Demographic Heterogeneities in a Stochastic Chikungunya Virus Model with Poisson Random Measures and Near-Optimal Control under Markovian Regime Switching

1  Introduction

The Aedes mosquito-borne alphavirus (family Togaviridae), known as the CHIKV, has recently resurfaced in several regions worldwide, leading to widespread epidemics. Chikungunya fever is primarily characterized by symptoms such as fever, arthralgia, and, in some cases, a maculopapular rash. Bioactive compounds found in hops (Humulus lupulus, Cannabaceae), particularly acylphloroglucinols—commonly referred to as α- and β-acids—have demonstrated significant antiviral activity against CHIKV without exhibiting cytotoxic effects. Vector-borne diseases are infections transmitted to humans via vectors harboring microbial organisms, viruses, or bacteria. CHIKV, an arthropod-borne alphavirus, is propagated predominantly by Aedes aegypti and Aedes albopictus mosquitoes—the same vectors responsible for Zika, dengue, and yellow fever transmission [1]. CHIKV is a positive-sense, single-stranded RNA virus (11.8 kb) that encodes four nonstructural proteins (nsP1, nsP2 [helicase], nsP3, nsP4 [RNA-dependent RNA polymerase]) and six structural proteins (C, E3, E2, E1, 6K, and TF) [2]. Characterized clinically by arthralgia, myalgia, and tachycardia, CHIKV has expanded from its African origins to South America, particularly Brazil. Between 2013 and 2019, the Pan American Health Organization (PAHO) recorded over 600 CHIKV-related deaths in Latin America, likely underestimating the true global mortality burden.

Recent outbreaks of CHIKV have been markedly distinct from previous ones, characterized by accelerated transmission and broader geographic spread, often originating in migrant communities from regions where the virus remains endemic. A nationwide epidemic in Malaysia in 1998, primarily driven by workforce migration, exemplifies this trend [3]. Following a 20-year hiatus, Indonesia experienced a significant resurgence from 2001 to 2003 [4]. The 2005 global outbreak in India, affecting 1.4 million individuals, represents one of the most severe in recent history, occurring 32 years after the region’s last major CHIKV event [5]. The national economic impact of this epidemic was estimated at 45.26 disability-adjusted life years per million, approximately equating to 26,000 cases. In 2006, Réunion Island, located southwest of Madagascar in the Indian Ocean, witnessed an epidemic that impacted one-third of its population [6]. This outbreak was particularly severe, with a notably high fatality rate and significant neurological, hepatocellular, and cardiovascular complications. CHIKV, while similar to Zika and dengue, shows distinct clinical features. In endemic areas, 10%–70% are infected and 50%–97% develop symptoms after 4–7 days [7]. Neonates are highly vulnerable, and individuals over 65 have a five-fold higher fatality rate. The disease has an acute phase (7 days) and a persistent/recurrent phase lasting months or years, with acute polyarthralgia affecting 30%–90% of cases; some patients may also experience coronary, neurological, or ocular complications [8]. Chronic CHIKV infection poses ongoing health challenges. During the Réunion Island epidemic, 43%–75% had symptoms one year post-infection, including arthritic pain, fatigue, and neuralgia. Up to 50% may experience persistent pain, often linked to osteoarthritis and ankylosing spondylitis, affecting joints and cartilage [9].

CHIKV transmission occurs in two distinct phases: the forest and city transmission cycles. In the urban cycle, propagation follows a mosquito-mediated human-to-human transmission, while the forest cycle involves a mosquito-borne transmission from animals to humans route [10]. The forest phase is the initial mechanism for the virus’s persistence in Africa [11], whereas the urban cycle dominates in densely populated areas, where Aedes mosquitoes, particularly Aedes aegypti and Aedes albopictus, act as the primary vectors, and humans are the principal hosts [9]. Although Ae. aegypti has traditionally been the dominant vector, the significance of Ae. albopictus has increased, particularly in recent outbreaks in Europe, Gabon, and Réunion [12]. However, as evidenced by the 2013 Caribbean epidemic, Ae. aegypti remains a major vector despite this shift [13]. The spread of Ae. albopictus as a prominent vector has been linked to a genetic variation in the virus’s E1 protein (A226V polymorphism), which enhances the virus’s transmissibility to this mosquito species [9]. Additionally, vertical transmission of CHIKV, particularly in cases following the 2005 epidemic, has been observed, though its validity remains debated [14]. This mode of transmission is considered particularly concerning when the mother remains infected in the days following childbirth [9,11,13].

In humans, the transmission process of CHIKV is well understood, with certain cell types exhibiting heightened susceptibility. Human endothelial and epidermal cells, primordial fibroblasts, and monocyte-derived lymphocytes are particularly prone to infection, facilitating viral propagation. Conversely, CHIKV replication is generally not supported by major lymphocytes, monocytes, lymphoma cells, monocytoid cells, or dendritic cells derived from monocytes [9]. However, this finding remains contentious, as CHIKV-confirmed fibroblasts have been observed in vivo in virus-contaminated individuals [11]. Following initial infection, the host mounts an immune activation, however the infection subsequently spreads to lymphocytes and circulates through various organs [13]. This parasitic phase, marked by widespread viral dissemination, is when mosquitoes can acquire the virus from infected hosts during feeding, facilitating further transmission.

Several aspects of the immune response to CHIKV remain unresolved. To eliminate the virus, CHIKV activates non-hematopoietic mitochondria, primarily fibroblasts, triggering a type I interferon response at the site of infection [15]. The interferon response is regulated by factors 3 and 7, which are less effective in older individuals but are crucial for defending against infection in neonates who lack these factors [15]. Brito and Teixeira [16] highlighted that CHIKV-related fatalities were often underreported within healthcare systems, largely due to limited awareness of the virus’s clinical manifestations and complications. This underrecognition is understandable, given that CHIKV had only recently emerged in the Americas. The authors noted that secondary causes of death—such as cardiac, respiratory, or metabolic conditions—were frequently documented without acknowledging CHIKV infection as the primary or contributing factor. In individuals with persistent symptoms, CD4+ T-cells are more prevalent in synovial tissues, while CD8+ T-cells are found in the skin during the critical phase of the illness [16]. In addition, bone marrow mesenchymal antigen-2 may help mitigate autoimmune responses triggered by CHIKV and protect lymphoid tissue from damage [17].

To better understand biophysical phenomena and the unpredictable nature of transmission of infections, mathematical models have been extensively used [18,19]. Additionally, they are essential instruments for assessing and choosing the best means of control [20]. Sir Ronald Ross’s groundbreaking research on plasmodium established the groundwork for viral infection modeling. Plenty of models have been created since then. Deterministic ordinary differential equations (ODEs) were commonly employed for initial simulations, which assumed consistent death rates and ignored the effects of demographic characteristics and density of the population on morbidity. In [21], Bellan showed that the consequences of vector prevention techniques might be greatly overestimated if age-independent morbidity is assumed. The propagation of many ailments, such as dengue fever, human liver, tumor cells in immunogenetic tumour, HIV with CD4+ t-cells, SARS (severe acute respiratory syndrome), Zika virus, and, most recently, Nipah virus, is also known to be significantly impacted by density of population [18,20]. Recently, Olayiwola and Oluwafemi [22] modeled Tuberculosis dynamics with awareness and vaccination using the Laplace-Adomian method. Olayiwola et al. [23] studied the impact of booster vaccines on managing a new COVID-19 variant using the Laplace-Adomian decomposition method. Mohamed Ali et al. [24] examined stability criteria and optimal control measures for a fractional-order model of human papillomavirus (HPV) infection and cervical cancer. El-Mesady et al. [25] investigated optimal control strategies to reduce HPV transmission in a fractional-order mathematical model.

Meanwhile, the effects of population density have been incorporated into various simulations of insect-borne disease transmission [20]. In [21], the authors considered density-dependent larval mortality rates, but did not account for density effects on transitions between aquatic developmental stages. Although density-dependent mortality during the aquatic phase was included in the basic model of Aedes aegypti population dynamics proposed in [9], the three distinct aquatic stages were merged into a single class. This simplification poses challenges when assessing the impact of specific vector control strategies, as pharmaceutical treatments may affect eggs and larvae differently [14].

Similarly, although Buonomo [26] incorporated mortality driven by population density in humans and mosquitoes within a malaria transmission model, the model did not differentiate between the various aquatic stages of mosquito development via critical aspect for designing effective mosquito control measures. In the same vein, Chitnis et al. [27] studied the propagation processes of the CHIKV by considering only the egg and larval stages of the mosquito population, modeling the effects of density solely on recruitment rates and developmental progression. However, they did not address the impact of larval density on mortality rates, instead focusing exclusively on density-dependent stage advancement during the aquatic maturation process.

Achieving probabilistic optimal control becomes particularly challenging when strict constraints are present. In simpler cases, it may even be impossible to realize true optimal control due to the inherent unpredictability of the system, as highlighted in certain studies [28]. Nevertheless, even in such complex environments, it is often possible to attain near-optimal solutions for stochastic processes [29]. In many applications—such as economics, technology, and beyond developing near-optimal control strategies within an appropriate framework is crucial for achieving desired outcomes. The growing demand for probabilistic near-optimal control has driven significant advancements in this field, establishing it as a key component of dynamic regulation [28,29].

To address these challenges, researchers have developed controlled frameworks based on continuous-time models and finite-state Markov chains, particularly for systems where stochastic differential equations (SDEs) experience abrupt changes phenomena frequently encountered in fields like financial markets, risk management, and asset allocation [30]. Moreover, necessary conditions for ensuring near-optimal control in such complex systems have been rigorously established [30]. It is also widely recognized that environmental perturbation contributes a crucial role in the spread of viral diseases in real-world settings [31]. Since SDEs can more accurately capture the randomness inherent in infection patterns, simulating epidemic outbreaks within a stochastic framework is considered a more reliable approach. Consequently, numerous studies have explored how external noise influences epidemic dynamics [32]. Dereich et al. [33] investigated the behavior of a multilevel Monte Carlo algorithm within an epidemiological framework where population growth remains stable, independent of disease incidence. Their findings demonstrated that the system sustains at least one robust and favorable recurring pattern, ensuring the persistence of the epidemic in a stable environment.

Beyond white noise, Poisson noise is a critical factor in epidemiological simulations, inducing transitions between distinct environmental states [33,34]. In this work, we account for the influence of Poisson noise [35], which drives regime shifts corresponding to varying environmental conditions. For example, fish population growth rates can differ substantially between dry and wet seasons. These transitions, typically following an exponential waiting time distribution, are characterized by a memoryless property [36].

Inspired by the aforementioned literature, we propose a novel model for CHIKV transmission that incorporates density-dependent mortality rates within the human population. This model is calibrated using demographic data from a major CHIKV outbreak in Florida, USA, through the MCMC method. To more effectively quantify the influence of key parameters, a global sensitivity analysis is conducted, focusing on the primary thresholds that govern the model’s dynamics. Incorporating a Poisson random measure, we extend the model to a stochastic framework that accounts for dual infection dynamics, environmental noise with random jumps, and a saturated transmission rate. This stochastic model establishes sufficient conditions for both the persistence and potential eradication of the virus, offering a robust foundation for analyzing long-term disease behavior under fluctuating environmental conditions. Additionally, we address the near-optimal control problem in random frameworks influenced by Poisson noise and regime-switching, without the restrictive requirement of second-order continuous differentiability in the drift and diffusion coefficients. Necessary and sufficient requirements for achieving near-optimal control strategies—such as targeted vaccination and optimized intervention timing—are derived, demonstrating that any ϵ-optimal insight minimizes the Hamiltonian function involving error of order ϵ3/16. Furthermore, the model establishes that an ϵ-minimum condition guarantees near-optimality with an error order of ϵ1/2. This comprehensive framework significantly enhances the ability to design efficient and effective public health interventions against vector-borne diseases such as Chikungunya.

The organization of the work is as follows: Section 2 introduces our autonomous model, which incorporates the adaptive immune response and latently infected cells, and is then extended to a stochastic CHIKV model involving Gaussian white noise and Poisson random measure. Section 3 focuses on the mathematical analysis of the model in relation to the Poisson random measure. Section 4 presents and analyzes the near-optimality control conditions, including numerical simulations and an efficiency analysis. Section 5 discusses model calibration, population stratification, and the contact matrix for the CHIKV model. Section 6 conducts a sensitivity analysis and further examines the optimal control model, including numerical simulations and supporting arguments. Finally, the paper concludes with a summary in the last section. By rigorously analyzing individual class behaviors and leveraging advanced methodologies like Possion random measure, the entire procedurehas been illustrated in the flowchart presented in Fig. 1.

Figure 1: Flow diagram of the overall methodology

2  Model Description

In what follows, the development of CHIKV is dependent on the transmission properties of the virus. Chikungunya fever, which is caused by the CHIKV, an alphavirus spread by mosquitoes, is distinguished by an acute fever that appears suddenly, along with eruption, extreme arthritis in the joints, and exhaustion. The Asian continent, the European Union, and the Americas have all seen occurrences of such illness, which is mostly propagated by Aedes aegypti and Aedes albopictus mosquitoes [1,3,4]. Research indicates a clear pattern in the spread of CHIKV [14]. To better understand this phenomenon, we propose two models as follows: Firstly, the interaction between the infectious agent, host cells, and immune components such as T cells and antibodies—is modeled within an adaptive immune response framework for CHIKV. This framework helps explain immune variability during infection and fluctuations in transmission. We proceed by presenting the adaptive immune response model, represented by the following system of ODEs:

{dS(t)dt=π−ϖ1S(t)−βS(t)V(t),dI(t)dt=βS(t)V(t)−ϖ2I(t)−ϵI(t)Z(t),dV(t)dt=ϖ3I(t)−ϖ4V(t)−ϖ5V(t)B(t),dB(t)dt=ϖ6+ϖ7B(t)IV(t)−δB(t),dZ(t)dt=ϖ8+ϖ9I(t)Z(t)−ϖ10Z(t),(1)

subject to the ICs: S(0)=S0,I(0)=I0,V(0)=V0,B(0)=B0 and Z(0)=Z0. However, system (1) operates on the following assertions:

(i)   The number of sensitive cells, S(t), decreases due to natural loss and interaction with virus particles V(t), and fresh vulnerable cells are injected at a steady rate (see Shu et al. [37]).

(ii)   Infected cells, I(t), are formed through virus-cell interaction and can be removed naturally or destroyed by cytotoxic T-lymphocytes, Z(t) (see Dubey et al. [38]).

(iii)   Infected cells emit V(t), which are neutralized through interactions with antibodies. B(t) cells are activated by infection and eventually die. The population of cytotoxic T cells, Z(t), grows and then decreases at a steady rate (see Handel and Antia [39]).

Secondly, by accounting for lymphocytes that have been infected but are not yet infectious, a CHIKV model incorporating latently infected cells offers a more accurate representation of transmission dynamics. This enhancement captures the latent period between viral entry and active replication, improving the precision of predictive algorithms. The incubation phase influences viral propagation potential, timing of immune responses, and viral load buildup. Including this stage strengthens estimates for outbreak control and therapeutic intervention strategies. The following extended framework of ODEs builds upon the previous model by incorporating latently infected cells:

{dS(t)dt=π−ϖ1S(t)−βS(t)V(t),dL(t)dt=(1−ρ)βS(t)V(t)−(Θ+φ)L(t),dI(t)dt=ρβS(t)V(t)+φL(t)−ϖ2I(t)−ϵI(t)Z(t),dV(t)dt=ϖ3I(t)−ϖ4V(t)−ϖ5V(t)B(t),dB(t)dt=ϖ6+ϖ7V(t)B(t)−δB(t),dZ(t)dt=ϖ8+ϖ9I(t)Z(t)−ϖ10Z(t).(2)

subject to the ICS: S(0)=S0, L(0)=L0, V(0)=V0, B(0)=B0, Z(0)=Z0. These values determine the initial population sizes of each compartment. The dynamics of the compartmental model, capturing the transitions between the five key compartments.

The formulation of the proposed CHIKV models is grounded in established immunological and epidemiological modeling frameworks. The first model (1) focuses on the interaction between susceptible host cells, infected cells, virus particles, and immune responses (T-cells and antibodies). Such compartmental structures have been widely adopted in viral dynamics research (see Shu et al. [37]; Dubey et al. [38]; Handel and Antia [39]) as they capture the essential mechanisms of infection progression and immune regulation.

However, to provide a more realistic description of CHIKV transmission, it is crucial to account for the latent phase of infection, where cells are infected but not yet producing new virions. This motivates the extension presented in (2), where a new compartment for latently infected cells is introduced. Similar extensions have been successfully employed in the study of other mosquito-borne viral infections (see e.g., Buonomo [26], Chitnis et al. [27], Shu et al. [37]; Dubey et al. [38]). Incorporating latency reflects the biological reality that the virus requires an incubation period before active replication, thereby improving predictive accuracy for outbreak size, timing, and intervention strategies.

Thus, the transition from the baseline model (1) to the extended model (2) is justified both biologically and mathematically, ensuring that the formulation aligns with state-of-the-art approaches in infectious disease modeling. This refinement allows the model to better capture observed CHIKV transmission dynamics and provides a more solid foundation for evaluating control measures.

This second model (2) modifies the initial system (1) by incorporating the latently infected cells class L(t). All the parameters are described in Table 1.

The basic reproductive number

R0=βδπϖ3ϖ10ϖ1(ϖ5ϖ6+δ)(ϵϖ8+ϖ2ϖ10),

serves as a threshold: if R0>1, infection can spread, while R0<1 indicates eventual eradication. Public health relevance lies in its components: reducing β (vector control), lowering ϖ3 (antivirals), or enhancing immune clearance terms in the denominator (vaccination, improved diagnostics) all decrease R0. Thus, R0 guides which interventions most effectively reduce transmission, although under regime switching and stochastic effects the refined threshold R0L provides a more realistic measure. Effective control requires a combined strategy, where vector reduction lowers transmission opportunities while immune-strengthening interventions (e.g., vaccination or therapeutics) enhance host resistance, together providing the most impactful reduction in spread.

For the latently infected cells, we have ℵ¯=S(t)+L(t)+I(t)+V(t)+B(t)+Z(t). Now we convert the classical latently infected model to a Gaussian white noise version as follows:

{dS(t)=[π−ϖ1S(t)−βS(t)V(t)]dt+σ1S(t)dB1(t),dL(t)=[(1−ρ)βS(t)V(t)−(Θ+φ)L(t)]dt+σ2L(t)dB2(t),dI(t)=[ρβS(t)V(t)+φL(t)−ϖ2I(t)−ϵI(t)Z(t)]dt+σ3I(t)dB3(t),dV(t)=[ϖ3I(t)−ϖ4V(t)−ϖ5V(t)B(t)]dt+σ4V(t)dB4(t),dB(t)=[ϖ6+ϖ7V(t)B(t)−δB(t)]dt+σ5B(t)dB5(t),dZ(t)=[ϖ8+ϖ9I(t)Z(t)−ϖ10Z(t)]dt+σ6Z(t)dB6(t).(3)

In spite of the inherent advantages associated with Poisson random measure perturbations, their utility may be further amplified if noise-modulated drift velocity remains inside an optimal threshold domain. The integration of both local and nonlocal Lipschitz criteria underscores that embedding Poisson random measure disturbances can significantly enhance the fidelity and variability of transmitted information, particularly in feedback-driven epidemic models governed by arbitrary SDEs. As highlighted in previous studies [40], Poisson random measure noise exhibits notable advantages over conventional Gaussian disturbances in the context of theoretical epidemiological modeling. Nevertheless, its implementation introduces elevated computational complexity, which poses additional analytical challenges. The jump-diffusion framework incorporating Poisson random measures extends classical formulations by providing a more precise depiction of phenomena such as the abrupt firing of neuronal membrane potentials. Furthermore, the incorporation of such noise mechanisms has been shown to improve the robustness and temporal precision of recurrent neural network (RNN) architectures. When applied to the system described in Eq. (3), the use of Poisson random measure noise leads to a system of compound Poisson-driven SDEs, as detailed in [41].

{dS(t)=[π−ϖ1S(t)−βS(t)V(t)+σ1S(t)dB1(t)]dt+∫ΦΘ1(Ψ)S(t−)ℵ¯(dt,dΨ),dL(t)=[(1−ρ)βS(t)V(t)−(θ+φ)L(t)]dt+σ2L(t)dB2(t)+∫ΦΘ2(Ψ)L(t−)ℵ¯(dt,dΨ),dI(t)=[ρβS(t)V(t)+φL(t)−ϖ2I(t)−ϵI(t)Z(t)]dt+σ3I(t)dB3(t)+∫ΦΘ3(Ψ)I(t−)ℵ¯(dt,dΨ),dV(t)=[ϖ3I(t)−ϖ4V(t)−ϖ5V(t)B(t)]dt+σ4V(t)dB4(t)+∫ΦΘ4(Ψ)V(t−)ℵ¯(dt,dΨ),dB(t)=[ϖ6+ϖ7V(t)B(t)−δB(t)]dt+σ5B(t)dB5(t)+∫ΦΘ5(Ψ)B(t−)ℵ¯(dt,dΨ),dZ(t)=[ϖ8+ϖ9I(t)Z(t)−ϖ10Z(t)]dt+σ6Z(t)dB6(t)+∫ΦΘ6(Ψ)Z(t−)ℵ¯(dt,dΨ),(4)

whereas

Ω1=σ1S(t)dB1(t)+∫ΦΘ1(Ψ)S(t−)ℵ¯(dt,dΨ),Ω2=σ2L(t)dB2(t)+∫ΦΘ2(Ψ)L(t−)ℵ¯(dt,dΨ),Ω3=σ3I(t)dB3(t)+∫ΦΘ3(Ψ)I(t−)ℵ¯(dt,dΨ),Ω4=σ4V(t)dB4(t)+∫ΦΘ4(Ψ)V(t−)ℵ¯(dt,dΨ),Ω5=σ5B(t)dB5(t)+∫ΦΘ5(Ψ)B(t−)ℵ¯(dt,dΨ),Ω6=σ6Z(t)dB6(t)+∫ΦΘ6(Ψ)Z(t−)ℵ¯(dt,dΨ).

A new scheme of the associated SDE model is built to clarify the dynamical interconnections across the CHIKV propagation system, specifically in the occurrence of latently infectious cell compartments. The chronological progression of every demographic compartment with deterministic and stochastic consequences involving Gaussian white noise and Poisson random measure disruptions. Nonlinear prevalence factors control compartment transitions, and stochastic stimuli are included to account for unpredictable interaction occurrences and modifications in the environment. In particular, the abrupt activation of persistent lymphocytes is represented by a Poisson jump mechanism and a delay differential term, which effectively characterize the pattern of latent infections. Transitions such as infection, latency stimulation, virus generation, immune system stimuli, and natural eradication are indicated in the aforementioned system.

However, the compensated Poisson random measure indicated by ℵ¯, it can be described as

ℵ¯(dt,dT)=ℵ(dt,dT)−α(dt)dΨ.

The expression Θ(t) represents the left limit. The terms Bȷ(t)∀ȷ∈{1,…,6} represent autonomous classical Brownian motions, started at Bȷ(0)=0. The parameters ξȷ, where ȷ∈{1,…,6}, indicate the intensity of the stochastic perturbations, which incorporate Gaussian white noise effects into the epidemiological model. All additional components retain the identical interpretation. The Poisson random measure in this case is ℵ¯, and the strength of its estimate becomes α(.). Also, the mapping α stated on measure set Ψ[0,∞) possessing the features α(Ψ)<∞ and Θω≥0, (ω={1,…,6}).

2.1 Basic Concept

This section presents the foundational definitions and lemmas pertinent to stochastic analysis. The notation and structural conventions adopted here align with those outlined in [42], and will be consistently employed throughout this study.

⟨Q(t)⟩=1t∫0tQ(ȷ)dȷ.(5)

Lemma 1: ([43]) To facilitate our investigation, we shall employ two essential presumptions, denoted (A1(t)) and (A2(t)). Such requirements are necessary for establishing existence-uniqueness for global non-negative outcome for framework (3).

(A1(t)): for all M>0 ∃LM>0 such that

∫Φ|ΛI(Ψ1,℧)−ΛI(Ψ2,℧)|2dα(Ψ)≤LM|℧1−℧2|2,ȷ={1,2,…,6},(6)

with |℧1|∨|℧2|≤S, where

Λ1(℧,Ψ)=Θ1(Ψ)℧ for ℧=S(t−),Λ2(℧,Ψ)=Θ2(Ψ)℧ for ℧=L(t−),Λ3(℧,Ψ)=Θ3(Ψ)℧ for ℧=I(t−),Λ4(℧,Ψ)=Θ4(Ψ)℧ for ℧=V(t−),Λ5(℧,Ψ)=Θ5(Ψ)℧ for ℧=B(t−),Λ6(℧,Ψ)=Θ6(Ψ)℧ for ℧=Z(t−),(7)

where t represents the compensated stochastic parameter.

(A2(t)):|log⁡(1+Θȷ(℧))|≤∁ for Θȷ(℧)>−1,ȷ={1,2,…,6}, where ∁ is a fixed non-negative number.

Theorem 1: For a stochastic model (4) (S(t),L(t),I(t),V(t),B(t),Z(t)) is a solution containing ICs (S(0),L(0),I(0),V(0),B(0),Z(0))∈R6+, then

limt→∞S(t)+L(t)+I(t)+V(t)+B(t)+Z(t)t=0.(8)

Furthermore, if max{ϖ10ιh,ϖ10ȷh{>σ12+σ22+σ32+σ42+σ52+σ62t, then

limt→∞∫0tS(s)dB1(s)t=0,limt→∞∫0tL(s)dB2(s)t=0,limt→∞∫0tI(s)dB3(s)t=0,limt→∞∫0tB(s)dB4(s)t=0,limt→∞∫0tV(s)dB5(s)t=0,limt→∞∫0tZ(s)dB6(s)t=0.(9)

Thus, the solutions of CHIKV model (4) possess the subsequent features;

limt→∞supS(t)=πϖ1,limt→∞supL(t)=0,limt→∞supI(t)=0,limt→∞supB(t)=ϖ6δ,limt→∞supV(t)=0,limt→∞supZ(t)=ϖ8ϖ10.(10)

Proof: The supporting arguments for this lemma are analogous to those provided for Lemmas 2.1 and 2.2 in [44]; therefore, they are omitted here for brevity in the context of this analysis. ◻

The previously outlined definitions of mean persistence are particularly noteworthy, as highlighted in [42].

Definition 1: ([45])To establish the property of extinction or persistence, the stochastic model (4) is required to satisfy the following conditions:

limt→∞inf1t∫0tW(s)ds>0, a.s.(11)

The criteria outlined herein are consistent with those detailed in [42], wherein the associated lemmas delineate the requisite conditions for the persistence of infection.

Lemma 2: (Strong law) ([45]) Considering a real-continuous mapping S={S}0≤t, the local martingale criterion occurs if it disappears as t→0 and

limt→∞⟨S,S⟩t=∞, a.s.⟹limt→∞Φt⟨S,S⟩t=0, a.s.limt→∞sup⟨S,S⟩tt<0, a.s.⟹limt→∞Stt=0 a.s.(12)

Lemma 3: ([45]) Consider h∈∁([0,∞)×B¯(0,∞)) and G∈∁([0,∞)×B¯R)∋limt→∞G(t)t=0, a.s. if t≥0, then

log⁡f2(t)≥Γ0(t)−Γ∫0th(S)dS+G(t), a.s.

Therefore, we have

limt→∞inf⟨g(t)⟩≥Γ0Γ, a.s.

Here, Γ and Γ0 indicate positive constants, respectively.

Definition 2: (Approximate optimal control) ([46]) A set of permissible control strategies {(ϝηδ,Vηδ)} indexed by δ>0 and any member (ϝηδ,Vηδ), or just Vηδ, within this set is termed as an approximate optimal control if the following condition holds:

|J(0,ϝ0,Vηδ)−J(0,ϝ0)|≤ϵ(δ),(13)

where

J(0,ϝ0)=infVη∈Vad[0,T]J(0,ϝ0,Vη).(14)

The function ϵ(δ) satisfies:

ϵ(δ)→0asδ→0.(15)

If ∃ a fixed 𝒞>0 such that:

ϵ(δ)=𝒞δp,(16)

then Vηδ is termed as approximate optimal control of order δp.

Definition 3: (Generalized gradient in Clarke’s sense) [47] Consider Φ(⋅):Φ→R as a locally Lipschitz function on a convex domain Φ⊂Rn. The generalized gradient for Clarke Ψ at ϝ^∈Φ is stated as:

∂ϝΨ(ϝ^)={ξ∈Rn∣ξ⋅q≤φ0(ϝ^;q),∀q∈Rn},(17)

where

Ψ0(ϝ^;q)=lim supϝ→ϝ^p→0+Φ(ϝ+pq)−Φ(ϝ)p,ϝ∈𝒟,ϝ+pq∈Φ.(18)

Lemma 4: ([47]) Suppose that a1,a2,…,an are real-valued functions. Then the sunsequent inequality holds:

|a1+a2+⋯+an|m≤Cm(|a1|m+|a2|m+⋯+|an|m),(19)

where m>0 and

Cm={1,0<m≤1,nm−1,m>1.(20)

Lemma 5: ([47]) Suppose that there is a complete metric space (A,d) and a bounded operator ℱ(⋅):A→R which is also lower-semicontinuous. If ∃q>0 and Vq(⋅)∈ϝ such that:

ℱ(Vq(⋅))≤infV(⋅)∈ϝℱ(V(⋅))+q.(21)

2.2 Global Positive Solution

To enhance the realism of the CHIKV transmission model, we incorporate a compartment for latently infected cells and introduce stochasticity via a Poisson random measure. The resulting stochastic model (4) serves as a physiological analogue for the population dynamics of CHIKV infection. This model formulation requires the presence of a global, bounded, and non-negative solution. To analyze the well-posedness of the system defined by (4), we adopt the standard assumptions (A1) and (A2) stated in Lemma 1. These hypotheses enable us to establish the existence-uniqueness of a global solution. The forthcoming analysis will rigorously demonstrate that, under these assumptions, the stochastic CHIKV model with latently infected cells driven by a Poisson random measure permits a unique global positive solution.

Theorem 2: Surmise that the stochastic model (4), defined for t≥0, permits a unique solution (S(t),L(t),I(t),V(t),B(t),Z(t)) corresponding to ICs (S(0),L(0),I(0),V(0),B(0),Z(0))∈R+6. Furthermore, the solution stays in R+6 having probability one, means that (S(t),L(t),I(t),V(t),B(t),Z(t))∈R+6 ∀ t≥0, almost surely.

Proof: Since the drift and diffusion are guaranteed to be locally Lipschitz by the requirement (A1), there will be a time t at which the suggested concern will have a locally specific solution in the range [0,πe). The explosion phase is represented here by πe; researchers are directed to [42] for more details. To establish that the outcome is global, it need to be shown that πe=∞ is adequate. In order to illustrate this, let K0 be a large enough non-negative real number such that each outcome falls inside the range [1K0,K0]. As a result, given K≥K0, permit

πK={inft∈[0,πe):1K≥min{℧}orK≤max{℧}},(22)

where ℧=(S(t),L(t),I(t),B(t),V(t),Z(t),(t),IΛ(t)). Here, inf∅=∞. Moreover, πK increases as K→∞. Suppose that πK possesses a limit of π∞, It is almost a given that π∞≤πe. Alternatively, it is necessary to demonstrate that π∞=∞. If this supposition is false, then the constant T>0 and Φ lie within (0, 1) such that

P{π∞≤T}>Φ.(23)

So that, there is a natural number K1≥K0, the following holds

ε≤P{T≥πK},∀K≥K1.(24)

To demonstrate the remaining part of the hypothesis, we proceed by constructing the mapping as follows:

dU(℧)=LU(℧)dt+σ1(S−Q)d℧1(t)+σ2(L−1)d℧2(t)+σ3(I−1)d℧3(t)+σ4(B−1)d℧4(t)+σ5(V−1)d℧5(t)+σ6(Z−1)d℧6(t)+∫Φ[Θ1(℧)S−𝒬log⁡(Θ1(Ψ)+1)]ℵ¯(dt,dΨ)+∫Φ[Θ2(℧)L−log⁡(Θ2(Ψ)+1)]ℵ¯(dt,dΨ)+∫Φ[Θ3(℧)I−log⁡(Θ3(Ψ)+1)]ℵ¯(dt,dΨ)+∫Φ[Θ4(℧)B−log⁡(Θ4(Ψ)+1)]ℵ¯(dt,dΨ)+∫Φ[Θ5(℧)V−log⁡(Θ5(Ψ)+1)]ℵ¯(dt,dΨ)+∫Φ[Θ6(℧)Z−log⁡(Θ6(Ψ)+1)]ℵ¯(dt,dΨ).(25)

The aforementioned relationship defines the LU operator as a mapping from R6+ to R+. To demonstrate the remaining part of the hypothesis, we proceed by explicitly constructing this mapping as follows:

LU=(1−𝒬S)(π−ϖ1S(t)−βS(t)V(t))+(1−1L)+((1−ρ)βS(t)V(t)−(Θ+φ)L(t))+(1−1I)(ρβS(t)V(t)+φL(t)−ϖ2I(t)−ϵI(t)Z(t))+(1−1B)(ϖ3I(t)−ϖ4V(t)−ϖ5V(t)B(t))+(1−1V)(ϖ6+ϖ7V(t)B(t)−δB(t))+(1−𝒬Z)(ϖ8+ϖ9I(t)Z(t)−ϖ10Z(t))+𝒬σ122+σ222+σ322+σ422+σ522+σ622+∫Φ[𝒬Θ1(Ψ)−𝒬log⁡(Θ1(Ψ)+1)]t(dΨ)+∫Φ[Θ2(Ψ)−log⁡(Θ2(Ψ)+1)]t(dΨ)+∫Φ[Θ3(Ψ)−log⁡(Θ3(Ψ)+1)]t(dΨ)+∫Φ[Θ4(Ψ)−log⁡(Θ4(Ψ)+1)]t(dΨ)+∫Φ[Θ5(Ψ)−log⁡(Θ5(Ψ)+1)]t(dΨ)+∫Φ[Θ6(Ψ)−log⁡(Θ6(Ψ)+1)]t(dΨ)≤π+ϖ1Q+Θ+φ+ϖ2+ϖ6−ϖ7+ϖ4+ϖ10+𝒬σ122+σ222+σ322+σ422+σ522+σ622+∫Φ[𝒬Θ1(Ψ)−𝒬log⁡(Θ1(Ψ)+1)]t(dΨ)+∫Φ[Θ2(Ψ)−log⁡(Θ2(Ψ)+1)]t(dΨ)+∫Φ[Θ3(Ψ)−log⁡(Θ3(Ψ)+1)]t(dΨ)+∫Φ[Θ4(Ψ)−log⁡(Θ4(Ψ)+1)]t(dΨ)+∫Φ[Θ5(Ψ)−log⁡(Θ5(Ψ)+1)]t(dΨ)+∫Φ[Θ6(Ψ)−log⁡(Θ6(Ψ)+1)]t(dΨ)≤π+ϖ1Q+Θ+φ+ϖ2+ϖ6−ϖ7+ϖ4+ϖ10+𝒬σ122+σ222+σ322+σ422+σ522+σ622+∫Φ[𝒬Θ1(Ψ)−𝒬log⁡(Θ1(Ψ)+1)]t(dΨ)+∫Φ[Θ2(Ψ)−log⁡(Θ2(Ψ)+1)]t(dΨ)+∫Φ[Θ3(Ψ)−log⁡(Θ3(Ψ)+1)]t(dΨ)+∫Φ[Θ4(Ψ)−log⁡(Θ4(Ψ)+1)]t(dΨ)+∫Φ[Θ5(Ψ)−log⁡(Θ5(Ψ)+1)]t(dΨ)+∫Φ[Θ6(Ψ)−log⁡(Θ6(Ψ)+1)]t(dΨ).

Furthermore, (22) ensures the inequality S+L+I+B+Z+V≤1, thus

LU≤π+ϖ1Q+Θ+φ+ϖ2+ϖ6−ϖ7+ϖ4+ϖ10+𝒬σ12+σ22+σ32+σ42+σ52+σ622+∫Φ[𝒬Θ1(Ψ)S−𝒬log⁡(Θ1(Ψ)+1)]t(dΨ)+∫Φ[Θ2(Ψ)L−log⁡(Θ2(Ψ)+1)]t(dΨ)+∫Φ[Θ3(Ψ)I−log⁡(Θ3(Ψ)+1)]t(dΨ)∫Φ[Θ4(Ψ)B−log⁡(Θ4(Ψ)+1)]t(dΨ)+∫Φ[Θ5(Ψ)V−log⁡(Θ5(Ψ)+1)]t(dΨ)∫Φ[𝒬Θ6(Ψ)Z−𝒬log⁡(Θ6(Ψ)+1)]t(dΨ)=O.

For a non-negative O and an automatons fixed (S,L,I,B,V,Z), we arrive at the following:

𝒰(℧)≤dt+σ1(S−1)dB1(t)+σ2(L−1)dB2(t)+σ3(I−1)dB3(t)+σ4(V−1)dB4(t)+σ5(B−1)dB5(t)+σ6(Z−1)dB6(t).(26)

By evaluating the integral of both sides of Eq. (26) over the interval from 0 to T∧πK, the following can be articulated as

E𝒰(S(T∧πK),L(T∧πK),I(T∧πK),B(T∧πK),V(T∧πK),Z(T∧πK))≤𝒰(S(0),L(0),I(0),V(0),B(0),Z(0))+KT<∞.(27)

Fix ΩK={πK≤t} for K≥K1 by (24), P(ΩK)≥ε. Observing each ϖ9∈ΩK, at least one of S(πK,ϖ9),L(πK,ϖ9),I(πK,ϖ9),V(πK,ϖ9),H(πK,ϖ9), also R(πK,ϖ9), which is equal either {k or 1k}. Consequently, S(πK,ϖ9),L(πK,ϖ9),I(πK,ϖ9),V(πK,ϖ9),H(πK,ϖ9) and R(πK,ϖ9) in no way is less than {K−1−log⁡K or 1K−1−log⁡K}.