + µ0)− c1c2 , bc2 (c2 + µ0)(c1 + µ0)− c1c2 , 0, 0, 0, 0 ) . M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 12 of 40 Endemic equilibrium: The unique endemic equilibrium point F∗ for the model (1) is also computed as in region ϕ, where S∗(t) = b(c1 + µ0)M∗ (kI∗ − (c2 + µ0)M∗)(c1 + µ0)− c1c2M∗ , V∗(t) = c2b(c1 + µ0)M∗ (c1 + µ0)[(kI∗ − (c2 + µ0)M∗)(c1 + µ0)− c1c2M∗] , E∗(t) = kb(c1 + µ0)I∗ (α+ r1 + µ0)[(kI∗ − (c2 + µ0)M∗)(c1 + µ0)− c1c2M∗]M∗ , I∗(t) = ((c2 + µ0)(c1 + µ0) + c1c2)M∗ k(c1 + µ0 + αkb k(c2 + d1 + µ0)(α+ r1 + µ0)M∗ , T∗(t) = θI∗ (r2 + d2 + µ0) , R∗(t) = r2θI∗ µ0(r2 + d2 + µ0) + r1kb(c1 + µ0)I∗ (α+ r1 + µ0)[(kI∗ − (c2 + µ0)M∗)(c1 + µ0)− c1c2M∗]M∗ . 4. Expression for R0 The basic reproduction number R0 is determined by analyzing the next-generation procedure in this section. The Basic Reproduction number R0 is a quantity used in epidemiology to describe how illness spreads and is controlled. We can determine the extent of the disease in the population and the best ways to protect community members from this deadly virus using R0. The following R0 is found using the next generation method, let Φ = (F , E), then form system (1), we have dΦ dt = F(x)− E(x) Here we have F(x) =  kSI M 0 0  , E(x) =  (α+ r1 + µ0)E (θ + d1 + µ0)I− αE (r2 + d2 + µ0)T− θI  When F and E are in disease-free equilibrium, their Jacobian at X 0 is calculated, F(x) =  0 kS0 M0 0 0 0 0 0 0 0  , E(x) =  (α+ r1 + µ0) 0 0 α (θ + d1 + µ0) 0 0 θ (r2 + d2 + µ0)  The basic reproduction number R0, is calculated as, R0 = ρ(FV −1)(X 0) = kα(c1 + µ0) (α+ r1 + µ0)(θ + d1 + µ0)(c1 + c2 + µ0) M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 13 of 40 Here, we arrive to the following conclusion based on the basic reproduction number R0. 5. Stability Analysis of DFE and EEP of model (1) Here in this part of our work consisted of the stability analysis of the system (1). Furthermore, this section is divided into two subsections: local stability of the model and global stability of the model. 5.1. Local stability of disease free equilibrium (DFE) In this section, we study the local stability of the measles model (1) in the next theorem by using the Jacobian matrix. Theorem 5. “If R0 ≤ 1, the DFE of the system (1) about an equilibrium point F 0 is locally asymptotically stable (LAS); if R0 ≥ 1, it is unstable. Proof. The Jacobian Matrix at J0 is given below J(F 0) =  −(c2 + µ0) c1 0 −kµ0 (c2+µ0)(c1+µ0)−c1c2 0 0 c2 −(c1 + µ0) 0 0 0 0 0 0 −(α+ r1 + µ0) kµ0 (c2+µ0)(c1+µ0)−c1c2 0 0 0 0 α −(θ + d1 + µ0) 0 0 0 0 0 θ −(r2 + d2 + µ0) 0 0 0 r1 0 r2 −µ0  . The local stability criteria of Meals-free equilibrium are ascertained by analyzing the eigen- values of the Jacobian matrix of the system (1) from columns C6 and C5, we determine two eigenvalues for direct expansion,and the required eigenvalue is given as: λ1 = −µ0, λ2 = −(r2 + d2 + µ0), (8) and the Jacobian matrix is reduced to 4X4 matrix as, J(F 0) =  −(c2 + µ0) c1 0 −kµ0 (c2+µ0)(c1+µ0)−c1c2 c2 −(c1 + µ0) 0 0 0 0 −(α+ r1 + µ0) kµ0 (c2+µ0)(c1+µ0)−c1c2 0 0 α −(θ + d1 + µ0)  . A characteristic equation of this Jacobian matrix is given by: λ4 + a1λ 3 + a2λ 2 + a3λ+ a4 = 0. (9) M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 14 of 40 Here a1 = c2 + c1 + α+ r1 + θ + d1 + 4µ0, a2 = µ0(c2 + c1 + µ0) (α+ r1 + µ0)(θ + d1 + µ0) + (c2 + c1 + 2µ0)(α+ r1 + θ + d1 + 2µ0) (α+ r1 + µ0)(θ + d1 + µ0) + (1−R0), a3 = µ0(α+ r1 + d1 + 2µ0)(c2 + c1 + µ0) (c2 + c1 + 2µ0)(α+ r1 + µ0)(θ + d1 + µ0) + (1−R0), a4 = µ0(c2 + c1 + µ0)(α+ r1 + µ0)(θ + d1 + µ0)(1−R0). (10) Using the Routh-Hurwitz condition, a1a2a3 > a23 + a21a4. Hence our conclusion is gained as λ1 = −µ0 < 0 and λ2 = −(r2 + d2 + µ0) < 0. Also the condition a1a2a3 > a23 + a21a4 is hold from (9). 5.2. Global stability of disease free equilibrium (DFE) Global stability of the model (1) is investigated by the following theorem. Theorem 6. “If k5 ≤ c2 and R0 ≤ 1, the DFE of the system (1) about an equilibrium point F 0 is globally asymptotically stable (GAS); if R0 ≥ 1, it is unstable. Proof. We define Lyapunov function as U(t) = A1E+ A2I+ A3T, Take derivative with respect to t, we have U̇(t) = A1Ė+ A2İ+ A3Ṫ. Putting the values from system (1), we get ˙U(t) ≤ A2k3 ( A1k + A3θ A2k3 − 1 ) I+ (A2α+ A1k2)E− (A3k4)T. (11) Where A1 = c1α, A2 = K2k5, A3 = kαµ0 θ . From (11), we have ˙U(t) ≤ kk3αµ0 θ (R0 − 1) I+ k2α(k5 − c1)E− ( kk4αµ0 θ ) T. So, we have ˙U(t) ≤ 0, if k5 ≤ c1 and R0 ≤ 1 which follow the conclusion. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 15 of 40 5.3. Local stability of endemic equilibrium point (EEP) F∗ Theorem 7. “If R0 ≥ 1, the DEE of the system (1) about an equilibrium point F∗ is (LAS) locally asymptotically stable otherwise unstable. Proof. The Jacobian Matrix at J∗ is given below J(F ∗) =  −(k0 + kI∗ M∗ )− c∗2 c1 0 −kS∗ M∗ 0 0 c2 −k1 − c∗2 0 0 0 0 kI∗ M∗ 0 −k2 − c∗2 kS∗ M∗ 0 0 0 0 α −k3 − c∗2 0 0 0 0 0 θ −k4 0 0 0 r1 0 r2 −µ0  , Local stability criteria of measles-endemic equilibrium are ascertained by analyzing the eigenvalues of the Jacobian matrix of the system (1) from columns C6 and C5, we determine two eigenvalues for direct expansion,and the required eigenvalue is given as: λ1 = −µ0, and λ2 = −k4 = −(r2 + d2 + µ0). Furthermore, the Jacobian matrix is reduced to 4X4 matrix. J(F∗) =  −(k0 + kI∗ M∗ )− c∗2 c1 0 −kS∗ M∗ c2 −k1 − c∗2 0 0 kI∗ M∗ 0 −k2 − c∗2 kS∗ M∗ 0 0 α −k3 − c∗2  . The characteristic equation of the above Jacobian matrix is given as: c∗2 + b1c ∗ 2 + b2c ∗ 2 + b3c ∗ 2 + b4 = 0. Here b1 = k0 + k1k2k3 + kI∗ M∗ , b2 = k0(k1 + k2 + k3) + kI∗ M∗ (k1 + k2 + k3) + k1(k2 + k3) + (k2k3)− kS∗ M∗ − c1, b3 = k1k2k3 − kk1αS∗ M∗ + k0(k1(k2 + k3) + k2k3 − kαS∗ M∗ ) + kI∗ M∗ (k1(k2 + k3) + k2k3 − kαS∗ M∗ )− c1(k2 + k3), b4 = k0k1k2k3 − kk0k1αS∗ M∗ + kk1k2k3αI∗ M∗ − k2k1αS∗I∗ M∗2 − c1k2k3 + kαS∗c1 M∗ . (12) The value of characteristic polynomial, will be negative, if it satisfied the Routh-Hurwitz Criteria that is n = 4, b1 > 0, b3 > 0, b4 > 0, and b1b2b3 > b23 + a21a4, so we can say that (EEP) of the model (1) will be local asymptotically stable for R0 > 1. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 16 of 40 5.4. Backward bifurcation analysis for Local stability of endemic equilib- rium (EE) Theorem 8. “[28] and [29] If R0 > 1, the DEE of the system (1) about an equilibrium point F∗ is (LAS) Locally asymptotically stable otherwise unstable and also satisfy the condition k1 < 1. Proof. Let S = x1,V = x2,E = x3, I = x4,T = x5, dx1 dt = h1, dx2 dt = h2, dx3 dt = h3, dx4 dt = h4, dx5 dt = h5, so the system (1) for first five equations with dx dt = H, where H = (h1, h2, h3, h4, h5) T can be written as: h1(x) = b+ c1x2 − kx1x4 M − µ0x1 − c2x1 = b+ c1x2 − kx1x4 M − (µ0 + c2)x1 = b+ c1x2 − kx1x4 M − k0x1, h2(x) = c2x1 − c1x2 − µ0x2 = c2x1 − (c1 + µ0)x2 = c2x1 − k1x2, h3(x) = kx1x4 M − αx3 − r1x3 − µ0x3 = kx1x4 M − (α− r1 − µ0)x3 = kx1x4 M − k2x3, h4(x) = αx3 − θx4 − (d1 + µ0)x4 = αx3 − (θ − d1 + µ0)x4 = αx3 − k3x4, h5(x) = θx4 − r2x5 − (d2 + µ0)x5 = θx4 − (r2 + d2 + µ0)x5 = θx4 − k4x5. (13) To prove LAS of the system (1), We follow the center Manifold theory in [28], so we choose that k = k∗ as bifurcation parameter when R0 = 1 and assume that equilibrium point exist around the bifurcation point. k∗ = k = (α+ r1 + µ0)(θ + d1 + µ0)(c2 + c1 + µ0) α(c1 + µ0) = k2k3(c2 + c1 + µ0) k1α . (14) So the Jacobian matrix can be calculated at disease-free equilibrium point F0 with k∗ as J(F 0, k∗) =  −(c2 + µ0) c1 0 −k∗µ0 (c2+µ0)(c1+µ0)−c1c2 0 c2 −(c1 + µ0) 0 0 0 0 0 −(α+ r1 + µ0) k∗µ0 (c2+µ0)(c1+µ0)−c1c2 0 0 0 α −(θ + d1 + µ0) 0 0 0 0 θ −(r2 + d2 + µ0)  . Since the Jacobian matrix having a simple eigenvalue at k∗.So it can be seen that the left and right eigenvectors that is denoted byW = (w1, w2, w3, w4, w5) and V = (v1, v2, v3, v4, v5) respectively and calculated as; M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 17 of 40 w1 = ( c1 + µ0 c2 )w2 > 0, w2 > 0, w3 = ( θ + eta1 + µ0 α )w4 > 0, w4 > 0, w5 = ( θ r2 + d2 + µ0 )w4 > 0, and v1 = ( k2k3(c2 + µ0)(c2 + c1 + µ0)− k∗α(c1 + µ0) k2k∗(c1 + µ0) )v4, = ( k0k2k3(c2 + c1 + µ0)− k1k ∗α k1k2k∗ )v4, v2 > 0, v3 = ( α k2 )v4, v4 > 0, v5 = 0. Evaluation of a: By using, ∂2g1 ∂x1∂x4 = −kµ0 b , ∂2g3 ∂x3∂x4 = kµ0 b , We calculate a from a = 2v1w1w4 ∂2g1 ∂x1∂x4 + 2v3w1w4 ∂2g3 ∂x3∂x4 , After simplification, we have a = 2 k1 c2 kµ0 b w2w4[( α k2 ) + ( k0k ∗α− k0k2k3(c2 + c1 + µ0) k1k2k∗ )]v4, Since, we seen that all the terms in expression of a is positive except k0k ∗α− k0k2k3(c2 + c1 + µ0), So a > 0, iff k0k ∗α− k0k2k3(c2 + c1 + µ0) > 0, or simplify as k1 < 1. Evaluation of b: ∂2g1 ∂x4∂k∗ = ∂2g1 ∂k∗∂x4 = − c1 + µ0 (c2 + µ0)(c2 + c1 + µ0) = − k1 k0(c2 + c1 + µ0) . As; b = v1w4 ∂2g1 ∂x4∂k∗ + v3w4 ∂2g3 ∂x4∂k∗ , M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 18 of 40 putting the values and simplify, we have b = w4 k1 k0(c2 + c1 + µ0) [( α k2 ) + ( k0k ∗α− k0k2k3(c2 + c1 + µ0) k1k2k∗ )]v4, Since as we seen that all the terms in expression of b is positive except k0k ∗α − k0k2k3(c2 + c1 + µ0), So b > 0, iff k0k ∗α− k0k2k3(c2 + c1 + µ0) > 0, or simplify as k1 < 1. According to Center Monifold Theory [28] and [29], the system (1) is locally asymp- totically stable (LAS) around the bifurcation point at the endemic equilibrium point, as a > 0 and b > 0 for k < 1. As a result, when R0 = 1, system (1) enters backward bifurcation mode. Because the endemic equilibrium point is LAS, illness will persist in the population for an extended period. 5.5. Global stability of endemic equilibrium (EE) Theorem 9. “If R0 ≥ 1, the DEE of the system (1) about an equilibrium point F∗ is (GAS) globally asymptotically stable otherwise unstable. Proof. Consider a Lyapunov function U1(t) : Q =⇒ R defined by, U1(t) = ( S− S∗ − S∗ ln S(t) S∗ ) + ( V− V∗ − V∗ lnV(t) V∗ ) + ( E− E∗ − E∗ lnE(t) E∗ ) + ( I− I∗ − I∗ ln I(t) I∗ ) + T∗ ( k2 + 1 θI∗ )( T− T∗ − T∗ lnT(t) T∗ ) , We want to show the global stability at point F∗. Differentiate of U1(t) with respect to t, gives us U ′ 1(t) = ( 1− S∗ S ) S ′ + ( 1− V∗ V ) V ′ + ( 1− E∗ E ) E ′ + ( 1− I∗ I ) I ′ +T∗ ( k2 + 1 θI∗ )( 1− T∗ T ) T ′ , (15) let consider first two terms from (15), we have( 1− S∗ S ) S ′ + ( 1− V∗ V ) V ′ = ( 1− S∗ S )( b+ c1V− kIS N − c2S− µ0S ) + ( 1− V∗ V ) (c2S− c1V− µ0V) , = ( 1− S∗ S )( b+ c1V− kIS N − c2S− µ0S− c1V∗ + kI∗S∗ N ∗ + c2S∗ − µ0S∗ ) , after simplification,( 1− S∗ S ) S ′ + ( 1− V∗ V ) V ′ = kI∗S∗ N ∗( 1− S∗ S − ISM∗ I∗S∗M ) + c2S∗ ( 2− S∗ S − SV∗ S∗V ) + µ0S∗ ( 2− S S∗ − S∗ S ) + c1V∗ ( S S∗ − S∗V SV∗ ) + kIS∗ M − µ0V+ µ0V∗, (16) M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 19 of 40 first we solve; −µ0V+ µ0V∗ consider, µ0V∗ − µ0V = µ0V∗ ( 1− V V∗ ) , (17) we know that, µ0V∗ = c2S∗ − c1V∗, So (17) becomes, µ0V∗ − µ0V = (c2S∗ − c1V∗) ( 1− V V∗ ) , = c2S∗ − c1V∗ + c2S∗ V∗ V + c1V, = c2S∗ ( 1− V∗ V ) + c1V∗ ( −1 + V V∗ ) , Putting these values in (16), so we get( 1− S∗ S ) S′ + ( 1− V∗ V ) V′ = kI∗S∗ M∗ ( 1− S∗ S − ISM∗ I∗S∗M ) + c2S∗ ( 2− S∗ S − SV∗ S∗V ) + µ0S∗ ( 2− S S∗ − S∗ S ) + c1V∗ ( S S∗ − S∗V SV∗ ) + c2S∗ ( 1− V∗ V ) + c1V∗ ( −1 + V V∗ ) + kIS∗ M = kI∗S∗ M∗ ( 1− S∗ S − ISM∗ I∗S∗M ) + c2S∗ ( 3− S∗ S − V∗ V − SV∗ S∗V ) + µ0S∗ ( 2− S S∗ − S∗ S ) + c1V∗ ( 1− S∗V SV∗ − ( 1− V∗ V ) − ( 1− S∗ S )) + kIS∗ M . (18) Next we consider,( 1− E∗ E ) E ′ + ( 1− I∗ I ) I ′ = ( 1− E∗ E )( kSI M − k2E ) + ( 1− I∗ I ) (αE− k3I) , = −kISE∗ ME + kIS M − k2E+ k2E∗ + αE− k3I− αE I∗ I + k3I∗. Now putting the value of k2 = α+ r1 + µ0 in above equation, so we get,( 1− E∗ E ) E ′ + ( 1− I∗ I ) I ′ = −kISE∗ ME + kIS M + αE− k3I− αE I∗ I + k3I∗ − (α+ r1 + µ0)E+ (α+ r1 + µ0)E∗ = −kISE∗ ME + kIS M + αE∗ ( 1− E E∗ − EI∗ E∗I ( 1− I I∗ )) + (r1 + µ0)E∗ ( 1− E E∗ ) + k3I∗ ( 1− I I∗ ) M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 20 of 40 we know that k3I∗ = αE∗, by putting in above equation, we get.( 1− E∗ E ) E′ + ( 1− I∗ I ) I′ = −kISE∗ ME + kIS M + αE∗ ( 1− E E∗ − EI∗ E∗I ( 1− I I∗ )) + (r1 + µ0)E∗ ( 1− E E∗ ) + αE∗ ( 1− I I∗ ) , = −kISE∗ ME + kIS M + αE∗ ( 2− E E∗ − I I∗ − EI∗ E∗I ( 1− I I∗ )) + (r1 + µ0)E∗ ( 1− E E∗ ) . (19) Adding equations (18) and (19), we get; ( 1− S∗ S ) S ′ + ( 1− V∗ V ) V ′ + ( 1− E∗ E ) E ′ + ( 1− I∗ I ) I ′ = kI∗S∗ M∗ ( 1− S∗ S − ISM∗E∗ I∗S∗ME ) + c2S∗ ( 3− S∗ S − V∗ V − SV∗ S∗V ) + µ0S∗ ( 2− S S∗ − S∗ S ) + c1V∗ ( 1− S∗V SV∗ − ( 1− V∗ V ) − ( 1− S∗ S )) + αE∗(2− E E∗ − I I∗ − EI∗ E∗I ( 1− I I∗ ) + (r1 + µ0)E∗ ( 1− E E ∗) + kIS∗ M , (20) last two terms of (r1 + µ0)(1− E E ∗ )E∗ + kIS∗ M = (r1 + µ0)E∗ − (r1 + µ0)E∗ + kIS∗ M (21) Since; k2E∗ = kI∗S∗ M∗ or k = k2E∗M∗ S∗I∗ , putting values in (21), so we get, (r1 + µ0) ( 1− E E ∗) E∗ + kIS∗ M = (r1 + µ0)E∗ − (r1 + µ0)E+ k2E∗M∗ MI∗ , = (r1 + µ0)E∗ − (r1 + µ0)E+ IM∗ I∗M (α+ r1 + µ0)E∗, = (r1 + µ0)E∗ − (r1 + µ0)E+ (r1 + µ0)E∗ ( IM∗ I∗M ) + αE∗ ( IM∗ I∗M ) , = (r1 + µ0)E∗ ( 1− E E∗ + IM∗ I∗M ) + αE∗ ( IM∗ I∗M ) , = (r1 + µ0)E∗ ( 1 + 1− E E∗ − 1 + IM∗ I∗M + αE∗ ( IM∗ I∗M )) , = (r1 + µ0)E∗ ( 2− E E∗ − ( 1− IM∗ I∗M ) + αE∗ ( IM∗ I∗M )) . M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 21 of 40 Putting this values in equation (20), we get; ( 1− S∗ S ) S ′ + ( 1− V∗ V ) V ′ + ( 1− E∗ E ) E ′ + ( 1− I∗ I ) I ′ = kI∗S∗ M∗ ( 1− S∗ S − ISM∗E∗ I∗S∗ME ) + c2S∗ ( 3− S∗ S − V∗ V − SV∗ S∗V ) + µ0S∗ ( 2− S S∗ − S∗ S ) + c1V∗(1− S∗V SV∗ − (1− V∗ V )− (1− S∗ S )) + αE∗ ( 2− E E∗ − I I∗ + IM∗ I∗M − EI∗ E∗I ( 1− I I∗ ) + (r1 + µ0)E∗ ( 2− E E ∗ − ( 1− IM∗ I∗M ))) . (22) Now, we consider last term from equation (15), T∗( k1 + 1 θI∗ )(1− T∗ T )T ′ = T∗( k1 + 1 θI∗ )(1− T∗ T )(θI− k4T) Since k4T∗ = θI∗ =⇒ k4 = θI∗ T∗ T∗ ( k1 + 1 θI∗ )( 1− T∗ T ) T ′ = T∗ ( k1 + 1 θI∗ )( 1− T∗ T )( θI− θI′ T∗T ) , = θ(T∗ ( k1 + 1 θI∗ )( 1− T∗ T )( I− I′ T∗T ) , = T∗ ( k1 + 1 I∗ )( 1− T∗ T )( I− I′ T∗T ) , = k1T∗ + T∗ I∗ ( I+ I∗ − TI∗ T∗ − T∗I T ) , = k1T∗I I∗ + k1T∗ − k1T− k1T∗2I TI∗ + T∗I I∗ + T∗ − T− T∗2I TI∗ , = T∗ ( 1− T∗ T − T∗I TI∗ + I I∗ ) + k1T∗ ( 1− T∗ T − T∗I TI∗ + I I∗ ) , = T∗ ( 1− T∗ T − T∗I TI∗ (1− T T∗ ) ) + k1T∗ ( 1− T∗ T − T∗I TI∗ ( 1− T T∗ )) . (23) Adding equations (22) and (23), so finally we get; U ′ 1(t) = ( 1− S∗ S ) S ′ + ( 1− V∗ V ) V ′ + ( 1− E∗ E ) E ′ + ( 1− I∗ I ) I ′ + T∗ ( k1 + 1 θI∗ )( 1− T∗ T ) T ′ = kI∗S∗ M∗ ( 1− S∗ S − ISM∗E∗ I∗S∗ME ) M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 22 of 40 + c2S∗ ( 3− S∗ S − V∗ V − SV∗ S∗V ) + µ0S∗ ( 2− S S∗ − S∗ S ) + c1V∗ ( 1− S∗V SV∗ − ( 1− V∗ V ) − ( 1− S∗ S )) + αE∗ ( 2− E E∗ − I I∗ + IM∗ I∗M − EI∗ E∗I ( 1− I I∗ )) + (r1 + µ0)E∗ ( 2− E E∗ − ( 1− IM∗ I∗M )) + T∗ ( 1− T∗ T − T∗I TI∗ ( 1− T T∗ )) + k1T∗ ( 1− T∗ T − T∗I TI∗ ( 1− T T∗ )) . (24) The following inequality from equation (24) is true since the arithmetic mean is greater than the geometric mean: Hence, we conclude that U′ 1(t) ≤ 0, because the geometric mean is smaller than the arithmetic mean, and exercise the LaSalle’s Invariance rule, we get that the measle model (1) is global asymptotically stable at EEP F∗, when R0 ≥ 1. 6. Sensitivity Analysis This section presents the sensitivity analysis that account for all variables that signif- icantly affect the basic reproduction number R0. Sensitivity analysis is recommended to determine the importance of the numerous factors affecting the occurrence and spread of disease. In addition to preventing the spread of diseases, obtainingR0 < 1 needs managing the system (??) variables. The sensitive index, which shows the ratio of modifications in an element to changes in a parameter, can be calculated using the formula. The sensitivity index for each parameter involved in the fundamental reproduction numbers R0 < 1 is shown in the next 1. g[y] = X R0 ∂R0 ∂X , where X average factor of R0 is the formula used to calculate the The variables of infection constant k and infection rate of measles α are the primary variables that boost the basic reproduction number R0, while the death rate c2 and vaccination rate c1 is the main factor that decreases the basic reproduction number R0. These parameters are directly related to R0. This is evident from Table (1) and Fig. (1). Nonetheless, the variables that have a minor impact on the fundamental reproduction number to increase are c1, while θ, d1 and µ0 are the factors which show decay in the infection flow. The next figure (2) describes the R0’s 3D dynamics. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 23 of 40 Variables Sensitivity Index g(k) 1 g(α) 0.6402 g(µ0) 0.0042 g(r1) -0.6139 g(θ) -0.8866 g(d1) -0.06795 g(c2) -0.5928 g(c1) 0.5168 Table 1: The parameters and their description involved in sensitivity analysis of the model for R0 (1). 7. Numerical Analysis The deterministic system (1) is numerically simulated in this part using the non- standard finite difference technique (NSFD) [37, 38]. First, the model equations are ex- pressed as follows: As an example, the first system equation (1), dS(t) dt = b+ c1V− kSI M − (c2 + µ0)S. (25) From the none-standard finite difference method, we decomposed as Sj+1 − Sj h = b+ c1Vj − kSjIj M − (c2 + µ0)Sj . (26) Further model equations (1) have been scaled down using the non-standard finite difference method, similar to (26). Vj+1 = Vj + h ( c2Sj − c1Vj − µ0Vj ) , Ej+1 = Ej + h ( kSjIj M − αEj − r1Ej − µ0Ej ) , Ij+1 = Ij + h ( αEj − θ)Ij − (d1 + µ0)Ij ) , Tj+1 = Tj + h ( θIj − r2Tj − (d2 + µ0)Tj ) , Rj+1 = Rj + h ( r2Tj + r1Ej − µ0Rj ) . We evaluate the system (1) using the NSFD technique and the numerical values from [ [37]] that are listed in 2. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 24 of 40 k al ph a c1 c2 m u0 r1 th et a d1 S e n s it iv it y I n d e x -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Sensitivity of R 0 to Model Parameters Figure 1: Sensitivity index. time t (days) 0 10 20 30 40 50 60 S u s c e p ti b le c la s s ×107 0 0.5 1 1.5 2 2.5 3 3.5 phi = 0.7 phi = 0.8 phi = 0.9 phi = 1.0 Figure 3: NSFD-based graph of computations for the deterministic model (1) susceptible compartment. Variables The Physical Representation Values b The rate at which new population entered susceptible 0.0784 k Infectious rate 0.9091 c2 The rate of vaccinated population entered from I to V 0.6 c1 The rate of transmission of exposed population entered from E to I 0.167 α The rate of transmission of exposed population entered from E to I 0.14286 θ The rate of transmission of infected population entered from I to T 0.03 r2 The rate of transmission of treatments population entered from T to R 0.125 r1 The rate of transmission of self recovered population entered from E to R 0.3425 d1 Infectious death rate in I 0.02202 d2 Disease-related death rate in T 0.0550 µ0 Natural death rate 0.0545 Table 2: The parameters and their description involved in the model (1). M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 25 of 40 Figure 2: 3D dynamics of R0 of COVID-19n two strains with sensitive parameters. time t (days) 0 10 20 30 40 50 60 V a c c in a ti o n c la s s ×107 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 phi = 0.7 phi = 0.8 phi = 0.9 phi = 1.0 Figure 4: NSFD-based graph of computations for the deterministic model (1) vaccination compartment. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 26 of 40 time t (days) 0 10 20 30 40 50 60 E x p o s e d c la s s 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 phi = 0.7 phi = 0.8 phi = 0.9 phi = 1.0 Figure 5: NSFD-based graph of computations for the deterministic model (1) exposed compartment. time t (days) 0 10 20 30 40 50 60 In fe c te d c la s s 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 phi = 0.7 phi = 0.8 phi = 0.9 phi = 1.0 Figure 6: NSFD-based graph of computations for the deterministic model (1) infected compartment. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 27 of 40 time t (days) 0 10 20 30 40 50 60 T re a tm e n t c la s s 0 100 200 300 400 500 600 700 800 900 1000 phi = 0.7 phi = 0.8 phi = 0.9 phi = 1.0 Figure 7: NSFD-based graph of computations for the deterministic model (1) treatment compartment. time t (days) 0 10 20 30 40 50 60 R e c o v e re d c la s s 0 2000 4000 6000 8000 10000 12000 14000 phi = 0.7 phi = 0.8 phi = 0.9 phi = 1.0 Figure 8: NSFD-based graph of computations for the deterministic model (1). recovered compartment Using data from Table (1), we plotted the model through the NSFD scheme. From Figure (3), we observe that susceptibility decreases over time and stabilizes after about one month. Implementing a vaccination campaign before infection helps reduce the disease flow M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 28 of 40 and achieves satisfactory stability (see 4). In the absence of clear symptoms, populations are exposed to the disease, and the dynamics of the compartments are shown in Figure (5). Initially, the infection class is at a high level, but after treatment and the vaccination campaign, a decline in the disease flow is seen (see 6). Applying treatment strategies to control the disease, Figure (7) shows an exponential growth in the recovered compartment, demonstrating the model’s accuracy and usefulness (see 8). time t (days) 0 10 20 30 40 50 60 S u s c e p ti b le c la s s ×107 0 0.5 1 1.5 2 2.5 3 3.5 Figure 9: Plot of fractional technique based numerical solutions for the deterministic model (1) susceptible compartment. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 29 of 40 time t (days) 0 10 20 30 40 50 60 V a c c in a ti o n c la s s ×107 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure 10: Plot of fractional technique-based numerical solutions for the deterministic model (1) vaccination compartment. time t (days) 0 10 20 30 40 50 60 E x p o s e d c la s s 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Figure 11: Plot of fractional technique-based numerical solutions for the deterministic model (1) (1) exposed compartment. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 30 of 40 time t (days) 0 10 20 30 40 50 60 In fe c te d c la s s 1000 2000 3000 4000 5000 6000 7000 8000 9000 Figure 12: Plot of fractional technique-based numerical solutions for the deterministic model (1) (1) infected compartment. time t (days) 0 10 20 30 40 50 60 T re a tm e n t c la s s 0 100 200 300 400 500 600 700 800 900 1000 Figure 13: Plot of fractional technique-based numerical solutions for the deterministic model (1) (1) treatment compartment. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 31 of 40 time t (days) 0 10 20 30 40 50 60 R e c o v e re d c la s s 0 2000 4000 6000 8000 10000 12000 14000 Figure 14: Plot of fractional technique-based numerical solutions for the deterministic model (1) (1) recovered compartment. Here, we model the illness dynamics using the RK2 (second-order Runge-Kutta) frac- tional numerical technique. We illustrate the model using this approach, which entails gradually approximating the solution of the system of differential equations, using data from Table (1). Because it strikes a balance between precision and computing efficiency, the RK2 approach is frequently used in fractional systems. In figures (9-14), we plotted the system (1) using the fractional order α = 0.7, 0.8, 0.9, 1. As we can see from Figure (9), the population’s susceptibility gradually declines and stabi- lizes after about a month, which is comparable to the outcomes of the NSFD scheme. But the RK2 approach captures these dynamics more accurately, particularly when it comes to fleeting behaviors. Once more, the stability attained here suggests that the disease’s development has been successfully modeled. A major factor in reducing the spread of disease is the introduction of a vaccination cam- paign prior to the commencement of infection, as shown in Figure (10). Over time, a more stable population results from the vaccination campaign’s reduction of the number of vulnerable people and greater suppression of the disease flow. The consequences of the immunization campaign can be precisely recorded because to the RK2 scheme’s ability to fine-tune these modifications. The number of exposed individuals declines over time under varying fractional order α values, as illustrated in figure (11). In contrast to greater values (e.g., 1.0), lower values of α (e.g., 0.7) cause a slower decline, suggesting that fractional- order models reflect memory effects that impact the course of disease. The decrease in the number of infected people over time is shown in figure (12), which shows a similar pattern. A slower decline occurs with a lower α, indicating that memory effects extend the population’s infection period. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 32 of 40 The number of people receiving therapy over time is depicted in figure (13). The number of people receiving treatment first increases, peaks, and then progressively decreases. Lower values of the fractional order α show a slower fall, indicating that treatment dynamics are influenced by memory effects. The model exhibits a more conventional integer-order trend with a quicker fall when α = 1.0. The reconstructed population’s dynamics throughout time are depicted in figure (14). As infected individuals recover, the number of recovered persons rises, peaking before leveling off. A longer-lasting memory impact in the system is suggested by a greater recovery peak at lower α values (e.g., 0.7). The typical integer-order model (α = 1.0) exhibits a quicker stabilization and a somewhat lower peak. 8. The stochastic form of the model (1) In this section, we incorporate the effect of atmospheric white noise by transforming the original deterministic eye disease system (1) into a stochastic system, as described in [38]. To achieve this, nonlinear perturbations are introduced into each equation of the system. Specifically, the rate of change is modified individually for each compartment to reflect the impact of stochastic fluctuations within that class. S(t) : −c1 −→ −β + (Φ11S+Φ12)B1(t), V(t) : −c2 −→ −µ+ (Φ21V+Φ22)B2(t), E(t) : −α −→ −β + (Φ31E+Φ32)B3(t), I(t) : −θ −→ −µ+ (Φ41I+Φ42)B4(t), T(t) : −r2 −→ −r + (Φ51T+Φ52)B5(t), R(t) : −µ0 −→ −d0 + (Φ61R+Φ62)B6(t). The following set of equations represents the stochastic form of system (1): dS = [b+ c1V− kSI M − (c2 + µ0)S]dt+ (Φ11S+Φ12)S dB1(t), dV = [c2S− (c1 + µ0)V]dt+ (Φ21V+Φ22)V dB2(t), dE = [kSIM − (α+ r1 + µ0)E]dt+ (Φ31E+Φ32)E dB3(t), dI = [αE− (θ + d1 + µ0)I]dt+ (Φ41I+Φ42)I dB4(t), dT = [θI− (r2 + d2 + µ0)T]dt+ (Φ51T+Φ52)T dB5(t), dR = [r2T+ r1E− µ0R]dt+ (Φ61R+Φ62)R dB6(t). (27) We are now present the numerical simulation of this stochastic model in the next section. 9. Stochastic model (27) numerical analysis 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. (28) M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 33 of 40 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 (27) must be appropri- ately separated for our suggested model in order to apply the Euler-Maruyama approach method similarly (28). This can be done by Sti+1 = Sti + [ b+ c1V− kSI M − (c2 + µ0)S ] ∆t+ √ ∆tη1, Vti+1 = Vti + [c2S− (c1 + µ0)V] ∆t+ √ ∆tη2, Eti+1 = Eti + [ kSI M − (α+ r1 + µ0)E ] ∆t+ √ ∆tη3, Iti+1 = Iti + [αE− (θ + d1 + µ0)I] ∆t+ √ ∆tη4, Tti+1 = Tti + [θI− (r2 + d2 + µ0)T]∆t+ √ ∆tη5, Rti+1 = Rti + [r2T+ r1E− µ0R]∆t+ √ ∆tη6. (29) Using this numerical scheme, we compare with NSFD scheme in next figures. Time (days) 0 10 20 30 40 50 60 S u s c e p ti b le 0 20 40 60 80 100 120 140 160 180 200 Stochastic Deterministic Figure 15: Comparison of stochastic and deterministic model (1) susceptible compartment S. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 34 of 40 Time (days) 0 10 20 30 40 50 60 V a c c in a te d 0 10 20 30 40 50 60 70 80 90 100 Stochastic Deterministic Figure 16: Comparison of stochastic and deterministic model (1) vaccination compartment V. Time (days) 0 10 20 30 40 50 60 E x p o s e d 0 5 10 15 20 25 30 35 40 45 50 Stochastic Deterministic Figure 17: Comparison of stochastic and deterministic model (1) exposed compartment E. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 35 of 40 Time (days) 0 10 20 30 40 50 60 In fe c te d 0 5 10 15 20 25 30 35 40 45 50 Stochastic Deterministic Figure 18: Comparison of stochastic and deterministic model (1) infected compartment I. Time (days) 0 10 20 30 40 50 60 T re a te d 0 5 10 15 20 25 30 Stochastic Deterministic Figure 19: Comparison of stochastic and deterministic model (1) treatment compartment T. M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 36 of 40 Time (days) 0 10 20 30 40 50 60 R e c o v e re d 0 10 20 30 40 50 60 70 80 90 100 Stochastic Deterministic Figure 20: Comparison of stochastic and deterministic model (1) recovered compartment R. 10. Conclusion This paper develops an SV EITR epidemiological model to investigate the dynamic movements of a novel of measles. By demonstrating the uniqueness and existences of a positively boundness of solution, we were able to demonstrate the dynamical system’s epidemiological significance and establish the problem’s well-posedness. The dynamic be- havior of the suggested model at its equilibrium states was assessed through analytical calculation of the threshold parameter. This threshold value, which denotes circumstances under which the disease may either continue or disappear, was an essential indicator in epidemic modeling. Furthermore local and global unsteady equilibrium stabilities of the solutions were established by applying the necessary constraints on the threshold value. The global consistency of states of equilibrium was demonstrated using the Lyapunov func- tion theory. The statistical dynamics of viral disease were investigated in detail using the NSFD numerical scheme, which was found to be compatible with the continuum model. The purpose of the study was to find out how well continuous treatment and immunization control strategies, which don’t change over time, can stop the spread of measles. The find- ings suggest that raising vaccination rates may lower the number of infectious cases and, if not less than 40% of the vulnerable population receives the shot, may even eradicate measles. Furthermore, increasing the rate of treatment can also help stop the transmission of measles because higher treatment levels result in a small increase in the vulnerable pop- ulation but a decrease in the total number of infected people. It was determined that in addition to vaccination, therapy is helpful. According to the research, if at least 15% of M. Mohiudin et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6517 37 of 40 affected people receive treatment, it is possible to eradicate disease from human beings once vaccination is accessible. All things considered, when combined, a comprehensive approach that includes vaccination a sizable percentage of vulnerable people, attaining as many as forty percent, and treating not less than 15% of the sick population can be quite successful in preventing measles. This joint strategy seeks to reduce infections, lessen the disease’s severity, and enhance general public health results. Evaluating the sensitivity of variables influencing a threshold factor in the proposed Measles model was the main goal of our investigation. A rate of contact k of susceptible and infected people was found to be the most significant parameter. Educating and organizing people who can act as advo- cates and increase public awareness is essential. These powerful people will be essential in motivating others to investigate various approaches meant to reduce transmission rates. They can significantly help reduce the virus’s transmission by promoting preventive mea- sures like social distance, vitamin A supplementation (particularly in youngsters), and adherence to good hygiene standards. These people’s awareness efforts will inspire the public to think creatively and proactively about other ways to reduce transmission rates. The aim of the mathematical analysis conducted in the present research was to gain a full understanding of the long-term evolution of measles. We think that those making decisions and health authorities will find the knowledge gathered from this study to be helpful in the fight against measles. Subsequent studies will examine general fractional and stochastic optimum control problems using actual data in order to identify the best controls for putting required measures into place. Acknowledgements The authors would like to acknowledge the support of Prince Sultan University for paying the Article Processing Charges (APC) of this publication. Declarations All authors have read and approved the final manuscript. Competing interests Not exist. References [1] Shinjini Bandopadhyay, Angana Das Gupta, Asesh Banerjee, and Prabuddha Gupta. Bitesize epidemiology for general awareness of all students-i. Resonance, 28(3):411– 432, 2023. [2] UNICEF, WHO, et al. Who warn of perfect storm of conditions for measles outbreaks, affecting children. New York: Organización Mundial de la Salud, 2022. 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