EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6750 ISSN 1307-5543 – ejpam.com Published by New York Business Global Analysis of Monkeypox Transmission Dynamics Incorporating Quarantine and Surface Contamination Julito A. Puebla Jr. Department of General Education, Caraga State University Cabadbaran Campus, 8605 Cabadbaran City, Philippines Abstract. Monkeypox (Mpox) is a zoonotic viral disease that has re-emerged as a global health concern since 2022. In countries with limited vaccine access, non-pharmaceutical interventions such as quarantine and environmental sanitation remain the primary control strategies. However, existing models rarely integrate both measures within a single analytical framework. This study develops and analyzes a deterministic SEIQR-C model that incorporates quarantine protocols and surface contamination to examine the transmission dynamics of monkeypox. The model divides the human population into susceptible, exposed, infectious, quarantined, and recovered compart- ments, alongside a contaminated-surface component. Using the next generation matrix approach, the basic reproduction number (R0) is derived, accounting for both direct and indirect transmis- sion. Analytical results show that the disease-free equilibrium is locally and globally asymptotically stable when R0 < 1, and that an endemic equilibrium exists when R0 > 1. Numerical simulations confirm that increasing quarantine efficiency and cleaning or decay rates significantly reduce infec- tion persistence, underscoring their vital roles in controlling monkeypox outbreaks in low-resource settings. 2020 Mathematics Subject Classifications: 92D30, 34D23, 37N25 Key Words and Phrases: Monkeypox, stability analysis, quarantine, surface contamination, SEIQR-C model, basic reproduction number 1. Introduction Monkeypox (Mpox) is a viral zoonotic disease caused by the monkeypox virus (MPXV), a double-stranded DNA virus belonging to the Orthopoxvirus genus of the Poxviridae family [1]. The clinical presentation of Mpox closely resembles that of smallpox, with symptoms including fever, lymphadenopathy, and a vesiculopustular rash, though the disease is generally less severe than smallpox in terms of case fatality rates [2]. Since the first human case was identified in 1970 in the Democratic Republic of Congo, Mpox has remained endemic in Central and West African countries, with sporadic outbreaks occurring primarily through zoonotic spillover events [1, 3]. However, the 2022 outbreak DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6750 Email address: julito.puebla@csucc.edu.ph (J. A. Puebla Jr.) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 2 of 19 marked a pivotal moment in the epidemiology of Mpox, with sustained human-to-human transmission reported in over 100 countries, prompting the World Health Organization (WHO) to declare the outbreak a public health emergency of international concern [4]. Transmission of Mpox occurs through three primary routes: animal-to-human (zoonotic), human-to-human (direct), and indirect environmental transmission [1, 5]. Zoonotic trans- mission typically occurs through direct contact with infected animals or their bodily flu- ids, particularly from rodents and non-human primates, or through consumption of un- dercooked bushmeat [6]. Human-to-human transmission results from close contact with lesions, body fluids, or respiratory droplets from infected individuals [7]. More recently, en- vironmental transmission has been recognized as an important secondary route, as MPXV remains viable on surfaces for several days after patient contact, particularly on bedding, clothing, and high-touch objects in both community and healthcare settings [8, 9]. Studies have detected viable MPXV DNA on household and hospital surfaces up to 15 days after contamination, highlighting the challenges of infection prevention in densely populated or poorly sanitized environments [9]. Although vaccines developed for smallpox, such as the Modified Vaccinia Ankara (MVA-BN), have demonstrated high protective efficacy against Mpox, immunization cam- paigns remain largely confined to high-income countries and target mainly high-risk popu- lations [10]. In contrast, many low and middle-income countries, including the Philippines which reported confirmed Mpox cases beginning in 2022 do not have routine Mpox vac- cination programs in place [11, 12]. In these regions, non-pharmaceutical interventions (NPIs), including case isolation, quarantine, and environmental sanitation, remain the primary tools for outbreak control [13, 14]. Recent modeling studies have underscored the importance of NPIs in mitigating Mpox transmission where vaccine coverage is limited, particularly in preventing sustained human-to-human and environmental transmission cy- cles [15–18]. Several recent studies have incorporated various strategies into the mathematical mod- eling of monkeypox transmission dynamics. For instance, Bolaji et al. [15] developed a compartmental model that integrates both quarantine and vaccination strategies, provid- ing insights into how combined pharmaceutical and non-pharmaceutical interventions can reduce the basic reproduction number. Similarly, Hassan et al. [16] formulated a model emphasizing the role of environmental contamination by considering the indirect trans- mission of monkeypox via contaminated surfaces, with optimal control analysis for sani- tation measures. While these models successfully captured critical facets of transmission, they either focused on vaccination-centric approaches with limited environmental dynam- ics [15], or addressed contaminated surfaces without integrating isolation or quarantine protocols [16]. Moreover, Ngungu et al. [13] examined non-pharmaceutical interventions based on real outbreak data but did not explicitly incorporate environmental reservoirs or structured isolation compartments. Therefore, a comparative gap remains in synthesizing both quarantine measures and surface contamination within a single unified framework, especially in the context of regions with low vaccination coverage. In this study, an extended SEIQR-C model is developed to describe the transmis- sion dynamics of monkeypox by integrating both quarantine and environmental surface J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 3 of 19 contamination. Earlier mathematical models have largely concentrated on direct human- to-human transmission, overlooking the combined effects of isolation measures and indirect transmission through contaminated surfaces. Incorporating quarantine and surface con- tamination into a single modeling framework provides a more comprehensive and realistic representation of the disease dynamics, as it captures both behavioral and environmental pathways of infection. The model further offers analytical and numerical insights into how variations in quarantine efficiency and cleaning or decay rate affect infection persis- tence and disease control, thereby enhancing the understanding of the mechanisms that influence the spread and management of monkeypox. 2. Model Formulation To investigate the transmission dynamics of monkeypox, a deterministic SEIQR-C compartmental model is formulated, incorporating both direct human-to-human trans- mission and indirect transmission through contaminated surfaces. The SEIQR-C model that divides the total human population into five health-related classes, along with an environmental compartment representing surface contamination: (i) Susceptible individuals (S) are at risk of acquiring monkeypox through direct contact with infectious individuals or indirect contact with contaminated surfaces in the environment. (ii) Exposed individuals (E) who have been infected but are not yet infectious. (iii) Infectious individuals (I) capable of transmitting the virus to others. (iv) Quarantined individuals (Q) who have been isolated and no longer contribute to transmission. (v) Recovered individuals (R) who have acquired permanent immunity. (vi) Contaminated surfaces (C) such as bedding, clothing, and shared objects—serve as environmental reservoirs contributing to the indirect transmission of the monkeypox virus. The total human population is given by N = S + E + I +Q+ R, while C represents a non-human compartment contributing to indirect transmission. The model is based on the following assumptions: (i) The population is replenished at a constant rate Λ, accounting for birth and immi- gration, and individuals are removed from all compartments through natural death at rate µh. (ii) Susceptible individuals become infected either by direct contact with infectious in- dividuals or indirectly through contact with contaminated surfaces. The force of infection is given by βhI + βcC, where βh and βc represent the direct and indirect J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 4 of 19 transmission rates, respectively. The force of infection is modeled using a standard mass-action form βhI + βcC, consistent with previous studies on monkeypox trans- mission [16, 18]. This formulation assumes proportional contact between susceptible individuals and both infectious humans and contaminated surfaces. The probability of infection increases proportionally with the number of infectious individuals and the level of environmental contamination. (iii) Exposed individuals are infected but not infectious. They progress to the infectious compartment at rate γ, or exit due to natural death. (iv) Infectious individuals either recover at rate γ0, are quarantined at rate θ, or die naturally. Quarantined individuals are isolated from the population and do not contribute to new infections. They recover at rate γq or die naturally. (v) Recovered individuals are assumed to gain permanent immunity and do not return to the susceptible class. (vi) Infectious individuals shed the virus into the environment at rate α, contributing to surface contamination. The environmental viral load decays at rate µp due to natural inactivation or cleaning. (vii) The model does not account for vertical transmission, re-infection, or disease impor- tation. All transitions between compartments follow standard mass-action incidence. This assumption is supported by epidemiological evidence showing that recovered individuals develop lasting immunity and that vertical transmission of monkeypox has not been observed in humans [2, 6, 7]. Moreover, the use of a standard mass- action incidence follows previous modeling approaches [16, 18], which apply this form to maintain analytical tractability while capturing the essential transmission mechanisms. Based on the aforementioned assumptions, Figure 1 presents the mathematical model depicting the transmission dynamics of monkeypox. Figure 1: SEIQR-C Model for the Transmission Dynamics of Monkeypox J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 5 of 19 The transmission dynamics of the monkeypox SEIQR-C model, shown in Figure 1, are described by the following system of ordinary differential equations dS dt = Λ− (βhI + βcC)S − µhS dE dt = (βhI + βcC)S − (γ + µh)E dI dt = γE − (γ0 + θ + µh)I dQ dt = θI − (γq + µh)Q dR dt = γ0I + γqQ− µhR dC dt = αI − µpC. (1) System (1) is defined with the following initial conditions S(0) > 0, E(0) ≥ 0, I(0) ≥ 0, Q(0) ≥ 0, and R ≥ 0, and C(0) ≥ 0. All parameters in system are described in Table 1. Parameter Description Value Source Λ Recruitment rate of the human population 5 Assumed βh Transmission rate between infectious and susceptible individuals 0.2084 [16, 17] βc Transmission rate between contaminated surfaces and susceptible individuals 7.6977x10−8 [17] γ Progression rate from exposed to infectious 0.2 [19] γ0 Recovery rate of infectious individuals 0.1490 [17, 19] γq Recovery rate of quarantined individuals 3.5430x10−4 [17] θ Quarantine rate of symptomatic individuals 2 [18] α Contamination rate from infectious individuals to surfaces 0.004 [18] µp Cleaning or decay rate of the virus on the contaminated surfaces 1.8 Assumed µh Natural human mortality rate 0.2 [18] Table 1: Description and Value of the Model Parameters Theorem 1. Given non-negative initial conditions, the solution of the SEIQR-C model (1) remains non-negative for all t ≥ 0. Proof. Assume that S(0) > 0, E(0) ≥ 0, I(0) ≥ 0, Q(0) ≥ 0, R(0) ≥ 0, and C(0) ≥ 0. Consider the first equation of the SEIQR-C model (1) dS dt = Λ− (βhI + βcC)S − µhS. (2) Let ΦS(t) = βhI(t) + βcC(t) + µh, and define the integrating factor θS(t) = exp (∫ t 0 ΦS(s)ds ) . (3) Multiply both sides by θS(t) and applying the Leibniz rule d dt ( θS(t)S(t) ) = ΛθS(t). (4) J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 6 of 19 Integrating from 0 to t θS(t)S(t) = θS(0)S(0) + ∫ t 0 ΛθS(s)ds. (5) Solve for S(t) yields S(t) = 1 θS(t) ( θS(0)S(0) + ∫ t 0 ΛθS(s)ds ) . (6) Since Λ > 0 and S(0) > 0, it implies that S(t) ≥ 0 for all t ≥ 0. From the second equation of the SEIQR-C model (1), we get dE dt + (γ + µh)E = (βhI + βcC)S. (7) Define the integrating factor, θE(t) = e(γ+µh)t. Multiply both sides by θE(t) and applying Leibniz rule d dt ( θE(t)E(t) ) = (βhI + βcC)SθE(t). (8) Integrating from 0 to t and solve for E(t), that is E(t) = 1 θE(t) [ E(0) + ∫ t 0 (βhI(s) + βcC(s))S(s) θE(s) ds ] ≥ 0. (9) A similar method is applied to the remaining compartmental variables of the equations in the SEIQR-C model (1). I(t) = 1 e(γ0+θ+µh)t [ I(0) + ∫ t 0 γE(s) e(γ0+θ+µh)s ds ] ≥ 0 (10) Q(t) = 1 e(γq+µh)t [ Q(0) + ∫ t 0 θI(s) e(γq+µh)s ds ] ≥ 0 (11) R(t) = 1 eµht [ R(0) + ∫ t 0 (γ0I(s) + γqQ(s)) eµhs ds ] ≥ 0 (12) C(t) = 1 eµpt [ C(0) + ∫ t 0 αI(s) µps ds ] ≥ 0 (13) Thus, the solution of the SEIQR-C model (1) remains non-negative for all t ≥ 0 ■ Theorem 2. The solutions of the SEIQR-C model (1) are bounded in the region Ω = { (S,E, I,Q,R,C) ∈ R6 + : S + E + I +Q+R ≤ Λ µh , C ≤ αΛ µhµp } . J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 7 of 19 Proof. Define the total human population as N(t) = S(t) +E(t) + I(t) +Q(t) +R(t). Adding the first five equations of the SEIQR-C model gives dN dt = Λ− µhN(t). (14) Multiplying both sides by the integrating factor eµht yields d dt ( N(t)eµht ) = Λeµht. (15) Integrating both sides gives N(t) = ( N(0)− Λ µh ) e−µht + Λ µh . (16) Since µh > 0, we conclude that N(t) ≤ max{N(0), Λ µh } for all t ≥ 0 , and limt→∞N(t) = Λ µh . Thus, the total human population is bounded. Now consider the viral concentration on contaminated surfaces C(t), which satisfies dC dt = αI − µpC. (17) Since I(t) ≤ N(t) ≤ Λ/µh, we obtain dC dt ≤ α · Λ µh − µpC. (18) Multiplying by the integrating factor eµpt, we get d dt ( C(t)eµpt ) ≤ α · Λ µh eµpt. (19) Solving this inequality, we find C(t) ≤ ( C(0)− αΛ µhµp ) e−µpt + αΛ µhµp , (20) and therefore, lim sup t→∞ C(t) ≤ αΛ µhµp . (21) Hence, all state variables of the SEIQR-C model remain bounded for all t ≥ 0, and the solution trajectories are contained in the compact, positively invariant region Ω = { (S,E, I,Q,R,C) ∈ R6 + : S + E + I +Q+R ≤ Λ µh , C ≤ αΛ µhµp } . ■ J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 8 of 19 3. Model Analysis 3.1. Monkeypox-Free Equilibrium and the Basic Reproduction Number To initiate the mathematical analysis of the monkeypox model, we first identify the monkeypox-free equilibrium (MFE), also known as the disease-free equilibrium (DFE), where no infection is present in the human population and the environment. This state represents the baseline condition in which the monkeypox is absent and all individuals are either susceptible or in non-infectious compartments. Theorem 3. The SEIQR-C model admits a monkeypox-free equilibrium given by E0 = (S0, 0, 0, 0, 0, 0) where S0 = Λ µh . Proof. Let E0 = (S0, E0, I0, Q0, R0, C0) be a monkeypox-free equilibrium of the SEIQR- C model. To solve the MFE, we set all the time derivatives in the model to zero. That is, Λ− (βhI0 + βcC0)S0 − µhS0 = 0 (βhI0 + βcC0)S0 − (γ + µh)E0 = 0 γE0 − (γ0 + θ + µh)I0 = 0 θI0 − (γq + µh)Q0 = 0 γ0I0 + γqQ0 − µhR0 = 0 αI0 − µpC0 = 0. Suppose that no infection is present in the population or the environment which means that E0 = I0 = Q0 = R0 = C0 = 0. Substituting these values into the system, we obtain S0 = Λ µh . Hence, the monkeypox-free equilibrium is given by E0 = ( Λ µh , 0, 0, 0, 0, 0 ) . (22) ■ Next, we will compute the basic reproduction number of the SEIQR-C model (1) using the next generation matrix (NGM) [20, 21]. The basic reproduction number denoted by R0 is defined as the average number of secondary infections that occurs when one infective is introduced into a completely susceptible population [20, 21]. Now, we first identified the infected compartments contributing to the generation of new cases: the exposed individuals E, the infectious individuals I, and the contaminated surfaces C. Then, the disease transmission terms were captured in the matrix F , represent- ing new infections, while the transition terms between compartments were incorporated into matrix V. These matrices are defined as F = (βhI + βcC)S 0 0  and V =  (γ + µh)E −γE + (γ0 + θ + µh)I −αI + µpC  J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 9 of 19 The Jacobian of F and V evaluated at E0 are F and V , respectively. F = 0 βhΛ µh βcΛ µh 0 0 0 0 0 0  and V = γ + µh 0 0 −γ γ0 + θ + µh 0 0 −α µp  Then, taking the inverse of V , that is V −1 =  1 γ+µh 0 0 γ (γ+µh)(γ0+θ+µh) 1 γ0+θ+µh 0 αγ (γ+µh)(γ0+θ+µh)µp α (γ0+θ+µh)µp 1 µp  . (23) Consequently, FV −1 = 0 βhΛ µh βcΛ µh 0 0 0 0 0 0   1 γ+µh 0 0 γ (γ+µh)(γ0+θ+µh) 1 γ0+θ+µh 0 αγ (γ+µh)(γ0+θ+µh)µp α (γ0+θ+µh)µp 1 µp  (24) =  Λγ(βhµp+αβc) µhµp(γ+µh)(γ0+θ+µh) Λ(βhµp+αβc) µhµp(γ0+θ+µh) Λβc µhµp 0 0 0 0 0 0  (25) Accordingly, the basic reproduction number of the SEIQR-C model (1) is the largest eigenvalue of FV −1. Hence, R0 = Λγ(βhµp + αβc) µhµp(γ + µh)(γ0 + θ + µh) . (26) The basic reproduction number R0 serves as a critical threshold parameter in epidemio- logical modeling [22]. In the SEIQR-C model, R0 captures both direct human-to-human transmission and indirect transmission via contaminated surfaces. A value of R0 > 1 indicates that the monkeypox can spread and persist, while R0 < 1 suggests that the monkeypox will eventually die out. This threshold informs the assessment of public health strategies such as quarantine, sanitation, and contact reduction, with the aim of reducing R0 below one to achieve disease control. 3.2. Monkeypox Endemic Equilibrium The monkeypox endemic equilibrium (MEE) point of system (1) refers to a state where monkeypox persists in the population. It is obtained by setting the system of equations in (1) to zero and solving for the equilibrium values of the variables. Theorem 4. The SEIQR-C model (1) admits a monkeypox endemic equilibrium given by E∗ = (S∗, I∗, E∗, Q∗, R∗, C∗). J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 10 of 19 Proof. Let E∗ = (S∗, E∗, I∗, Q∗, R∗, C∗) be a monkeypox endemic equilibrium of the SEIQR-C model. To solve the MEE, we set all the time derivatives in the system (1) to zero. That is, Λ− (βhI∗ + βcC∗)S∗ − µhS∗ = 0 (βhI∗ + βcC∗)S∗ − (γ + µh)E∗ = 0 γE∗ − (γ0 + θ + µh)I∗ = 0 θI∗ − (γq + µh)Q∗ = 0 γ0I∗ + γqQ∗ − µhR∗ = 0 αI∗ − µpC∗ = 0. Suppose that E∗ = (S∗, E∗, I∗, Q∗, R∗, C∗) is a nontrivial equilibrium point, where all components are nonzero, indicating that the environment is not monkeypox-free. To solve for S∗, we can express the third and sixth equations in terms of I∗: E∗ = (γ0 + θ + µh)I∗ γ and C∗ = αI∗ µp . Substituting E∗ and C∗ into the second equation equation and solving for S∗, we obtain S∗ = µp(γ + µh)(γ0 + θ + µh) γ(βhµp + βcα) . (27) Next, to solve for I∗, we substitute the expressions for C∗ and S∗ into the first equation. After algebraic manipulation, we get I∗ = Λγ(βhµp + βcα)− µhµp(γ + µh)(γ0 + θ + µh) (γ + µh)(γ0 + θ + µh)(βhµp + βcα) . (28) Moreover, the remaining variables E∗, Q∗, R∗ and C∗ are obtained by substituting I∗ into the third, fourth, fifth, and sixth equations, respectively. E∗ = Λγ(βhµp + βcα)− µhµp(γ + µh)(γ0 + θ + µh) γ(γ + µh)(βhµp + βcα) (29) Q∗ = Λγθ(βhµp + βcα)− µhµpθ(γ + µh)(γ0 + θ + µh) (γq + µh)(γ + µh)(γ0 + θ + µh)(βhµp + βcα) (30) R∗ = γ0I∗ + γqQ∗ µh (31) C∗ = Λγα(βhµp + βcα)− µhµpα(γ + µh)(γ0 + θ + µh) µp(γ + µh)(γ0 + θ + µh)(βhµp + βcα) (32) This completes the proof. ■ Theorem 5. If R0 > 1, then the SEIQR-C model (1) has a unique positive endemic equilibrium. J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 11 of 19 Proof. Let R0 > 1. Note that R0 = Λγ(βhµp+αβc) µhµp(γ+µh)(γ0+θ+µh) . Then, S∗ = µp(γ + µh)(γ0 + θ + µh) γ(βhµp + βcα) = Λ µhR0 (33) E∗ = Λγ(βhµp + βcα)− µhµp(γ + µh)(γ0 + θ + µh) γ(γ + µh)(βhµp + βcα) = Λ(R0 − 1) (γ + µh)R0 (34) I∗ = Λγ(βhµp + βcα)− µhµp(γ + µh)(γ0 + θ + µh) (γ + µh)(γ0 + θ + µh)(βhµp + βcα) = Λγ(R0 − 1) (γ + µh)(γ0 + θ + µh)R0 (35) R∗ = γ0I∗ + γqQ∗ µh = (θR0 − 1)Λγ(βhµp + βcα) γq(γ + µh)(γ0 + θ + µh)(βhµp + βcα)R0 (36) C∗ = Λγα(βhµp + βcα)− µhµpα(γ + µh)(γ0 + θ + µh) µp(γ + µh)(γ0 + θ + µh)(βhµp + βcα) = α(R0 − 1)Λγ(βhµp + βcα) µp(γ + µh)(γ0 + θ + µh)(βhµp + βcα)R0 (37) Hence, the unique positive endemic equilibrium of the SEIQR-C model (1) exists when R0 > 1. ■ 3.3. Stability Analysis This section presents the local stability analysis of the equilibrium points of the SEIQR- C model for monkeypox. Theorem 6. If R0 < 1, then the SEIQR-C model (1) is locally asymptotically stable at the monkeypox-free equilibrium E0 and unstable otherwise. Proof. The Jacobian matrix of the system (1) is given by J =  −(βhI + βcC)− µh 0 −βhS 0 0 −βcS βhI + βcC −(γ + µh) βhS 0 0 βcS 0 γ −(γ0 + θ + µh) 0 0 0 0 0 θ −(γq + µh) 0 0 0 0 γ0 γq −µh 0 0 0 α 0 0 −µp  . (38) The Jacobian matrix evaluated at E0 is of the form JE0 =  −µh 0 −βhΛ µh 0 0 −βcΛ µh 0 −(γ + µh) βhΛ µh 0 0 βhΛ µh 0 γ −(γ0 + θ + µh) 0 0 0 0 0 θ −(γq + µh) 0 0 0 0 γ0 γq −µh 0 0 0 α 0 0 −µp  . (39) J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 12 of 19 Then, the first four eigenvalues obtained are λ1 = −µh, λ2 = −µp, λ3 = −µh, and λ4 = −(γq + µh), while the remaining two eigenvalues can be determined as the roots of the following second-degree polynomial: λ2 + (γ + 2µh + γ0 + θ)λ+ (γ + µh)(γ0 + θ + µh)− Λγβh µh . Suppose R0 < 1. Then R0 can be express as Λγβhµp µhµp(γ + µh)(γ0 + θ + µh) + Λγαβc µhµp(γ + µh)(γ0 + θ + µh) < 1. It follows that, Λγβh µh(γ + µh)(γ0 + θ + µh) < 1. Multiply both sides by (γ + µh)(γ0 + θ + µh), we have Λγβh µh < (γ + µh)(γ0 + θ + µh). Subsequently, (γ + µh)(γ0 + θ + µh)− Λγβh µh > 0. By Routh-Hurwitz criterion [23], all eigenvalues of the second-degree polynomial are neg- ative when R0 < 1. Thus, the SEIQR-C model (1) is locally asymptotically stable at the monkeypox-free equilibrium E0 when R0 < 1 and unstable otherwise. ■ Theorem 7. If R0 < 1, then the SEIQR-C model (1) is globally asymptotically stable at the monkeypox-free equilibrium E0 and unstable otherwise. Proof. Consider the Lyapunov function below L = E + γ + µh γ I + Λβc µhµp C. The time derivative of L calculated along the solution of the system (1) is negative definite. That is, dL dt = dE dt + ( γ + µh γ ) dI dt + ( Λβc µhµp ) dC dt (40) = βhSI + βcSC − (γ + µh)E + (γ + µh γ )( γE − (γ0 + θ + µh)I ) + αβc µhµp ( αI − µpC ) (41) J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 13 of 19 = βhSI + (γ + µh γ )( − (γ0 + θ + µh)I ) + αβcα µhµp I (42) = ( βhΛ µh + Λβhα µhµp − (γ + µh)(γ0 + θ + µh) γ ) I (43) = ( µpβhΛ + Λβhα µhµp − (γ + µh)(γ0 + θ + µh) γ ) I (44) = (γ + µh)(γ0 + θ + µh) γ ( R0 − 1 ) I (45) ≤ 0 for R0 < 1. (46) The time derivative dL dt ≤ 0 whenever the basic reproduction number R0 < 1. The equality dL dt = 0 holds only when E = I = C = 0, that is, at the monkeypox-free equilibrium E0. Therefore, by LaSalle’s Invariance Principle [24], it follows that the largest positively invariant set where dL dt = 0 is the singleton E0, and thus all solutions in Ω converge to E0. This establishes that the monkey-free equilibrium of the SEIQR-C model is globally asymptotically stable whenever R0 < 1. ■ 3.4. Sensitivity Analysis The sensitivity index quantifies the relative change in R0 in response to a relative change in a parameter [25]. Given the basic reproduction number for the SEIQR-C model (1) R0 = Λγ(βhµp + αβc) µhµp(γ + µh)(γ0 + θ + µh) (47) the sensitivity index of R0 with respect to a parameter p is defined as Sp = ∂R0 ∂p · p R0 . (48) The sensitivity index for each parameter of the SEIQR-C model (1) with respect to R0 is given by: SΛ = ∂R0 ∂Λ · Λ R0 = 1 (49) Sγ = ∂R0 ∂γ · γ R0 = R0 ( 1 γ − 1 γ + µh ) · γ R0 = µh γ + µh (50) Sβh = ∂R0 ∂βh · βh R0 = βhµp βhµp + αβc (51) Sβc = ∂R0 ∂βc · βc R0 = αβc βhµp + αβc (52) Sα = ∂R0 ∂α · α R0 = αβc βhµp + αβc (53) J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 14 of 19 Sµp = ∂R0 ∂µp · µp R0 = − αβc βhµp + αβc (54) Sµh = ∂R0 ∂µh · µh R0 = −1− µh(γ + γ0 + θ + 2µh) (γ + µh)(γ0 + θ + µh) (55) Sγ0 = ∂R0 ∂γ0 · γ0 R0 = − γ0 γ0 + θ + µh (56) Sθ = ∂R0 ∂θ · θ R0 = − θ γ0 + θ + µh . (57) (58) Figure 2: Sensitivity indices of the SEIQR-C model parameters Figure 2 presents the sensitivity indices of the SEIQR-C model. The results reveal that the recruitment rate (Λ), the human-to-human transmission rate (βh), the progression rate from exposed to infectious individuals (γ), the transmission rate from contaminated surfaces (βc), and the contamination rate (α) exhibit a positive impact on the basic repro- duction number (R0). This indicates that increases in these parameters contribute to a higher potential for monkeypox transmission. Conversely, the cleaning or decay rate (µp), the natural death rate (µh), the recovery rate (γ0), and the quarantine rate (θ) show nega- tive sensitivity indices. This suggests that increasing these parameters effectively reduces R0, leading to a decrease in the number of individuals infected with monkeypox. 4. Simulation In this section, we present numerical simulations to illustrate the theoretical findings. The parameter values used in the simulations are listed in Table 1. Simulation 1. The parameter values utilized in this study are presented in Table 1. Using these parameters, we compute the basic reproduction number as R0 = 0.917, and determine the monkeypox-free equilibrium to be E0 = (25, 0, 0, 0, 0, 0). The initial J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 15 of 19 conditions for the compartments of the SEIQR-C model are considered as follows: (a) (100, 70, 50, 40, 30, 55), (b) (200, 130, 90, 100, 90, 130), (c) (500, 300, 190, 150, 125, 220), and (d) (1000, 565, 455, 225, 155, 350). As shown in Figure 3, for all initial conditions, the solution trajectories converge to the monkeypox-free equilibrium E0 = (25, 0, 0, 0, 0, 0). This result indicates that the SEIQR-C model exhibits local asymptotic stability at the monkeypox-free equilibrium when R0 < 1. Figure 3: (Simulation 1) The SEIRQ-C model is locally asymptotically stable at E0 when R0 < 1. Simulation 2. Consider the same parameter values used in Simulation 1, except for an increased recruitment rate of Λ = 10, and decreased values of the quarantine rate θ = 0.5 and the cleaning or decay rate µp = 0.3. As a result, the basic reproduction J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 16 of 19 number increases to R0 = 21.786, while the disease-free equilibrium remains at E0 = (25, 0, 0, 0, 0, 0). Using the same initial conditions as in Simulation 1, it is observed from Figure 4 that the solution trajectories do not converge to E0, indicating that the SEIQR- C model is unstable at the disease-free equilibrium when R0 > 1. Furthermore, since R0 = 21.786 and the endemic equilibrium point E∗ = (15, 18, 4, 6, 9, 1) now exists, the solution trajectories are seen to converge to E∗. Therefore, the SEIQR-C model is locally asymptotically stable at the endemic equilibrium whenever R0 > 1. Figure 4: (Simulation 2) The SEIRQ-C model is locally asymptotically stable at E∗ when R0 > 1. Figures 3 and 4 further demonstrate the contrasting epidemic dynamics of the SEIQR- C model under varying control parameters while maintaining identical initial values for J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 17 of 19 all compartments. In Figure 3 (R0 < 1), higher quarantine (θ) and cleaning or viral decay rate (µp) lead to a progressive reduction in the number of infectious individuals and environmental contamination, confirming the stability of the disease-free equilibrium. Conversely, in Figure 4 (R0 > 1), the same initial conditions are applied but with lower values of θ and µp, resulting in persistent infection and convergence toward an endemic equilibrium. These findings highlight that the effectiveness of quarantine measures and environmental sanitation plays a critical role in reducing the basic reproduction number and ensuring the long-term stability of the human population against monkeypox trans- mission. Although no previous model has simultaneously incorporated quarantine and contaminated surfaces, the present results are comparable to those in [15] and [16], which also emphasized the importance of quarantine and environmental sanitation in controlling the spread of monkeypox. 5. Conclusion This study formulated and analyzed a deterministic SEIQR-C model to describe the transmission dynamics of monkeypox by incorporating both quarantine protocols and en- vironmental contamination. The model ensured non-negative and bounded solutions, and equilibrium analyses identified both disease-free and endemic states. Stability analyses demonstrated that the disease-free equilibrium is locally and globally asymptotically sta- ble when R0 < 1, while the endemic equilibrium becomes locally stable when R0 > 1. Numerical simulations further validated these theoretical findings, showing that the in- fection dies out when quarantine and cleaning measures are adequately implemented. The results emphasize that increasing the efficiency of quarantine and the rate of surface cleaning or viral decay are among the most effective strategies for controlling monkeypox transmission, particularly in low-resource settings where vaccine coverage remains limited. While the SEIQR-C model provides valuable insights into the combined effects of quarantine and surface contamination, it is based on several simplifying assumptions. The current model does not include vaccination dynamics, disease-induced mortality, or parameter estimation using real outbreak data. Furthermore, the global stability of the endemic equilibrium was not established. Future studies may focus on constructing a Lyapunov function to determine the global stability of the endemic equilibrium, extending the model to incorporate disease-induced mortality and vaccination, and validating the model through real-data fitting and parameter estimation. These directions will enhance the model’s applicability and improve its predictive capability for guiding public health policies against monkeypox transmission. Conflict of Interest The author affirms that no conflict of interest exists with respect to the research, authorship, and/or publication of this article. J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 18 of 19 Acknowledgments The author would like to thank all individuals and institutions who supported the completion of this study. References [1] World Health Organization. Monkeypox, 04 2023. [2] Eveline Bunge, Bernard Hoet, Liddy Chen, Florian Lienert, Heinz Weidenthaler, L. R. Baer, and Robert Steffen. The changing epidemiology of human monkeypox—a po- tential threat? a systematic review. PLOS Neglected Tropical Diseases, 16:e0010141, 02 2022. [3] Z. Jezek and F. Fenner. Human monkeypox. Karger Publishers, 17, 10 1988. [4] World Health Organization. Disease outbreak news, 2023. [5] S. A. Boone and C. P. Gerba. Significance of fomites in the spread of respiratory and enteric viral disease. Applied and Environmental Microbiology, 73:1687–1696, 01 2007. [6] L. D. Nolen, J. Osadebe, Katomba, J. Likofata, D. Mukadi, B. Monroe, J. Doty, C. M. Hughes, J. Kabamba, J. Malekani, P. L. Bomponda, J. I. Lokota, M. P. Balilo, T. Likafi, R. S. Lushima, B. K. Ilunga, F. Nkawa, E. Pukuta, S. Karhemere, J. J. M. Tamfum, B. Nguete, E. O. Wemakoy, A. McCollum, and M. Reynolds. Extended human-to-human transmission during a monkeypox outbreak in the democratic re- public of the congo. Emerging Infectious Diseases, 22:1014–1021, 06 2016. [7] A. Yinka-Ogunleye, O. Aruna, D. Ogoina, N. Aworabhi, W. Eteng, S. Badaru, A. Mo- hammed, J. Agenyi, E. N. Etebu, T. W. Numbere, A. Ndoreraho, E. Nkunzimana, Y. Disu, M. Dalhat, P. Nguku, A. Mohammed, M. Saleh, A. McCollum, K. Wilkins, O. Faye, A. Sall, C. Happi, N. Mba, O. Ojo, and C. Ihekweazu. Reemergence of human monkeypox in nigeria, 2017. Emerging Infectious Diseases, 24(6), 06 2018. [8] C. Morgan, F. Whitehill, J. Doty, J. Schulte, A. Matheny, J. Stringer, L. Delaney, R. Esparza, A. Rao, and A. McCollum. Environmental persistence of monkeypox virus on surfaces in household of person with travel-associated infection, dallas, texas, usa, 2021. Emerging Infectious Diseases, 28, 10 2022. [9] D. Nörz, S. Pfefferle, T. Brehm, G. Franke, I. Grewe, B. Knobling, M. Aepfelbacher, S. Huber, E. Klupp, S. Jordan, M. Addo, J. Schulze zur Wiesch, S. Schmiedel, M. Lüt- gehetmann, and J. Knobloch. Evidence of surface contamination in hospital rooms occupied by patients infected with monkeypox, germany, june 2022. Eurosurveillance, 27, 06 2022. [10] M. Christodoulidou and N. Mabbott. Efficacy of smallpox vaccines against mpox infections in humans. Immunotherapy Advances, 3, 10 2023. [11] A. Hossain, Md. A. Monem, M. Rahman, and R. Raza. Mpox (monkeypox): a comprehensive update of current epidemic evidence. Science in One Health, 4:100100, 12 2024. [12] E. Roxas, P. J. Acacio-Claro, M. M. Lota, A. Abeleda, S. N. Dalisay, M. Landicho, J. A. Puebla Jr. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6750 19 of 19 Y. Fujimori, J. Z. Rosuello, J. Kaufman, M. Danchin, V. Belizario, and F. Vogt. Enablers and barriers of covid-19 vaccination in the philippines. Vaccines, 13:719– 719, 07 2025. [13] M. Ngungu, E. Addai, A. Adeniji, A. Umar Muhammad, and Kayode O. Math- ematical epidemiological modeling and analysis of monkeypox dynamism with non- pharmaceutical intervention using real data from united kingdom. Frontiers in Public Health, 11, 02 2023. [14] J. Puebla, J. M. Macalalag, and K. Pajaron. A mathematical model for covid-19 and tuberculosis coinfection with the effect of a quarantine measure. European Journal of Pure and Applied Mathematics, 18:6149–6149, 08 2025. [15] B. Bolaji, A. Ibrahim, F. Ani, B. Omede, and G. Acheneje. A model for the control of transmission dynamics of human monkeypox disease in sub-saharan africa. Journal of the Nigerian Society of Physical Sciences, pages 1800–1800, 05 2024. [16] A. Hassan, A. Dipo, and Muhamad. Optimal control and stability analysis of mon- keypox transmission dynamics with the impact of contaminated surfaces. Frontiers in Applied Mathematics and Statistics, 10, 03 2024. [17] A. Alshehri and S. Ullah. Optimal control analysis of monkeypox disease with the impact of environmental transmission. AIMS Mathematics, 8:16926–16960, 01 2023. [18] O. J. Peter, S. Kumar, N. Kumari, F. A. Oguntolu, K. Oshinubi, and R. Musa. Transmission dynamics of monkeypox virus: a mathematical modelling approach. Modeling Earth Systems and Environment, 10 2021. [19] C. Madubueze, I. Onwubuya, G. Nkem, and Z. Chazuka. The transmission dynamics of the monkeypox virus in the presence of environmental transmission. Frontiers in Applied Mathematics and Statistics, 8, 11 2022. [20] O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz. On the definition and the computation of the basic reproduction ratio r 0 in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology, 28, 06 1990. [21] V. D. Driessche and J. Watmough. Reproduction numbers and sub-threshold en- demic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180:29–48, 11 2002. [22] C. Castillo-Chávez, Z. Feng, W. Huang, P. Driessche, D. Kirschner, and Y. Abdul- Aziz. On the computation of r0 and its role in global stability, 01 2022. [23] T. Roskilly and R. Mikalsen. Marine systems identification, modeling and control. Elsevier/Bh, Butterworth-Heinemann Is An Imprint Of Elsevier, 2015. [24] J. P. LaSalle. The Stability of Dynamical Systems. SIAM, 01 1976. [25] M. Martcheva. An Introduction to Mathematical Epidemiology. Springer US, 2015.