Modeling Highly Random Dynamical Infectious Systems

Wanduku, Divine

doi:10.1007/978-3-319-99918-0_17

Cited by 5 publications

(16 citation statements)

References 37 publications

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“…Applying standard techniques in [31,33,35], it is proven that there exists a unique positive SSP fYðtÞ, t ! t 0 g that satisfies (4.3)-(4.9).…”

Section: Model Validation Resultsmentioning

confidence: 99%

“…Proof. The proof of the existence and uniqueness of a positive solution for the system (4.3)-(4.9) follows the similar technique in [31,33,35], and is left out to maximize space. In the following the existence of the positive self-invariant space B R nþ6 ð0, rÞ is shown.…”

Section: Model Validation Resultsmentioning

confidence: 99%

“…The properties of H j are given in Assumption 2.1. Using ideas in [32,33,35] we adopt assumptions for the nonlinear function H j , j 2 Ið0, nÞ:…”

Section: Description Of Modelmentioning

confidence: 99%

“…That is, the proportions per unit time e j ¼ 1 À e j and e j , j 2 Ið0, nÞ are random variables, and likewise the natural death rates l k , k 2 S j , I, T, A, R È É , j 2 Ið0, nÞ are random variables per unit time. Denote these random variables, respectively, by e j ,ẽ j , j 2 Ið0, nÞ and l k , k 2 S j , I, T, A, R È É , j 2 Ið0, nÞ: We represent the environmental variabilities as independent white noise processes applying similar techniques in the earlier studies [33,35].…”

Section: Derivation Of the Stochastic Hiv/aids Modelmentioning

confidence: 99%

“…To avoid repetition, the reader is referred to Wanduku [35] [Pp. 520, Equation 22], where it is shown that for any state k 2 S j , I, T, A, R È É , j 2 Ið0, nÞ, even when the natural death rates are influenced by random fluctuations of white noise types exhibited in (B.1) and (B.3), it follows that under the assumption of independent natural deaths in any interval ½t 0 , t 0 þ t of length t, the number of deaths that occur in state k 2 S j , I, T, A, R È É , j 2 Ið0, nÞ are best described by independent Poisson processes with means Eðl k Þ ¼ l k , wherel k and l k are defined in (4.1).…”

Section: Notesmentioning

confidence: 99%

See 4 more Smart Citations

On the almost sure convergence of a stochastic process in a class of nonlinear multi-population behavioral models for HIV/AIDS with delayed ART treatment

Wanduku

2020

Stochastic Analysis and Applications

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The success in reducing global HIV prevalence rates is attributed to control measures like information and education campaigns (IECs), antiretroviral therapy (ART), and national, multinational and multilateral support providing official developmental assistance (ODAs) to combat HIV. A class of stochastic nonlinear multi-population behavioral HIV/AIDS models is studied, where behavioral change is inspired by the IECs safe sex education. Exponential almost sure stability analysis of the model is conducted, and the results are employed to determine the impacts of the supply of ODAs, IECs, early treatment and poverty rates on the epidemic. The behavioral change and the noise induced basic reproduction numbers are obtained.

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“…Applying standard techniques in [31,33,35], it is proven that there exists a unique positive SSP fYðtÞ, t ! t 0 g that satisfies (4.3)-(4.9).…”

Section: Model Validation Resultsmentioning

confidence: 99%

Section: Model Validation Resultsmentioning

confidence: 99%

“…The properties of H j are given in Assumption 2.1. Using ideas in [32,33,35] we adopt assumptions for the nonlinear function H j , j 2 Ið0, nÞ:…”

Section: Description Of Modelmentioning

confidence: 99%

Section: Derivation Of the Stochastic Hiv/aids Modelmentioning

confidence: 99%

Section: Notesmentioning

confidence: 99%

See 3 more Smart Citations

On the almost sure convergence of a stochastic process in a class of nonlinear multi-population behavioral models for HIV/AIDS with delayed ART treatment

Wanduku

2020

Stochastic Analysis and Applications

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The stationary distribution and stochastic persistence for a class of disease models: Case study of malaria

Wanduku

2020

Int. J. Biomath.

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This paper presents a nonlinear family of stochastic SEIRS models for diseases such as malaria in a highly random environment with noises from the disease transmission and natural death rates, and also from the random delays of the incubation and immunity periods. Improved analytical methods and local martingale characterizations are applied to find conditions for the disease to persist near an endemic steady state, and also for the disease to remain permanently in the system over time. Moreover, the ergodic stationary distribution for the stochastic process describing the disease dynamics is defined, and the statistical characteristics of the distribution are given numerically. The results of this study show that the disease will persist and become permanent in the system, regardless of (1) whether the noises are from the disease transmission rate and/or from the natural death rates or (2) whether the delays in the system are constant or random for individuals in the system. Furthermore, it is shown that “weak” noise is associated with the existence of an endemic stationary distribution for the disease, while “strong” noise is associated with extinction of the population over time. Numerical simulation examples for Plasmodium vivax malaria are given.

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Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays

Wanduku¹

2022

DCDS-S

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A multipopulation HIV/AIDS deterministic epidemic model is studied. The population structure is a multihuman behavioral structure composed of humans practicing varieties of distinct HIV/AIDS preventive measures learnt from information and education campaigns (IEC) in the community. Antiretroviral therapy (ART) treatment is considered, and the delay from HIV exposure until the onset of ART is considered. The effects of national and multilateral support providing official developmental assistance (ODAs) to combat HIV are represented. A separate dynamics for the IEC information density in the community is derived. The epidemic model is a system of differential equations with random delays. The basic reproduction number (BRN) for the dynamics is obtained, and stability analysis of the system is conducted, whereby other disease control conditions are obtained in a multi-and a finite dimensional phase space. Numerical simulation results are given.

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Modeling Highly Random Dynamical Infectious Systems

Cited by 5 publications

References 37 publications

On the almost sure convergence of a stochastic process in a class of nonlinear multi-population behavioral models for HIV/AIDS with delayed ART treatment

On the almost sure convergence of a stochastic process in a class of nonlinear multi-population behavioral models for HIV/AIDS with delayed ART treatment

The stationary distribution and stochastic persistence for a class of disease models: Case study of malaria

Finite- and multi-dimensional state representations and some fundamental asymptotic properties of a family of nonlinear multi-population models for HIV/AIDS with ART treatment and distributed delays

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