Volume 2024, Issue 1 2130429

Research Article

Open Access

The Significance of Stochastic CTMC Over Deterministic Model in Understanding the Dynamics of Lymphatic Filariasis With Asymptomatic Carriers

Corresponding Author

Mussa A. Stephano

[email protected]

orcid.org/0000-0003-2070-3443

Department of Applied Mathematics and Computational Science, The Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania nm-aist.ac.tz

Department of Mathematics, Physics and Informatics, Mkwawa University College of Education, P.O. Box 2513, Iringa, Tanzania muce.ac.tz

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Jacob I. Irunde,

Jacob I. Irunde

orcid.org/0000-0003-0640-2169

Department of Mathematics, Physics and Informatics, Mkwawa University College of Education, P.O. Box 2513, Iringa, Tanzania muce.ac.tz

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Maranya M. Mayengo,

Maranya M. Mayengo

orcid.org/0000-0002-1745-5509

Department of Applied Mathematics and Computational Science, The Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania nm-aist.ac.tz

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Dmitry Kuznetsov,

Dmitry Kuznetsov

orcid.org/0000-0001-7177-0991

Department of Applied Mathematics and Computational Science, The Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania nm-aist.ac.tz

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Mussa A. Stephano,

Corresponding Author

Mussa A. Stephano

[email protected]

orcid.org/0000-0003-2070-3443

Department of Applied Mathematics and Computational Science, The Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania nm-aist.ac.tz

Department of Mathematics, Physics and Informatics, Mkwawa University College of Education, P.O. Box 2513, Iringa, Tanzania muce.ac.tz

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Jacob I. Irunde,

Jacob I. Irunde

orcid.org/0000-0003-0640-2169

Department of Mathematics, Physics and Informatics, Mkwawa University College of Education, P.O. Box 2513, Iringa, Tanzania muce.ac.tz

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Maranya M. Mayengo,

Maranya M. Mayengo

orcid.org/0000-0002-1745-5509

Department of Applied Mathematics and Computational Science, The Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania nm-aist.ac.tz

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Dmitry Kuznetsov,

Dmitry Kuznetsov

orcid.org/0000-0001-7177-0991

Department of Applied Mathematics and Computational Science, The Nelson Mandela African Institution of Science and Technology, P.O. Box 447, Arusha, Tanzania nm-aist.ac.tz

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First published: 04 May 2024

https://doi.org/10.1155/2024/2130429

Citations: 1

Academic Editor: Arvind Kumar Misra

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Abstract

Lymphatic filariasis is a leading cause of chronic and irreversible damage to human immunity. This paper presents deterministic and continuous-time Markov chain (CTMC) stochastic models regarding lymphatic filariasis dynamics. To account for randomness and uncertainties in dynamics, the CTMC model was formulated based on deterministic model possible events. A deterministic model’s outputs suggest that disease extinction is feasible when the secondary threshold infection number is below one, while persistence becomes likely when the opposite holds true. Furthermore, the significant contribution of asymptomatic carriers was identified. Results indicate that persistence is more likely to occur when the infection results from asymptomatic, acutely infected, or infectious mosquitoes. Consequently, the CTMC stochastic model is essential in capturing variabilities, randomness, associated probabilities, and validity across different scales, whereas oversimplification and unpredictability of inherent may not be featured in a deterministic model.

1. Introduction

The nematode family of microfilarial parasites transmitted by mosquito-vector causing lymphatic filariasis commonly results in the chronic manifestation known as elephantiasis [1]. This disease is prevalent in tropical regions, especially in developing countries, where it often receives insufficient attention [2, 3]. Lymphatic system weakening, physical deformities, and long-term disability significantly impact the quality of life of affected individuals [4, 5].

Humans and mosquitoes serve as the definitive-primary and intermediate hosts of the parasites, respectively [4, 6]. In definitive-primary host, it presents three different stages, namely, asymptomatic carriers, acute, and chronic infected as illustrated in Figure 1. The asymptomatic stage exhibits nonvisible symptoms of infection, yet the parasites can still cause an impact on the immunity, facilitating the infection. In the acute stage, adenolymphangitis, a localized inflammation of lymph nodes and vessels, accompanied by skin problems is characterized. In the chronic stage, it progresses to hydrocele, lymphedema, compromised mental well-being, and elephantiasis [1, 4].

Details are in the caption following the image — **Figure 1**
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The life cycle of parasites, clinical manifestations, and physical condition [1, 4, 16, 17].

Lymphatic filariasis is a global health concern, exhibiting varying prevalence rates across different regions. It is transmitted by different mosquito vectors depending on the nematode species involved. The World Health Organization (WHO) in collaboration with governments recommended the implementation of mass drug administration (MDA) programs targeting both infected populations and those at risk [1]. Despite these efforts, the persistent prevalence of the disease emphasizes an urgent need for comprehensive research on its transmission dynamics to support eradication initiatives, aligning with Sustainable Development Goal 3 and the roadmap for eliminating neglected tropical diseases by 2030.

Mathematical modelling has been used to address lymphatic filariasis in different aspects. The key and fundamental studies include Mwamtobe et al. [7], who introduced a model on humans and mosquitoes and concluded that treatment is more effective than quarantine. Jambulingam et al. [8] worked on a statistical model with a hypothesis on MDA with respect to the burden in India, highlighting that the MDA duration is proportional to the endemicity baseline. Michael et al. [9] addressed the MDA efficacy and control with albendazole and diethylcarbamazine in combination. Swaminathan et al. [10] worked on challenges with the aspect of prospects targeting the elimination of the disease, and Cromwell et al. [11] developed a geospatial distribution between 2000 and 2018.

Regarding the convolution of the life cycle of the microfilarial parasites in both two hosts, modelling the interaction and spread mode is of significant importance when including stochasticity. Nevertheless, the existing models did not incorporate stochastic behavior in dynamics. The significance of the CTMC model is that it captures variabilities and uncertainties that arise due to environmental variations and demographics and also predicts the probabilities of extinction or outbreak in finite time and studies the random sample path solutions near disease-free and endemic equilibria [12–15]. This paper presents a stochastic CTMC model based on the modified existing deterministic models of lymphatic filariasis dynamics.

2. Model Formulation

2.1. A Deterministic Model

The formulation of the model that depicts the dynamics of lymphatic filariasis in both definitive-primary and intermediate hosts is based on assumptions that the vertical transmission is not accounted for by the exclusion of immigration considerations, adherence to the mass action principle for the rate of infection [17], and the occurrence of induced mortality in the class of infected chronic only. Figure 2 visually represents the dynamic modification of the work by Jambulingam et al. [8], Mwamtobe et al. [7], Stolk et al. [18], and Swaminathan et al. [10].

The human population is divided into five groups: susceptible S₁(t), exposed E₁(t), asymptomatic A(t), infected in the acute state I₁(t), and infected in the chronic state I₂(t). Similarly, the mosquito population comprises susceptible S₂(t), exposed E₂(t), and infected I₃(t).

S₁(t) recruited at a rate π₁, move to E₁(t) being bitten by I₃(t) at a rate β. A considerable number of individuals progress to A(t) at a rate α, while the remaining proportion progress to either I₁(t) at a rate ψ or I₂ at a rate ρ. Moreover, A(t) progress to either I₁(t) at a proportion of ξ or I₂(t) at a rate ϕ. I₁(t) progress to I₂(t) at a rate σ. I₂(t) induce death at a rate δ, and humans die naturally at a rate of μ_h.

π₂ is the recruitment rate of S₂(t), exposed to lymphatic filariasis with either A(t), I₁(t), or I₂(t) defined by general rate (Equation (1)):

(1)

where β₁, β₂, and β₃ are transmission rates for S₂ from A(t), I₁(t), and I₂, respectively. E₂(t) progress to I₃(t) at a pace ω, and the natural mortality rate of mosquitoes is μ_m.

Model (2) represents the epidemiological interactions and transitions using a nonlinear system from Figure 2.

(2)

Bounded by

S₂(0) > 0, E₁0) ≥ 0, A(0) ≥ 0, I₁(0) ≥ 0, I₂(0) ≥ 0, S₁(0) > 0, E₂(0) ≥ 0, and I₃(0) ≥ 0

2.2. Model Analysis

The dynamical system described by Equation (2) is subjected to evaluate its mathematical well-posedness and biological feasibility within the region .

2.2.1. Positivity

To demonstrate the positivity of model solutions for all t ≥ 0, let {t > 0 : S₁ > 0, E₁ > 0, A > 0, I₁ > 0, I₂ > 0, S₂ > 0, E₂ > 0, I₃ > 0}, N₁ = S₁ + E₁ + A + I₁ + I₂, and N₂ = S₂ + E₂ + I₃.

From the equation governing susceptible humans in Model (2),

( )

Assuming S₁(0) > 0 and ∃t = t₀ > 0, where S₁(t₀) = 0, dS₁(t₀)/dt < 0, E₁(t₀) ≥ 0, A(t₀) ≥ 0, I₁(t₀) ≥ 0, I₂(t₀) ≥ 0, S₂(t₀) > 0, E₂(t₀) ≥ 0, and I₃(t₀) ≥ 0, it follows:

(3)

This contradicts the assumption, hence S₁(t) > 0 for all t > 0. Applying a similar method as used to get Equation (3), then Model (2) is positive invariant ∀t > 0.

2.2.2. Boundedness

By considering the total population of humans and mosquitoes separately, we have

(4)

From Equation (4), the first population using integration techniques leads

(5)

Applying the standard comparison theorem [19], when N₁(0) > π₁/μ_h and N₁(0) < π₁/μ_h, we obtain

(6)

With time approaching to infinity, it follows that

(7)

Consequently, Model (2) is bounded within the region .

2.3. Filariasis Free Equilibrium E₀ and Basic Reproduction Number

2.3.1. Filariasis Free Equilibrium

The context involves the extinction of parasites from populations, leading to a disease-free equilibrium denoted as E₀, which signifies the absence of lymphatic filariasis.

(8)

2.3.2. Basic Reproduction Number

The computation of

, with the help of the next-generation matrix technique evaluated at E₀, is given by

(9)

whereby

( )

where

represents the threshold value of infections after initial states generated by one infection in susceptible individuals throughout the period of infection [20].

2.3.3. Global Stability of E₀

Theorem 1. The filariasis-free equilibrium E₀ is globally asymptotically stable when and unstable otherwise.

We decompose System (2) following the Metzler matrix technique as used in Stephano et al. and Castillo-Chavez and Song [17, 21]. Let u₁ stands for (S₁, S₂), while u₂ stands for (E₁, A, I₁, I₂, E₂, I₃). Therefore, Model (2) can be decomposed to:

(10)

where

( )

Matrix B₁ has negative eigenvalues which determine that matrix B₃ is a Metzler matrix. Given that the nondiagonal components of matrix B₃ exhibit positivity, it is clear that E₀ is globally asymptotically stable.

2.4. Filariasis Endemic Equilibrium E^∗

In the situation of persistence of lymphatic filariasis, the nontrivial point is computed from Model (2). It is denoted by

, such that

( )

2.4.1. The Global Stability Analysis of Endemic Equilibrium E^∗

Theorem 2. Model System (2) exhibits a unique endemic equilibrium E^∗ whenever , which is globally asymptotically stable.

Proof 1. We consider vector components and define a Lyapunov function G as follows:

(11)

By differentiating Equation (11) with respect to t, we have

(12)

Let , , c = A/A^∗, , , , , and t = λ/λ^∗.

Substituting equations in Equation (2) into Equation (12) gives the following expression of each term:

(13)

Summing up all expressions in Equation (13) and evaluating the values of k_i, i = 1, 2, ⋯, 8 gives

( )

Back substitution into Equation (13) and adding up gives

(14)

If an algebraic inequality 1 − z ≤ − lnz, for any z > 0, then we express all terms in Equation (14) as follows:

Considering the third term, we have:

(15)

A similar approach is applied to all terms. It is evident that dG/dt ≤ 0. Moreover, applying the Krasovkii–Lasalle theorem in Equation (14), dG/dt = 0, if and only if Y = E^∗. Hence, the set consists of the singleton E^∗. This concludes the proof.

2.5. Model Sensitivity Analysis

In this subsection, we utilize Latin hypercube sampling-partial rank correlation coefficient (LHS-PRCC) as a tool for sensitivity and uncertainty. The LHS-PRCC algorithm is robust and particularly suitable for nonlinear systems with a consistent relationship between input and output [17]. The algorithm integrates both uniform and normal probability density functions. This method is recognized for its effectiveness in managing nonlinear ordinary differential equations [22]. Moreover, it quantifies the level of the linear relationship between inputs and outputs to furnish PRCC indices, following the methodology described by [23]. The variable I₃(t) in Model (2) is used as a showcase to demonstrate the change of PRCC indices with time.

The findings from the LHS-PRCC analysis offered valuable insights into the impact of all parameters against the chosen variable and associated uncertainties on System (2). In the context of statistical inference, PRCC indices close to or equal to zero have no significance. Figure 3(a) visually represents the PRCC indices, highlighting that π₂, β, and β₂ on outputs have strong directly proportion relationship. Conversely, parameters like μ_h and μ_m demonstrating the relationship in inverse proportionally with the model outputs. Figure 3(b) provides a comprehensive view of their sensitivity and illustrates how changes in all parameters influence Model (2). These findings suggest the important variables to be considered for effective control strategies.

2.6. CTMC Stochastic Model Formulation

To assess probabilities of extinction or persistence, the CTMC model was formulated using the technique of events from the process of multitype branching. Capturing the real-world phenomena in the dynamics of lymphatic filariasis, incorporating randomness, is necessary for uncertainties and variabilities [24, 25]. The disease can vanish even when the threshold exceeds one, depending on the initial number of infections available in the population. This study considers individual movement to be discrete for realistic depiction [13, 14].

The formulation of the CTMC stochastic model is derived from System (2) events.

If S₁, E₁, A, I₁, I₂, S₂, E₂, and I₃ represent discrete random variables, then the vector associated is

( )

which follow the Markov chain property and adhere to homogeneity in time. Moreover, the total event is defined by

(16)

We define S₁(0) = π₁/μ_h and S₂(0) = π₂/μ_m, where infectives of type i, denoted as I_i, give rise to infectives of type j, denoted as I_j. It is assumed that the number of offspring produced by an individual of type i is independent of the number of offspring produced by either type i or type j, where j ≠ i [14, 26].

This calculation assumes there is initially only one infectious individual at the beginning of the disease outbreak (I(0) = 1), with all other types being zero (I_j = 0).

(17)

where p_i(k₁, k₂, ⋯, k_n) represents the likelihood that an infectious individual of type i produces k_j individuals of type j.

Filariasis extinction is defined by probability , where (r₁, r₂, ⋯, r_k) is the unique fixed point of the offspring pgf, p_i(r₁, r₂, ⋯, r₆) = r_i, and 0 < r_i < 1 for i = 1, 2, ⋯, k, whereas 1 − ℙ₀ is an outbreak probability.

If E₁(0) = 1, A(0) = 0, I₁(0) = 0, I₂(0) = 0, E₂(0) = 0, and I₃(0) = 0, the offspring probability generating function for the infected class E₁ is defined as follows:

(18)

where α/(α + ϕ + ρ + μ_h) stands for the probability of E₁(t) progress to the A(t), ψ/(α + ϕ + ρ + μ_h) represents the probability of E₁(t) progress to I₁(t), ρ/(α + ϕ + ρ + μ_h) denotes the probability of E₁(t) progress to I₂(t), and μ_h/(α + ϕ + ρ + μ_h) is the probability of E₁(t) die before any progress

Using a methodology employed in Equation (18), we derive the offspring probability-generating functions for A(0), I₁(0), I₂(0), E₂(0), and I₃(0) as p₂, p₃, p₄, p₅, and p₆, respectively. These functions are defined as follows:

(19)

The expectation matrix

computed from the number of offspring of type j resulted from type i as presented by evaluating at 1:

(20)

Using Equation (20) in conjunction with Equations (18) and (19), we derive

( )

where G₁ = α + ρ + ψ + μ_h,

, and

. From matrix

, it is straightforward to determine the maximum eigenvalue which represents the stochastic threshold. The correlation between the deterministic and stochastic thresholds for parasite extinction is expressed as

[27, 28]. If

, there exists the potential for a microfilarial parasite outbreak or extinction, contingent upon the number of infectives at the onset of the disease outbreak [13, 26, 29]. Therefore, when

, a fixed point (r₁, r₂, r₃, r₄, r₅, r₆) ∈ (0, 1)⁶ is determined from the offspring-generating functions (Equations (18) and (19)), which are utilized to compute the probability of disease extinction [14, 25]. To ascertain this fixed point, the equations p_i(r₁, r₂, r₃, r₄, r₅, r₆) = r_i for i = 1, 2, ⋯, 6 are solved.

(21)

The probability-generating functions r₁, r₂, ⋯, r₆ exhibit nonlinearity, rendering them difficult to compute analytically. Consequently, numerical simulations, as demonstrated by Stephano et al. [15], are employed for their computation.

3. Model Simulations

In this section, simulations are conducted using both deterministic and CTMC stochastic models to explore the dynamics of lymphatic filariasis. Parameter values, drawn from a Gaussian distribution, are based on baseline values obtained from existing literature, as detailed in Table 1. The simulations involve ten thousand sample trajectories, concurrently depicted with corresponding deterministic solutions to facilitate comparison. Euler’s and Gillespie’s algorithms are utilized alongside arbitrary initial conditions.

( )

Table 1. Parameter values month⁻¹.

Parameter	Baseline	Reference	Range
π₁	20.0000	[16]	[10.000030.0000]
β	0.0005	[17]	[0.000100.0050]
β₁	0.0015	[17]	[0.00100.0030]
β₂	0.0035	[17]	[0.00150.0045]
β₃	0.00025	[17]	[0.000150.00035]
ρ	0.00032	[17]	[0.000250.0075]
ψ	0.0015	[30]	[0.00250.0085]
δ	0.000015	[9]	[0.000010.00003]
ω	0.0055	[17]	[0.00350.0065]
ϕ	0.00045	[17]	[0.00030.0009]
π₂	100,000	[16]	[50000150000]
μ_h	0.0142	[15, 17]	[0.01000.0200]
α	0.0200	[9]	[0.01000.0500]
μ_m	0.050000	[30]	[0.01001.5000]
ξ	0.00030	[16, 18]	[0.00250.0055]
σ	0.0012	[10]	[0.00100.0030]

Figure 4, depicting the solutions of the deterministic model, corresponds to Figures 5 and 6, showing both deterministic and CTMC stochastic solutions. It is evident that the count of susceptible individuals decreases following infections, stabilizing after approximately 100 months. Similarly, the population of susceptible mosquitoes diminishes, reaching a steady state. The results reveal that both deterministic and CTMC stochastic models exhibit similar behavior, with stochastic outputs reflecting randomness and deterministic outputs representing the average trend observed across CTMC random sample paths. A negative correlation exists between the susceptible groups and the acutely infected, chronically infected, and asymptomatic classes. Initially, the count of exposed humans rises, peaking within the first 60 months, before declining and eventually stabilizing after 150 months. Similarly, the numbers of asymptomatic, acutely infected, and chronically infected humans also peak before stabilizing.

4. Conclusion

Based on the findings presented in this paper, it is evident that lymphatic filariasis poses a significant threat to human health, causing chronic and irreversible damage to the immune system. Through the development of both deterministic and CTMC stochastic models, this study is aimed at capturing the complexity of lymphatic filariasis dynamics. The deterministic model highlights the potential for disease extinction when the secondary threshold infection number falls below one, contrasting with scenarios where persistence is more likely to occur. Additionally, the role of asymptomatic carriers in disease transmission was identified as significant. These results emphasize the importance of employing stochastic models, such as CTMC, to capture variabilities, randomness, and associated probabilities more accurately, especially considering the limitations of deterministic models in capturing inherent complexities and unpredictabilities.

Conflicts of Interest

The authors declare no conflicts of interest.

Funding

The authors received no specific funding for this work.

Open Research

Data Availability Statement

All data is included in the manuscript.

References

1 WHO, Lymphatic filariasis, 2022, World Health Organization, https://www.who.int/news-room/fact-sheets/detail/lymphatic-filariasis/.
Google Scholar
2 Addiss D. and Eliminate Lymphatic Filariasis, The 6th meeting of the Global Alliance to Eliminate Lymphatic Filariasis: a half-time review of lymphatic filariasis elimination and its integration with the control of other neglected tropical diseases, 2010, 3, no. 1, https://doi.org/10.1186/1756-3305-3-100, 2-s2.0-77958007166.
10.1186/1756?3305?3?100
Google Scholar
3 Lustigman S., Prichard R. K., Gazzinelli A., Grant W. N., Boatin B. A., McCarthy J. S., and Basáñez M. G., A research agenda for helminth diseases of humans: the problem of helminthiases, PLoS Neglected Tropical Diseases. (2012) 6, no. 4, article e1582, https://doi.org/10.1371/journal.pntd.0001582, 2-s2.0-84861212955, 22545164.
10.1371/journal.pntd.0001582
PubMed Web of Science® Google Scholar
4 CDC, Parasites - lymphatic filariasis: centers for diseases control and prevention, 2022, https://www.cdc.gov/parasites/lymphaticfilariasis/epi.html.
Google Scholar
5 Centers for Disease Control and Prevention, Lymphatic filariasis, 2019, January 2023, https://www.cdc.gov/dpdx/lymphaticfilariasis/index.htm.
Google Scholar
6 Castillo J. C., Reynolds S. E., and Eleftherianos I., Insect immune responses to nematode parasites, Trends in Parasitology. (2011) 27, no. 12, 537–547, https://doi.org/10.1016/j.pt.2011.09.001, 2-s2.0-82855165053.
10.1016/j.pt.2011.09.001
CAS PubMed Web of Science® Google Scholar
7 Mwamtobe P. M., Simelane S. M., Abelman S., and Tchuenche J. M., Mathematical analysis of a lymphatic filariasis model with quarantine and treatment, BMC Public Health. (2017) 17, no. 1, 1–13, https://doi.org/10.1186/s12889-017-4160-8, 2-s2.0-85015160071.
10.1186/s12889-017-4160-8
PubMed Web of Science® Google Scholar
8 Jambulingam P., Subramanian S., De Vlas S., Vinubala C., and Stolk W., Mathematical modelling of lymphatic filariasis elimination programmes in India: required duration of mass drug administration and post-treatment level of infection indicators, Parasites & Vectors. (2016) 9, no. 1, 501–518, https://doi.org/10.1186/s13071-016-1768-y, 2-s2.0-84992187396, 27624157.
10.1186/s13071-016-1768-y
PubMed Web of Science® Google Scholar
9 Michael E., Malecela-Lazaro M. N., Simonsen P. E., Pedersen E. M., Barker G., Kumar A., and Kazura J. W., Mathematical modelling and the control of lymphatic filariasis, The Lancet Infectious Diseases. (2004) 4, no. 4, 223–234, https://doi.org/10.1016/S1473-3099(04)00973-9, 2-s2.0-1642420527.
10.1016/S1473-3099(04)00973-9
PubMed Web of Science® Google Scholar
10 Swaminathan S., Subash P. P., Rengachari R., Kaliannagounder K., and Pradeep D. K., Mathematical models for lymphatic filariasis transmission and control: challenges and prospects, Parasites & Vectors. (2008) 1, no. 1, 1–9, https://doi.org/10.1186/1756-3305-1-2, 2-s2.0-65349186696, 18275593.
10.1186/1756?3305?1?2
PubMed Web of Science® Google Scholar
11 Cromwell E. A., Schmidt C. A., Kwong K. T., Pigott D. M., Mupfasoni D., Biswas G., Shirude S., Hill E., Donkers K. M., Abdoli A., Abrigo M. R. M., Adekanmbi V., Adetokunboh Sr O. O., Adinarayanan S., Ahmadpour E., Ahmed M. B., Akalu T. Y., Alanezi F. M., Alanzi T. M., Alinia C., Alipour V., Amit Sr A. M. L., Anber N. H., Ancuceanu R., Andualem Z., Anjomshoa M., Ansari F., Antonio C. A. T., Anvari D., Appiah S. C. Y., Arabloo J., Arnold B. F., Ausloos M., Ayanore Sr M. A., Badirzadeh A., Baig Jr A. A., Banach Sr M., Baraki Sr A. G., Bärnighausen T. W., Bayati M., Bhattacharyya Sr K., Bhutta Z. A., Bijani A., Bisanzio D., Bockarie M. J., Bohlouli S., Bohluli M., Butt Z. A., Cano J., Carvalho F., Chattu V. K., Chavshin A. R., Cormier N. M., Damiani G., Dandona L., Dandona R., Darwesh A. M., Daryani A., Dash A. P., Deribe K., Deshpande A., Dessu B. K., Dhimal M., Dianatinasab M., Diaz D., do H. T., Earl L., el Tantawi M., Faraj A., Fattahi N., Fernandes E., Fischer F., Foigt N. A., Foroutan M., Guo Y., Hailu G. B., Hasaballah A. I., Hassankhani H., Herteliu C., Hidru H. D., Hole M. K., Hon J., Hossain N., Hosseinzadeh M., Househ M., Humayun A., Ilesanmi O. S., Ilic I. M., Ilic M. D., Iqbal U., Irvani S. S. N., Islam M. M., Jha R. P., Ji J. S., Johnson K. B., Jozwiak J. J., Kabir A., Kalankesh L. R., Kalhor R., Karami Matin B., Karch A., Karimi S. E., Kasaeian A., Kayode G. A., Kazemi Karyani A., Kelbore A. G., Khafaie M. A., Khalilov R., Khan J., Khatab K., Khater M. M., Khodayari M. T., Kianipour N., Kim Y. J., Kinyoki D. K., Kumar G. A., Kusuma D., la Vecchia C., Lansingh V. C., Lee P. H., LeGrand K. E., Levine A. J., Li S., Maleki S., Mansournia M. A., Martins-Melo F. R., Massenburg B. B., Mayala B. K., Meitei W. B., Mendoza W., Mengistu D. T., Mereta S. T., Mestrovic T., Mihretie K. M., Miller-Petrie M. K., Mohammadian-Hafshejani A., Mohammed S., Mokdad A. H., Moradi M., Moradzadeh R., Moraga P., Morrison S. D., Mosser J. F., Mousavi S. M., Munro S. B., Muthupandian S., mwingira U. J., Naderi M., Nagarajan A. J., Naik G., Negoi I., Nguyen T. H., Nguyen H. L. T., Olagunju A. T., Omar Bali A., Osarenotor O., Osei F. B., Pasupula D. K., Pirsaheb M., Pourjafar H., Rathi P., Rawaf D. L., Rawaf S., Rawassizadeh R., Reiner Jr R. C., Reta M. A., Rezapour A., Ribeiro A. I., Rostami A., Sabesan S., Sadeghi E., Sajadi S. M., Samy A. M., Sartorius B., Schaeffer L. E., Shaikh M. A., Sharafi K., Sharafi Z., Sharifi H., Shibuya K., Shin J. I., Soheili A., Soltani S., Spotin A., Stolk W. A., Tesfay B. E., ThekkePurakkal A. S., Topor-Madry R., Tran K. B., Tran B. X., Ullah I., Unnikrishnan B., Vasseghian Y., Vinkeles Melchers N. V. S., Violante F. S., Yamada T., Yaya S., Yazdi-Feyzabadi V., Yip P., Yonemoto N., Zaki L., Zaman S. B., Zamanian M., Zangeneh A., Zhang Z. J., Zhang Y., Ziapour A., King J. D., and Hay S. I., The global distribution of lymphatic filariasis, 2000–18: a geospatial analysis, The Lancet Global Health. (2020) 8, no. 9, e1186–e1194, https://doi.org/10.1016/S2214-109X(20)30286-2.
10.1016/S2214-109X(20)30286-2
PubMed Web of Science® Google Scholar
12 Allen L. and Burgin A., Comparison of deterministic and stochastic SIS and SIR models, 1999, Dept. Math. Technical report, Texas Tech. University.
Google Scholar
13 Maliyoni M., Probability of disease extinction or outbreak in a stochastic epidemic model for West Nile virus dynamics in birds, Acta Biotheoretica. (2021) 69, no. 2, 91–116, https://doi.org/10.1007/s10441-020-09391-y.
10.1007/s10441-020-09391-y
PubMed Web of Science® Google Scholar
14 Maliyoni M., Chirove F., Gaff H. D., and Govinder K. S., A stochastic tick-borne disease model: exploring the probability of pathogen persistence, Bulletin of Mathematical Biology. (2017) 79, no. 9, 1999–2021, https://doi.org/10.1007/s11538-017-0317-y, 2-s2.0-85023755257, 28707219.
10.1007/s11538-017-0317-y
PubMed Web of Science® Google Scholar
15 Stephano M. A., Irunde J. I., Mwasunda J. A., and Chacha C. S., A continuous time Markov chain model for the dynamics of bovine tuberculosis in humans and cattle, Ricerche di Matematica. (2022) 1–27, https://doi.org/10.1007/s11587-022-00696-3.
10.1007/s11587?022?00696?3
Web of Science® Google Scholar
16 Stephano M. A., Irunde J. I., Mayengo M. M., and Kuznetsov D., The randomness and uncertainty in dynamics of lymphatic filariasis: CTMC stochastic approach, The European Physical Journal Plus. (2024) 139, no. 2, https://doi.org/10.1140/epjp/s13360-024-04945-2.
10.1140/epjp/s13360-024-04945-2
Web of Science® Google Scholar
17 Stephano M. A., Mayengo M. M., Irunde J. I., and Kuznetsov D., Sensitivity analysis and parameters estimation for the transmission of lymphatic filariasis, Heliyon. (2023) 9, no. 9, article e20066, https://doi.org/10.1016/j.heliyon.2023.e20066, 37810166.
10.1016/j.heliyon.2023.e20066
PubMed Web of Science® Google Scholar
18 Stolk W. A., Stone C., and de Vlas S. J., Modelling lymphatic filariasis transmission and control: modelling frameworks, lessons learned and future directions, Advances in Parasitology. (2015) 87, 249–291, https://doi.org/10.1016/bs.apar.2014.12.005, 2-s2.0-84924515137.
10.1016/bs.apar.2014.12.005
PubMed Web of Science® Google Scholar
19 Lakshmikantham V., Leela S., and Martynyuk A. A., Stability Analysis of Nonlinear Systems, 1989, Springer.
Google Scholar
20 Dietz K., The estimation of the basic reproduction number for infectious diseases, Statistical Methods in Medical Research. (1993) 2, no. 1, 23–41, https://doi.org/10.1177/096228029300200103, 2-s2.0-0027736832.
10.1177/096228029300200103
CAS PubMed Google Scholar
21 Castillo-Chavez C. and Song B., Dynamical models of tuberculosis and their applications, Mathematical Biosciences & Engineering. (2004) 1, no. 2, 361–404, https://doi.org/10.3934/mbe.2004.1.361.
10.3934/mbe.2004.1.361
PubMed Web of Science® Google Scholar
22 Gomero B., Latin hypercube sampling and partial rank correlation coefficient analysis applied to an optimal control problem, [M.S. thesis], 2012, University of Tennessee.
Google Scholar
23 Marino S., Hogue I. B., Ray C. J., and Kirschner D. E., A methodology for performing global uncertainty and sensitivity analysis in systems biology, Journal of Theoretical Biology. (2008) 254, no. 1, 178–196, https://doi.org/10.1016/j.jtbi.2008.04.011, 2-s2.0-45449094504, 18572196.
10.1016/j.jtbi.2008.04.011
PubMed Web of Science® Google Scholar
24 Allen L. J., An Introduction to Stochastic Processes with Applications to Biology, 2010, CRC press, https://doi.org/10.1201/b12537.
10.1201/b12537
Google Scholar
25 Allen L. J. and LahodnyG. E.Jr., Extinction thresholds in deterministic and stochastic epidemic models, Journal of Biological Dynamics. (2012) 6, no. 2, 590–611, https://doi.org/10.1080/17513758.2012.665502, 2-s2.0-84868312013.
10.1080/17513758.2012.665502
PubMed Web of Science® Google Scholar
26 Allen L. J. and van den Driessche P., Relations between deterministic and stochastic thresholds for disease extinction in continuous-and discrete-time infectious disease models, Mathematical Biosciences. (2013) 243, no. 1, 99–108, https://doi.org/10.1016/j.mbs.2013.02.006, 2-s2.0-84875803481.
10.1016/j.mbs.2013.02.006
CAS PubMed Web of Science® Google Scholar
27 Allen L. J., An introduction to stochastic epidemic models, Mathematical Epidemiology, 2008, Springer, 81–130.
10.1007/978-3-540-78911-6_3
Google Scholar
28 Van den Driessche P. and Watmough J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences. (2002) 180, no. 1-2, 29–48, https://doi.org/10.1016/S0025-5564(02)00108-6, 2-s2.0-0036845274, 12387915.
10.1016/S0025-5564(02)00108-6
CAS PubMed Web of Science® Google Scholar
29 Lahodny G. E. and Allen L. J., Probability of a disease outbreak in stochastic multipatch epidemic models, Bulletin of Mathematical Biology. (2013) 75, no. 7, 1157–1180, https://doi.org/10.1007/s11538-013-9848-z, 2-s2.0-84879715009, 23666483.
10.1007/s11538-013-9848-z
PubMed Web of Science® Google Scholar
30 Michael E., Malecela-Lazaro M. N., and Kazura J. W., Epidemiological modelling for monitoring and evaluation of lymphatic filariasis control, Advances in Parasitology. (2007) 65, 191–237, https://doi.org/10.1016/S0065-308X(07)65003-9, 2-s2.0-36549012662, 18063097.
10.1016/S0065-308X(07)65003-9
PubMed Web of Science® Google Scholar

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The Significance of Stochastic CTMC Over Deterministic Model in Understanding the Dynamics of Lymphatic Filariasis With Asymptomatic Carriers

Abstract

1. Introduction