A novel mathematical analysis and threshold reinforcement of a stochastic dengue epidemic model with Lévy jumps

Kiouach, Driss; El-idrissi, Salim El Azami; Sabbar, Yassine

doi:10.1016/j.cnsns.2023.107092

Cited by 7 publications

(6 citation statements)

References 34 publications

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“…that is imposed in Theorem 3.1 to obtain the persistence in the mean of model (1.3) includes many incidence functional responses adopted usually in the literature. For example, -The bilinear incidence rate [13]:…”

Section: S(t) V(t) I(t) S(t) V(t)mentioning

confidence: 99%

“…Remark Besides its interesting mathematical interpretation, the importance of the last theorem lies basically in its ability to make model () rationally and biologically meaningful [28]. …”

Section: Existence and Unicity Of The Positive Global‐in‐time Solutionmentioning

confidence: 99%

“…Remark Condition

\left({\mathbf{H}}_{\mathbf{0}}&amp;amp;amp;#x0005E;{\prime}\right)

that is imposed in Theorem 3.1 to obtain the persistence in the mean of model () includes many incidence functional responses adopted usually in the literature. For example, The bilinear incidence rate [13]:

I{F}_1\left(S,I\right)&amp;amp;amp;#x0003D;{\beta}_1 SI

and

I{F}_2\left(V,I\right)&amp;amp;amp;#x0003D;{\beta}_2 VI

. The Holling type II incidence rate [32]:

I{F}_1\left(S,I\right)&amp;amp;amp;#x0003D;\frac{\beta_1 SI}{m_1&amp;amp;amp;#x0002B;S}\kern0.20em \left({m}_1&amp;amp;gt;0\right)

and

I{F}_2\left(V,I\right)&amp;amp;amp;#x0003D;\frac{\beta_2 VI}{m_2&amp;amp;amp;#x0002B;V}\kern0.20em \left({m}_2&amp;amp;gt;0\right)

. The Holling type III incidence rate [33]:

I F_{1} false(S, I false) = \frac{β_{1} S^{2} I}{((), m_{1} + S) ((), m_{1}^{})}

…”

Section: Extinction Persistence In the Mean And Weak Persistencementioning

confidence: 99%

“…The bilinear incidence rate [13]:

I{F}_1\left(S,I\right)&amp;amp;amp;#x0003D;{\beta}_1 SI

and

I{F}_2\left(V,I\right)&amp;amp;amp;#x0003D;{\beta}_2 VI

.…”

Section: Extinction Persistence In the Mean And Weak Persistencementioning

confidence: 99%

“…One of the mostly important of these latter is the well-known SVIR model which is none other than the usual SIR but with a consideration of the vaccination strategy, the reason that makes it very appropriate to analyze many diseases such as tuberculosis, avian influenza, diphtheria, polio, and pertussis [11,12]. However, the spread of any infectious disease is naturally influenced by environmental disturbances, and this makes it surely more difficult to predict its behavior [13,14]. In such cases, the introduction of the random effect into the dynamical model and the use of mathematical formulations based on stochastic differential equations is very convenient, the thing that has motivated several researchers in recent years to adopt this vision for many epidemic models including the SVIR one.…”

Section: Introduction and Model Formulationmentioning

confidence: 99%

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The impact of Lévy noise on the threshold dynamics of a stochastic susceptible‐vaccinated‐infected‐recovered epidemic model with general incidence functions

Kiouach,

Azami El‐idrissi,

Sabbar

2023

Math Methods in App Sciences

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Compartmental systems like the well‐famed SIR, SEIR, SQIR, SVIR, and their variants are efficient tools for the mathematical modeling of infectious illnesses, and they permit us to get a clear picture of how they proliferate. In actuality, the aleatory fluctuations factors present in the natural environment like storm surges, weather changes, and seismic tremors make the dissemination of epidemics susceptible to some randomness. This calls for a stronger mathematical formulation that takes into consideration this stochasticity effect. From this perspective, and in order to highlight in the same time the effect of the vaccination strategy, we survey in this paper an SVIR model with general incidence rates that is disturbed by both Brownian motions and Lévy jumps. Initially, we establish its well‐posedness in the sense that it has a unique positive and global‐in‐time solution. Then, we rely on some assumptions and nonstandard analytic techniques, to derive sufficient and almost necessary conditions for extinction, persistence in the mean, and also weak persistence. More explicitly, we identify firstly under some hypotheses a threshold between extinction and persistence in the mean. In other phrases, if , the infected population dies out while it persists in the mean when . Then, and by modifying the hypothetical framework in order to cover more incidence rates, we prove that can act also as a threshold between extinction and weak persistence. At last, we provide some numerical simulations to corroborate our findings and cover some particular cases of response functions.

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Section: S(t) V(t) I(t) S(t) V(t)mentioning

confidence: 99%

“…Remark Besides its interesting mathematical interpretation, the importance of the last theorem lies basically in its ability to make model () rationally and biologically meaningful [28]. …”

Section: Existence and Unicity Of The Positive Global‐in‐time Solutionmentioning

confidence: 99%

“…Remark Condition