key: cord-0107682-edffzoq9
authors: Kiouach, Driss; El-idrissi, Salim El Azami; Sabbar, Yassine
title: A novel mathematical analysis and threshold reinforcement of a stochastic dengue epidemic model with L'evy jumps
date: 2021-05-19
journal: nan
DOI: nan
sha: a546911d2d2e7bc8c8e06711024461ddae85e0ed
doc_id: 107682
cord_uid: edffzoq9

The rampant phenomenon of overpopulation and the remarkable increase of human movements over the last decade have caused an aggressive re-emergence of dengue fever, which made it the subject of several research fields. In this regard, mathematical modeling, and notably through compartmental systems, is considered as an eminent tool to obtain a clear overview of this disease's prevalence behavior. In reality, and like all epidemics, the dengue spread phenomenon is often subject to some randomness due to the different natural environment fluctuations. For this reason, a mathematical formulation that considers suitably as much as possible the external stochasticity is indeed required. By this token, we strive in this work to present and analyze a generalized stochastic dengue model that incorporates both slight and huge environmental perturbations. More precisely, our proposed model is represented under the form of an It^o-L'evy stochastic differential equations system that we demonstrate its mathematical well-posedness and biological significance. Based on some novel analytical techniques, we prove, and under appropriate hypothetical frameworks, of course, two important asymptotic properties, namely: extinction and persistence in the mean. The theoretical findings show that the dynamics of our disturbed dengue model are mainly determined by the parameters that are narrowly related to the small perturbations' intensities and the jumps magnitudes. In the end, we give certain numerical illustrative examples to support our theoretical findings and to highlight the effect of the adopted mathematical techniques on the results.

Since ancient times, mankind has had to deal with various very dangerous epidemics which are characterized by rapid spread and high death rate [1, 2] . Often caused by some kind of bacteria or viruses unknown in their time, these epidemics killed millions of people, and thus marked the history of several countries, societies and even Humanity in general [3] . By looking a little into the past, we can find in this context many examples such as smallpox, tuberculosis, plague, cholera, typhus, the "Spanish flu" of 1918, and closer to us SARS, Ebola, Zika virus, HIV and most recently COVID-19. Seemingly, the list of all these diseases is very long, and we cannot write it all down here, but what we can guarantee is that the name of Dengue fever will undoubtedly appear in it [4] . The dengue fever is a mosquito-borne viral infection caused mainly by one of the four dengue virus stereotypes (DENV-1 to DENV-4). According to the World Health Organization (WHO) [5] , a significant number of dengue infections produce just mild illness, but many others can lead to an acute flu-like illness which later turns into a potentially fatal complication named severe dengue. With thousands of mortalities and nearly four hundred million infections annually around the globe, this disease is considered to be the deadliest vector-borne epidemic after Malaria [6] . Despite the existence of some suggested developments regarding a remedy for the dengue virus [7] , until now, no effective vaccine or treatment against it are available in the market [8] . So, early-stage detection and access to appropriate medical care still the only possible solutions at hand to face this murderous disease. At present, more than one hundred countries are under the threat of dengue fever [9] , and what makes matters worse is the ability of this epidemic to affect almost all age groups, that is why a good comprehension of its evolution dynamics is firmly required. In this vein, mathematical modeling can be presented as the most useful, efficient and applicable tool for appropriately describing the dengue fever prevalence and perceiving its effects on a host population, especially in the long term.

In order to comprehend and supervise the running of dengue infection, a considerable number of mathematical models, notably compartmental ones, have been suggested and treated in details by several works [10, 11, 12] . The first attempt to describe the dengue spread was introduced by Newton and Reiter [10] in the form of an SEIR model that does not take into consideration the mosquitoes populations. Later, and in the same context, Focks et al. [11, 12] closed this loophole by using dynamic table models to illustrate the evolution of these populations. Because of its continued re-emergence [13] , the study of the dengue's spread has not stopped at this stage, and even it remains until now an active research subject that inspires several recent papers, see [14, 15, 16, 17] and the references given there. For example, in [14] the authors constructed a deterministic dengue's propagation model and parametrized it by employing real data from the 2017 dengue outbreak in Pakistan. In [15] , Wang and Zhao studied the vaccination effect on the prevalence of dengue fever under the framework of coinfection with Zika virus. A brief discussion on the dengue modeling in both deterministic and stochastic levels is presented in [16] . The dengue dissemination dynamics with the mosquitoes control, temperature elevation and human mobility restriction are detailedly treated in [17] . In [18] , Cai et al. drew up a dengue epidemic model with bilinear saturated incidence before going to investigate the global stability of the disease-free and the endemic equilibria. The formulation of their model can be presented by the following ordinary differential equations system:

with initial conditions S(0) > 0, I(0) > 0, R(0) > 0, S(0) > 0 and I(0) > 0. Clearly, the previous system describes the simultaneous evolution of two populations, the first is the host population denoted by N (t) at time t and subdivided into three compartments of susceptible, infected, and recovered individuals, with densities indicated respectively by S(t), I(t) and R(t). The second population is the mosquitoes one which is denoted at time t by N (t) and partitioned in turn into two classes of susceptible and infected individuals with densities S(t) and I(t) respectively. The recovered mosquitoes class is not considered in this model because of the short lifespan of these insects and the relatively long time required to recover from the dengue disease [18] . The eleven parameters appearing in system (1.1) are summarized in the following list:

• Λ andΛ are the susceptible humans and mosquitoes recruitment rates respectively.

• β is the transmission rate from mosquitoes to humans.

• a is a parameter measuring the inhibitory effect due to psychological and behavioral changes in the susceptible human population.

• ρ 0 is the disease-induced death rate for human individuals.

• r 1 and r 2 are respectively the human individuals recovery and treatment rates. For convenience in writing we denote from now on r 1 + r 2 by ρ 1 .

•β is the transmission rate from humans to mosquitoes.

• µ andμ are, in this order, the humans and mosquitoes natural death rates.

All parameters listed above are assumed to be in the positive real axis R + := {x ∈ R | x > 0} except a that is supposed to be just nonnegative. Since R(t) does not appear in the other equations of system (1.1), the dynamical behavior of dengue infection can be deduced just from the following simplified model:

According to the mathematical analyses effected in [18] , the spread dynamics of the aforementioned dengue model is completely determined by the basic reproduction number which is expressed in this case by

More precisely, if R 0 1, the system (1.2) admits a unique disease-free equilibrium E o = Λ µ , 0, 0, 0 , and it is globally asymptotically stable in the invariant region D = S, I, S, I ∈ R + | S + I Λ µ and S + I Λ µ . On the other hand if R 0 > 1, the disease-free equilibrium is still present, but it becomes unstable. In this case, the system (1.2) admits another steady state E ⋆ = S ⋆ , I ⋆ , S ⋆ , I ⋆ called the endemic equilibrium, and it is globally asymptotically stable in the interior of D. The endemic equilibrium components that we have just mentioned are given as follows:

µ(µa+bβ)(µ+ρ0+ρ1)+b 2 ββΛ . It is well known that environmental uncertainties influence the spread of any disease and make the divination of its behavior more difficult [19] . In such a situation deterministic models, and despite their ability in providing highly informative results and predictions, are not suitable enough. Therefore, a new or modified mathematical formulation that takes into account this effect of randomness is really required, especially in the framework of dengue fever dissemination analysis. In this regard, several authors have proposed and developed many stochastic models that are presenting the dengue infection dynamics from different viewpoints and perspectives. For example, in [20] Otero and Solari investigated a probabilistic model depicting the evolution of dengue fever within a spatially fixed humans population, that is to say that mosquitoes dispersal is the only one responsible for the disease's spreading. In order to highlight the role of human mobility on dengue prevalence, Barmak et al. [21] treated a perturbed model incorporating the infected mosquitoes and humans dynamics with different human mobility patterns. In [22] , Liu at al. took into account the randomness's effect on the dengue fever pervasion by assuming that the solution of the model (1.2) fluctuates normally around its value. Following this point of view, they explored a disturbed version of the dengue epidemic model (1.2), in which the environmental perturbations are in the form of proportional whites noises to the variables. Hence, the stochastic dengue model that they studied is presented as follows:

where, the nonnegative constants σ i (i = 1, 2, 3, 4) denote the intensities of the mutually independent Brownian motions B i (i = 1, 2, 3, 4). These latter, and even all the stochastic processes or random variables that will be met in this paper, are supposed to be defined on a probability space (Ω, F , P) equipped with a filtration (F t ) t 0 that satisfies the usual conditions (it is increasing and right continuous while F 0 contains all P-null sets). The insertion of white noises is a very reasonable and prominent approach to model phenomena in which a given quantity is constantly subjected to slight variable fluctuations, for example, those of the environment during a certain disease's spread [23, 24, 25] . But unfortunately, this method is neither appropriate nor sufficient to represent the effect of strong and sudden external disturbances like climate changes, floods, earthquakes, tornadoes, etc [26, 27] . For this reason, we will make recourse to the well known Lévy processes which are able to formulate adequately this kind of cases. By taking into account this type of random perturbations, we can extend the model (1.3) to the following system of stochastic differential equations with Lévy jumps (SDELJ for brevity):

(1.4)

Here and thereafter, S(t − ), I(t − ), S(t − ) and I(t − ) are respectively standing for the left limits of S(t), I(t), S(t) and I(t). N is a Poisson counting measure independent of B i (i = 1, 2, 3, 4) with compensating martingale N and finite characteristic measure ν that is defined on a measurable set U ⊂ R + . Also, it is assumed that ν is a Lévy measure such that N (dt, du) = N (dt, du) − ν(du)dt and we suppose that the jumps intensities ξ i : U → R are continuous functions on U. In order to provide an additional degree of realism to our study, we will suppose that the contact between humans and mosquitoes is homogeneously mixed. In other words, and in analogy with a chemical reaction, we will adopt the mass action rates βS I and β SI as the incidence functions of our epidemic model. The last fact will lead us to the following system which is none other than system (1.4) but in the case of the inhibitory effect absence:

For the reader's convenience, we summarize and illustrate the transmission mechanisms of the aforementioned dengue model with the help of the flowchart depicted in figure 1.

Our principal objective in this manuscript is to explore sufficient conditions for extinction and persistence in the mean of the dengue model (1.5). These two asymptotic properties are considered to be sufficient for having an excellent idea of the future dengue pandemic situation. The originality of our work lies essentially in the method that we adopt to estimate the limit values of the temporal averages

where Ψ and Ψ are respectively the positive solutions of the following systems:

(1.7)

Our method permits us to bridge the gap left by the use of classical approaches presented for example in [28, 29] . Also, we use a new and non-standard analytical technique to obtain a sharper threshold for the extinction case. The performed analysis in this paper seems to be very encouraging to study other related epidemic models and especially those which are perturbed with Lévy noises. The remainder of this article is organized as follows: in Section 2, we demonstrate the existence and uniqueness of a positive global-in-time solution to the stochastic dengue model (1.5). In Section 3, we provide some sufficient conditions for the dengue disease extinction, whereas those of its persistence in the mean are presented in Section 4. In Section 5, we support our theoretical results with the help of some numerical simulations before drawing the main conclusions of the article in Section 6.

The first step in exploring the dynamical characteristics of a mathematical population system is to know if it is well-posed or not, where the well-posedness here designates that the system admits a unique and positive global-in-time solution. In what follows, we will provide some conditions and assumptions under which the well-posedness of the dengue disease model (1.5) is ensured. But before doing so, let us first introduce the following hypotheses: Proof. As supposed in the statement of the theorem, let the hypotheses (A 1 ) and (A 2 ) hold. From (A 1 ), we can easily observe that the coefficients of the system (1.5) are locally Lipschitz continuous. Hence, by using Theorem 1.19 of [30] , one can immediately conclude that for any given initial value S(0),

where τ e is the explosion time [31] . At this point, our objective will be to show the globality in time of this solution, in other words, τ e = ∞ almost surely.

To this end, let k 0 be a sufficiently large positive integer such that S(0), I(0), S(0), I(0) ∈ 1 k 0 , k 0 , and consider for any integer k k 0 the following quantity, which is well defined by the adoption of the convention inf ∅ = ∞:

is an open subset of R 4 , so it follows from Theorem 3.1 of [31] that τ k is a stopping time for all τ k is a stopping time, then so is τ ∞ . Evidently, τ ∞ τ e (see [33] for more details), so τ e = ∞ will follow immediately once we show that τ ∞ = ∞ a.s., and this is exactly what we are going to do for accomplishing our proof. Suppose that τ ∞ = ∞ a.s. is false, then there is necessarily two positive constants ε and η such that

where α is a positive constant to be determined suitably later. The nonnegativity of this function can be observed from the inequality a − 1 − ln(a) 0, ∀a > 0. According to the general multi-dimensional Itô's formula see [30, page 8] , we have for all

and A here is to the positive constant given by

The remainder of the proof runs on the same lines as the demonstration of Theorem 2.1 in [34] , so we omit it here for the sake of space.

In mathematical epidemiology, our prime concern after proving the well-posedness is to know if the disease will disappear or it will continue to exist. In this section, we will do our utmost to find some conditions for the dengue disease extinction expressed in terms of noises intensities, jumps coefficients and system parameters. For the reader's convenience, the persistence in the mean will be covered and analyzed separately in the next section. For the sake of brevity, we will adopt from now on the following notations:

• θ p (u) := max θ p (u), θ p (u) .

• ρ := ρ 0 + ρ 1 .

Before stating the main result of this section, we must firstly give the following useful lemma: Lemma 3.1. Let S(t), I(t), S(t), I(t) be the solution of (1.5) that starts from a given initial value S 0 , I 0 , S 0 , I 0 ∈ R 4 + . Let also Ψ(t) and Ψ(t) be respectively the solutions of (1.6) and (1.7) that satisfy Ψ(0) = S(0) and Ψ(0) = S(0). If the following conditions are satisfied: Proof. The proof of this lemma is similar in spirit to that of Lemma 2.5 in [27] . Hereby, it is omitted here. 

Proof. Let Ψ(0) be a positive initial datum, by integrating the both sides of equation (1.6) from 0 to t, and then dividing by t, we get

Letting t go to infinity in the last equality and then using Lemma 3.1 yields

Hence, the item (b) of the lemma is proved. We are now in a position to show the statements (a) and (c). By applying the generalized Itô's formula see [30] to Ψ 2 we obtain

Integrating both sides of (3.2) from 0 to t and then dividing by t gives

Clearly, the constant Υ := 2µ − σ 2 1 − U ξ 2 1 (u)ν(du) is not zero, since if it is not the case, we will tend t to infinity and come across the contradictory equality 2Λ 2 µ = 0. For this reason, one can easily divide the both sides of (3.3) by Υ and deduce that

Using Lemma 3.1 together with (3.1) implies that

and by observing the positivity of t 0 Ψ 2 (s) ds t , we can conclude at the same time that Υ > 0. Hence, the claims (a) and (c) of the lemma are proved. By an analogous argument, the assertions (c), (d) and (e) can be drawn, and this finishes the proof. [35] ), we can easily assert ,and for almost all w ∈ Ω, that    S(t, w) Ψ(t, w)

S(t, w) Ψ(t, w) for all t > 0. (3.4)

In the lévy jumps case, the previous lemma can be seen as an alternative method to overcome the inexistence of an explicit expression for the stationary distribution of (1 .6) and (1.7) . This problem remains an open question until now, and we can find in the literature several works (see for example [28] and [29] ) that present the threshold analysis of their epidemiological model with a formulation incorporating an unknown stationary distribution. 

It is easily seen that the stochastic exponential extinction implies the ordinary stochastical one (see [36] ), but the converse is not true in general.

In order to simplify the writing of the next theorem, we introduce these conventions:

• For all x ∈ R, x + := max{0, x} = x + |x| 2 this function is commonly known as the ramp function, see [37] .

. 

Then lim sup

In particular, if the condition κ < 0 is verified, then the disease will die out exponentially almost surely.

Proof. First of all, let us define a C 2 function W :

Applying the Itô's formula to W I(t), I(t) shows that for all t 0 we have

By virtue of the well-known Cauchy-Schwartz inequality (see for example [38] ), we have

At the same time, it follows from the monotonicity of the function a → ln(1 + a) − a on (−1, +∞) it is increasing over (−1, 0) and decreasing over [0, +∞) that U ln 1 + λ 1 ξ 2 (u)I(t) + λ 2 ξ 4 (u) I(t)

Combining (3.6), (3.7) and (3.4) with (3.5) yields

Hence, we obtain

Integrating the last inequality from 0 to t, and then dividing by t on both sides gives

.

(3.9)

On the other hand, we can see by employing the classical Hölder's inequality that

This last fact together with Lemma 3.2 implies that

By a similar argument, we can also assert that

a.s. (3.11) It is fairly easy to see that M 1 (t) is a local martingale with finite quadratic variation, and from the hypothesis (A 5 ) we can affirm that M 2 (t) will be also so. Therefore, we conclude by the strong law of large numbers for local martingales that lim t→∞ M 1 (t) t and lim t→∞ M 2 (t) t a.s. (3.12) Taking the superior limit on both sides of (3.9) and combining the resulting inequality with (3.10), (3.11) and (3.12) lead us to

which is exactly the desired conclusion. In addition, it goes without saying that if κ < 0 then the disease will die out exponentially almost surely. Thus, the theorem is proved.

Compared to several existing works (see for instance [22, 39, 40, 41, 42, 43] ), the statement of the last theorem is indeed stronger because it offers a sharper threshold that weakens the disease extinction condition. The precision of our threshold κ comes back essentially to inequality (3.8) in the previous proof where we used the ramp function (x → x + ) instead of the absolute value always adopted in the literature to our best knowledge.

After having studied the extinction of the dengue fever, we turn now to explore its persistence in the mean, but before doing so, let us first recall the definition of this notion. For the sake of greater clarity and readability, we use from now on these notations: Proof. Let us consider the function W defined by

where θ 2 = 1 and θ 1 , θ 3 , θ 4 are three positive constants to be chosen suitably later. From Itô's formula, we have and for all t 0

− θ 1 σ 1 dB 1 (t) − σ 2 dB 2 (t) − θ 3 σ 3 dB 3 (t) − θ 4 σ 4 dB 4 (t) − θ 1 U ln 1 + ξ 1 (u) N (dt, du) − U ln 1 + ξ 2 (u) N (dt, du) − θ 3 U ln 1 + ξ 3 (u) N (dt, du) − θ 4 U ln 1 + ξ 4 (u) N (dt, du),

After some simplifications, we get

From the relation between geometric and arithmetic means (the first is less than or equal to the second), it follows that LW S(t), I(t), S(t), I(t) −4 4 θ 1 θ 3 θ 4 b 2 ββΛΛ + θ 1 bβ I(t) + θ 3 bβI(t)

.

, we obtain:

θ 1 bβ I(t) + θ 3 bβI(t) + M 2 − b 2 ββΛΛ M 1 M 3 M 4 = b θ 1 β I(t) + θ 3β I(t) − M 2 R0 b 2 ββΛΛ M 1 M 2 M 3 M 4 −1 .

Integrating (4.1) from 0 to t, and then dividing by t on both sides, we get

Since ln(a) a − 1 a for any a > 0, one can conclude that

θ i x i for all x ∈ R 4 + . Combining this fact with (4.2) leads to

. (4.3) Needless to say, M 1 (t) is a local martingale with finite quadratic variation, and from the assumption (A 5 ) we can assert that M 2 (t) is also so. Therefore, we deduce by the strong law of large numbers for local martingales that Taking the inferior limit on both sides of (4.3) and combining the resulting inequality with (4.4) and (4.5) yields lim inf

So if R 0 > 1, then the disease will persist in the mean as claimed, which completes the proof.

In this section, and by taking the parameter values from the theoretical data presented in Table 1 , we set forth some numerical simulations to belay the various results proved in this paper. The solution of our Dengue model, is simulated in our case with the initial condition given by S(0) = 0. 

In this case, and by selecting the parameter values appearing in the third column of Table 1 , and also taking U = R + with ν(U) = 1, we will shed some light on the theoretical results of Section 3. By a few simple calculations, it can be verified that we have the numerical values presented in Table 2 . From the latter, we observe easily that the condition κ < 0, and hypothesis (A 1 ), (A 2 ), (A 3 ) and (A 4 ) hold. So, and by the virtue of Theorem 3.1 the dengue epidemic dies out exponentially almost surely, which is exactly depicted in Figure 2 .

Expression Value The latter, and in the case of the numerical values taken in Figure 2, Table  1 ( R 0 = 1.0862 > 1).

Dengue fever is an arthropod-borne viral epidemic conveyed to humans through the bite of an infected mosquito with one of the dengue virus's serotypes which all belong to the Flaviviridae viruses family [8] . The dangerousness of this disease lies mainly in its ability to affect almost all age groups, ranging from infants to adults, and in its capacity to re-emerge rapidly in almost every human host population. The absence of an approved treatment or effective vaccine against dengue makes a good comprehension of its prevalence dynamics the only remaining solution to mitigate its severity. In this context, our work presents a mathematical compartmental model that describes the dengue disease dissemination under environmental small disturbances and unexpected massive external perturbations. More explicitly, we have formulated the dengue spread mechanisms by using an SIR-SI stochastic differential equations system that includes both proportional white noises and Lévy jumps. After having drawn up our model, a rigorous mathematical analysis is performed to get an insight into the dengue fever propagation behavior, especially in the long-term. The principal mathematical and epidemiological findings of our paper are listed as follows:

• We have demonstrated the existence and uniqueness of a positive global-in-time solution to our proposed dengue model.

• We have provided some sufficient conditions for the dengue fever extinction.

• An appropriate hypothetical framework for the persistence in the mean of the dengue disease is also established.

Compared to the existing works, the originality of our article resides in the following mathematical techniques and amelioration that we have used to accomplish our analysis:

• Our study adopted an alternative approach to estimate the time averages t 0 Ψ(s) ds t , t 0 Ψ 2 (s) ds t , t 0 Ψ(s) ds t and t 0 Ψ 2 (s) ds t , without resorting to the explicit stationary distribution's formula which still unknown until now (see [28, 29] ).

• In our analysis, we have used the inequality (3.7) instead of ln(x + 1) x for all x > −1, adopted usually in the literature (see for example [44] ), which allowed us to obtain a sharper threshold for the dengue extinction case.

• Our paper is distinguished from several previous works [22, 39, 40, 41, 42, 43] by the use of the ramp function in place of the absolute value in inequality (3.8) , which remarkably improved the extinction threshold and hence strengthened the statement of Theorem 3.1.

Epidemics and pandemics: their impacts on human history

Murderous contagion:a human history of disease

Epidemics and society:from the black death to the present

Dengue and severe dengue, WHO official website

The global distribution and burden of dengue

Dengue vaccine research, WHO official website

Dengue infection modeling and its optimal control analysis in east Java

Refining the global spatial limits of dengue virus transmission by evidence-based consensus

A model of the transmission of dengue fever with an evaluation of the impact of ultra-low volume (ulv) insecticide applications on dengue epidemics

Dynamic life table model for aedes aegypti (diptera: Culicidae): analysis of the literature and model development

Dynamic life table model for aedes aegypti (diptera: Culicidae): simulation results and validation

Dengue: the continual re-emergence of a centuries-old disease

Optimal control strategies for dengue transmission in Pakistan

Dynamics analysis of a Zika-dengue co-infection model with dengue vaccine and antibodydependent enhancement

Comparison of stochastic and deterministic frameworks in dengue modelling

Effects of human mobility, temperature and mosquito control on the spatiotemporal transmission of dengue

Global dynamics of a dengue epidemic mathematical model

Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model

Stochastic eco-epidemiological model of dengue disease transmission by aedes aegypti mosquito

Modelling dengue epidemic spreading with human mobility

Stationary distribution and extinction of a stochastic dengue epidemic model

The threshold of a deterministic and a stochastic SIQS epidemic model with varying total population size

Stability of delay evolution equations with fading stochastic perturbations

The long-time behaviour of a stochastic SIR epidemic model with distributed delay and multidimensional Lévy jumps

Threshold of a stochastic SIR epidemic model with Lévy jumps

New results on the asymptotic behavior of an SIS epidemiological model with quarantine strategy, stochastic transmission, and Lévy disturbance

Sharp conditions for the existence of a stationary distribution in one classical stochastic chemostat

Stochastic dynamics of the delayed chemostat with Lévy noises

Stochastic differential equations and applications

Brownian Motion and Stochastic Calculus

Advanced and comprehensive research on the dynamics of covid-19 under mass communication outlets intervention and quarantine strategy: a deterministic and probabilistic approach

Analysis of a predator-prey model with Lévy jumps

Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations

Threshold behaviour of a stochastic SIR model

Advanced topics in applied mathematics: for engineering and the physical sciences

A new generalization on cauchy-schwarz inequality

Dynamics of a stochastic SIR epidemic model with distributed delay and degenerate diffusion

Dynamical behavior of a higher order stochastically perturbed SIRI epidemic model with relapse and media coverage

Stationary distribution and extinction of a stochastic model of syphilis transmission in an MSM population with telegraph noises

Dynamical behavior of a higher order stochastically perturbed hiv/aids model with differential infectivity and amelioration

Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation

A dynamics stochastic model with hiv infection of CD4+ t-cells driven by Lévy noise

Roughly speaking, our theoretical results show that the extinction and persistence conditions depend mainly on the white noise intensities, Lévy jumps magnitudes, and the system parameters of course. In order to elucidate the theoretical results and exhibit the effect of replacing the absolute value by the ramp function in the inequality (3.8), we have presented some numerical simulation examples. In the end, we point out that the obtained results generalize several previous works (for instance, [18] and [22] ), and improve our understanding of the dengue's spreading comportment, which makes this work a good basis for future studies, especially with the continuous reappearance of the dengue fever disease in many regions around the globe.

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

The theoretical data used to support the findings of this study are already included in the article.

The authors declare that the study was conducted in collaboration with the same responsibility. All authors read and approved the final manuscript.

On behalf of all authors, the corresponding author states that there is no conflict of interest.

In this subsection, we will turn to the dengue's persistence in the mean case. By keeping in mind the numerical values appearing in the last column of Table 1, we can easily draw up the following list:   Quantity Expression Value On the basis of this last table's numerical values, the assumptions (A 1 ) − (A 5 ) hold, and the quantity R 0 outnumbers one. So, and by Theorem 4.1, the dengue fever is persistent in the mean, which agrees well with the pictorial curves presented in Figure 3 . Remark 5.2. If we kept the same approach as [22, 39, 40, 41, 42, 43] in our treatment, the threshold that we will obtain in Theorem 3.1 will be of the following form K := D − C − B +μ × R 0 2µ Υ − 1 + (µ + ρ) × R 0 2μ Υ − 1 .