key: cord-1012658-aq85hqj7 authors: Sabbar, Yassine; Kiouach, Driss; Rajasekar, S.P.; El-idrissi, Salim El Azami title: The influence of quadratic Lévy noise on the dynamic of an SIC contagious illness model: New framework, critical comparison and an application to COVID-19 (SARS-CoV-2) case date: 2022-04-25 journal: Chaos Solitons Fractals DOI: 10.1016/j.chaos.2022.112110 sha: eeae2c0935223bf82b86359f00c6bdaf570a0dd0 doc_id: 1012658 cord_uid: aq85hqj7 This study concentrates on the analysis of a stochastic SIC epidemic system with an enhanced and general perturbation. Given the intricacy of some impulses caused by external disturbances, we integrate the quadratic Lévy noise into our model. We assort the long-run behavior of a perturbed SIC epidemic model presented in the form of a system of stochastic differential equations driven by second-order jumps. By ameliorating the hypotheses and using some new analytical techniques, we find the exact threshold value between extinction and ergodicity (persistence) of our system. The idea and analysis used in this paper generalize the work of N. T. Dieu et al. (2020), and offer an innovative approach to dealing with other random population models. Comparing our results with those of previous studies reveals that quadratic jump-diffusion has no impact on the threshold value, but it remarkably influences the dynamics of the infection and may worsen the pandemic situation. In order to illustrate this comparison and confirm our analysis, we perform numerical simulations with some real data of COVID-19 in Morocco. Furthermore, we arrive at the following results: (i) the time average of the different classes depends on the intensity of the noise (ii) the quadratic noise has a negative effect on disease duration (iii) the stationary density function of the population abruptly changes its shape at some values of the noise intensity. Mathematics Subject Classification 2020: 34A26; 34A12; 92D30; 37C10; 60H30; 60H10. Infectious illnesses are health disruptions caused by tiny components like viruses, germs, bacteria, fungi, or parasites. These microorganisms live in and on our bodies and propagate through many ways of dissemination such as direct physical contact (horizontal spread), mother to baby (vertical transfer), airborne particles, water, food, animal, or insect bite [1] . Minor infections may respond the rest and home treatments, while life-threatening diseases may require hospitalization. In late 2019, a deadly disease (Coronavirus disease 2019, abbreviated as COVID- 19) obstructed ordinary lifestyles [2] and highly strained the healthcare professionals and medical structures [3] . Biologically, the great contagiousness and quick spread of this epidemic following its insertion into the host population is due to the shortage of pre-existing immunity against the virus [4] . This latter is transmitted between humans by direct contact with frequently touched objects and surfaces or by small water droplets produced by the exhalation of infected individuals [5] . Respiratory problems, change in the sense of smell or taste, fever, and dry cough represent the principal symptoms of the disease [6] . In order to partially prevent and control the spread of this epidemic, the government and its authorities have ordered everyone to wear a mask or muzzle, clean hands frequently and take measures such as general curfews, isolation and vaccination [7, 8] . In mathematical biology, dynamical systems have long been employed for analyzing and understanding the behavior of diseases in the population and examining the impact of intervention strategies [9] . The SIC (Susceptibles-Infected-Constantly recovered) model is one of the most substantial systems in epidemiological patterns and malady control which was originally proposed and treated by Kermack and McKendrick [1] in 1927 . From then on, various formulations of the SIC model with different factors have been investigated by many researchers due to their theoretical and functional value [10, 11] . In this model, the overall population is typically divided where the positive constants  , d  ,   ,   and  are respectively the insertion or inflow average into the population, the normal death rate, the mortality rate due to the disease, the transmission rate, and the recovery rate of the infected individuals. For the simplicity of notation, we define = d S   . The model (1.1) and its general shapes were extremely applied in the COVID-19 case and several authors demonstrated that the first phase of COVID-19 (March -May 2020) followed the SIC dynamic. For easier reference, let us quote the following works: • In [12] , Prodanov presented some novel analytical findings and numerical algorithms for parametric estimation of the SIC model (1.1) with COVID-19 data. • In [13] , the authors proposed a two-parameters SIC model and offered a general analytical solution of the model with an application to COVID-19 case. • In [14] , the authors analyzed an SIC epidemic model for COVID-19 spread with fuzzy parameters and presented an application to the case of Indonesia. The deterministic model (1.1) offers a systematic way to investigate transmission dynamics and produce long-term predictions [9] . Based on the expression 0 = ( ) to distinguish between the suppression and the continuation of the illness. More interesting results on the SIC model can be found in [15] where the stability properties of equilibrium are studied. It is worth noting that model (1.1) can be improved by taking into account some realistic assumptions such as randomness. We introduce this idea in the next subsection. predicting its long-run [16] . The stochastic approach highlights many hypotheses and preoccupations by investigating the dynamics of a system. For this reason, multiplicative and additive noise sources carry out a significant role in the transient dynamics of biological and physical systems [17] . Technically, if the noise level is abnormally high, the signal can be drowned out, the similar logic for biological systems where noises help reduce infection. Regarding the physical grasp of biological models, the above two types of random noise have been extensively greatly used [18, 19, 20, 21] . Clearly, additive noise is characterized by its proactive role in the transient dynamics of dynamical systems, and multiplicative noise is accountable for noise-induced transitions [22] . In this context, Spagnolo et al. [23] studied the dynamics of an ecosystem with multiplicative noise and a stochastic interference between the species. They showed that noise plays a pivotal role in population dynamics and its presence is responsible for the generation of quasi-deterministic temporal oscillations. Furthermore, they demonstrated that noise affects the extinction of species, which confirms the role of including stochasticity. By considering white and colored noise sources, Guarcello et al. [24] analyzed the phase dynamics in ballistic graphene-based short Josephson junctions. They explored the effects of thermal and correlated fluctuations on the escape time from these metastable states in the case of the aforementioned noises. More investigation on stabilization effects of dichotomous noise on the lifetime of the superconducting state, geometric approach to quantum phase transitions, and other intriguing studies can be found in [25, 26, 27, 28, 29, 30] . Concerning bio-mathematical systems, we can also show that casual and extrinsic perturbations have a considerable effect on the infection dynamic [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] . In [42] , the author showed that due to potential [57, 58, 59] . Moreover, the impact of human interventions, economic crises, and uncontrolled flow of people may have cruel consequences on epidemiological systems and this cannot be described by using differential systems driven by white noise [60, 61, 62, 63, 64] . Consequently, we should employ the stochastic differential equation with significant discontinuities, so-called jumps [65] . Based on some nice properties: (i) stationary and independent increments (ii) sample paths which are almost surely right continuous with left limits, Lévy processes can be applied to many concrete and real situations [54, 66, 67, 68, 69, 70] . To depict this randomness, Zhang and Wang in [57] proposed the following SIC epidemic model with Lévy perturbation: where Jumps noise part White noise part To describe the stochastic components of this system, we consider firstly a probability triple . Concerning the results on the asymptotic behavior of the model (2), we cite the following works: • In [58] , the authors established the threshold of the disappearance and perseverance of the disease under the following key parametric condition • In [71] , the authors provided a critical review indicating that the condition (1.3) is actually a restricted hypothesis in the sense that there are many cases where it cannot be verified. Without making (1.3), they proposed an alternative approach to determine the threshold of (1.2). • In [72] , the authors proposed a general class of Lévy-jumps perturbation by considering the system (1.2) with infinite Lévy measures ( ) =   and possible correlation between stochastic components. They established the threshold of (1.2). Lévy jumps are also used in physical domains and allow precise characterization of random processes with some discontinuities. In [73] , Guarcello et al. explored the effects of Lévy noise on the dynamics of sine-Gordon solitons in long Josephson junctions. In [74] , the authors investigated the influence of Gaussian and non-Gaussian noises on the ballistic graphene-based small Josephson junctions. Notably, they provided a comparative study and some useful outcomes. In this article, we introduce a generalization of the work presented in [71] . We consider a more realistic situation of the epidemics dissemination in the case of sudden environmental catastrophes and some human interventions. These disorders greatly affect the number of individuals and the random perturbation may be dependent on the square of the variables S , I , and C respectively. In view of this, we include the second-order Lévy jumps into the epidemic model (1.1) as follows: J o u r n a l P r e -p r o o f . That is to say that the system (1.4) is well-posed biologically and mathematically. In this case, we define 2, = {( , ) : > 0, > 0} x y x y  and we merely examine the asymptotic behavior of the susceptible and infected compartments. disturbed epidemic models is to establish the conditions which guarantee the extinction and the permanence of the infection. Since the stochastic model (1.6) is perturbed by a quadratic stochastic perturbation, the threshold analysis is not only a complicated but also an intriguing question. In addition, Liu et al. [75] in 2017 indicated that the quadratic stochastic disturbance will be studied in the future due to technical difficulties and the feasibility of the system in mathematical epidemiology. Recently, some authors have analyzed the dynamical behavior of epidemic models with white noises in the quadratic form (see for example, [76, 77, 78, 79] ). To the best of our knowledge, to this day, stochastic epidemic systems with quadratic Lévy jumps perturbation have not been treated due to their complexity. In this paper, we try to deal with the extinction of the infection and the existence of a single stationary distribution (stochastic positive equilibrium state) of system (1.6). Roughly speaking, ergodic stationary distribution means the permanence of the epidemic. In the literature, one of the standard approaches to prove ergodicity is the Lyapunov-candidate-function, which provides just sufficient conditions in the majority of cases [80, 81] . Thus, the main problematic of this article can be stated as follows: • Is it possible to provide the sufficient and necessary condition for the ergodic property of the system (1.6) and illness extinction under only (A) without adding more assumptions? Accurately, this current study proposes a novel method to deal with stochastic models driven by quadratic Lévy jumps. We present the enough and almost requisite criterion for the extinction of the epidemic and the ergodic property of the model (1.6). Based on some nice properties of an auxiliary equation with quadratic jump-diffusion, we establish the exact expression of the threshold 0 . In other words, if 0 <0, then the number of infected individuals will quickly converge to zero, and if 0 >0, then system (1.6) admits a single stationary ergodic distribution. To examine the effect of the nonlinear jump-diffusion, we illustrate its impact on the threshold value 0 and generally on the asymptotic behavior of the epidemic. The remnant of this study follows the following planning: in the second part, we present some asymptotic properties of an auxiliary equation with quadratic jump-diffusion, then we introduce the value of 0 . In Section 3, we prove that 0 is the threshold of the stochastic In this section, we will briefly present some characteristics and properties of the model in the case of no infection. For this reason, we consider this auxiliary equation 22 11 12 The following two properties can be proven effortlessly: • The equation (2.1) is well posed, that is to say that for any initial data (0) > 0  , (2.1) has a unique positive and global solution. • By the stochastic comparison theorem [82] , we deduce that ( ) ( ) for any 0 t  almost surely (briefly, a.s.). To proceed further, we start with the following estimation. Proof. We choose the Lyapunov-candidate-function , we have the following inequality 22 11 12 By using the identical arguments exposed in the proof of Lemma Proof. As stated in [86] , to prove the ergodic property of (2.1), it be sufficient to check the existence of a positive function V and constants 12 Then, we get To illustrate the significance of the result (2.6), let's compare it to that of [58] . In fact, according to work [71] , we can get the results in [58] without considering (1.3) . Usually, this clause is widely used to prove the following long-time estimates: a.s. 1 0 Note that without using the result (2.6), the threshold takes the following form Since the expression of  is unknown, our alternative method offers an exact value of the threshold value of the model (1.6). In the next section, we will show analytically that 0 is the real threshold among suppression and tenacity of the disease. As stated in the introduction, the central question related to the analysis of epidemiological J o u r n a l P r e -p r o o f systems is to predict what will happen in the long term? So, the main purpose of this part is to process this query. We integrate both sides from 0 to t , then after dividing by t , we get In this subsection, we use Feller's property for Markov processes and the mutually exclusive possibilities result to establish the condition of the ergodic property of system (1.6), which can deal with the gap left by employing the Khasminskii analysis [88] used in [78, 79] . Let's start with the following lemma. ( ln ( )) = ( ) ( ) 0.5( (ln ( ) ln ( )) ( ) 0.5(( ( )) ( ( )) ) ( ) ( ) Once again, applying Itô's formula to 1 (1 ) p pS   and (1 ( )) = (1 ( )) ( ( ) ( ) ( )) 0.5( 1)(1 ( )) ( ( ) ( )) (1 ) x x t p S t p I t S t p p p S t I t p I t p S t I t x                                              K Km K K K   (d ) (                                         K Km K Km Km Km K Km K 2 1. Now, let us work out some simulations to illustrate the impact of quadratic jump-diffusion on the dynamic of an SIC epidemic model, and to infer the future of the ongoing COVID-19 pandemic under the assumption of stochasticity. Here, we apply the algorithm presented in [93] to discretize the disturbed system (1.4). By using the software Matlab2015b and the parameter values listed in Table 1 , we treat the COVID-19 Morocco case till May 2021 under unexpected and higher-order fluctuations. We mention that we have combined two types of data: 1. Estimated values which are established using a long time series of cross-sections of actual data. We notice that, in Morocco, the COVID-19 pandemic persists up to now and the situation is relatively stable without complete extinction. 2. Assumed value (   in Test 2 and Test 3) which is selected according to two criteria: (a) Appropriately verify the analytical result obtained in the case of extinction. (b) To numerically show the sharpness of our threshold. For illustrative goals, in some cases, we simulate the model without noise (deterministic solution) besides the stochastic one, and we choose this initial condition: ( (0), (0), (0)) = (0.5, 0.1, 0.1) S I C . Furthermore, we consider that the unity of time is one day and the number of individuals is expressed in ten million population. To fully understand the results of this subsection and for the convenience of the reader, we will divide it into three parts. Table 1 -Test 1, diverse colors represent distinct sizes of the density. The right-hand column presents the 3D graph of the joint 2 -dimensional densities of ( ( ), ( ), ( )) S t I t C t . Journal Pre-proof For the sake of simplicity, we assume that ( ) = 1  and we choose the deterministic parameter values from Table 1 (Test 1) . Regarding the intensities of the noises, we select 11 experimental two-dimensional densities of ( ( ), ( ), ( )) S t I t C t in Figure 2 in order to give a good overview of the stationarity property. Obviously, the endemic equilibrium P of the corresponding deterministic version is no longer the steady state of the stochastic model (1.4) . Therefore, in the following, we will numerically explore how the solutions of the stochastic model (1.4) behave around the deterministic equilibrium. For a sufficiently large time, we will calculate the time average of quantities () St , () It , and () Ct, for different noise intensities, and we discuss the asymptotic behaviors around P . From Table 2 , we observe that the intensity of linear white noise affects the fluctuation of the solution around the equilibrium. By way of explanation, the time average is close to P when the noise intensities are low. Most importantly, as noise intensity increases, the time-average of susceptible and recovered individuals raises, while the time-average of the infected population reduces. This phenomenon is observed for all types of noise with some variations (see Tables 3, 4, 5) . Therefore, this fact illustrates the need to clearly integrate the influence of environmental fluctuations in the phenomenological description of biological systems. It is already clear that the quadratic fluctuations can explain the generalized effect of the worsening of the epidemiological situation, but in certain critical cases, the quadratic noise drastically affects the time extinction of the disease (we will discuss this case in Test 2). In fact, noise is responsible for noise-induced transitions such that a stationary probability density function can suddenly change shape at certain noise intensity values (we will discuss this case in Test 3). From Remark 2.1, we mentioned that the explicit stationary distribution of a one-dimensional random differential equation with Lévy jumps is unknown. Same thing for a multidimensional systems. So, in this situation, we cannot analytically calculate the probability density function of some quantities during the transient dynamics. This randomness has a huge impact on the extinction time of COVID-19. Since quadratic noise do not appear in the threshold value, we explore its influence on the duration of infection. From In the above two subsections, we have studied numerically the dynamical bifurcation (D-bifurcation), which is caused by the abrupt change in the sign of the threshold 0 . In this part, we will explore the stochastic phenomenological bifurcation (SP-bifurcation), which principally depends on the abrupt change in the shape of the stationary probability density function of the model (1.6). Explicitly, we will show that the joint stationary probability density function of the classes S and I abruptly changes its shape at some values of the noise intensity. Firstly, we take deterministic parameter values from is an absorbing state and the conditions of extinction and persistence of the infection are not so clear from physical point of view (see Figure 9 ). Now we make slight changes to the stochastic noise intensities as follows: 11 , and the shape of the density of ( , ) SI is depicted in Figure 12 . That is, the system (1.6) is strongly persistent. Environmental factors and unexpected phenomena have significant impacts on the spread of epidemics. This paper takes into account these two factors with quadratic representation. Specifically, we have analyzed an SIC epidemic model that incorporates quadratic Lévy jumps. Compared to previous works, many authors have considered the quadratic white noise perturbation (without Lévy jumps) in the various kinds of systems [94, 75, 95, 96] . But there are some limitations of these papers, which can be explained as: 1. The complex and brutal random fluctuations are simulated by the white noise or Lévy jumps? 2. Can we obtain the exact value of the threshold between extinction and persistence? 3. Are the techniques and analysis presented in mentioned papers general for other stochastic models? J o u r n a l P r e -p r o o f Salim El Azami El-idrissi: Review. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.  Constructed and probed the inuence of quadratic L_evy noise on the dynamic of an SIC contagious illness model.  By ameliorating the hypotheses and using some new analytical techniques, we find the exact threshold value between extinction and ergodicity (persistence) of our system.  The existence of a unique ergodic stationary distribution implies stochastic weak stability.  Gene mutation of SARS-CoV-2 has been rapidly emerging and transmitting all over the world during COVID-19 pandemic.  To illustrate the comparison and con_rm our theoretical analysis, we perform numerical simulations with some real data of COVID-19 in Morocco. A contribution to the mathematical theory of epidemics Covid-19: what is next for public health? The effect of the Covid-19 on sharing economy activities Combating COVID-19: health equity matters Challenges and potential solutions in the development of COVID-19 pandemic control measuresl Immune determinants of COVID-19 disease presentation and severity Targeted adaptive isolation strategy for COVID-19 pandemic Effects of information-dependent vaccination behavior on coronavirus outbreak: insights from a SIRI model Population biology of infectious diseases: part I Effects of predation on host-pathogen dynamics in SIR models Global stability of the endemic equilibrium of multigroup SIR epidemic models Analytical parameter estimation of the SIR epidemic model. applications to the COVID-19 pandemic Analytical features of the SIR model and their applications to COVID-19 An sir epidemic model for covid-19 spread with fuzzy parameter: the case of indonesia Stability of a stochastic SIR system Stability and threshold of a stochastic SIRS epidemic model with vertical transmission and transfer from infectious to susceptible individuals Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology Nonlinear relaxation in the presence of an absorbing barrier Self-regulation mechanism of an ecosystem in a non-gaussian fluctuation regime Dynamics of two picophytoplankton groups in mediterranean sea: Analysis of the deep chlorophyll maximum by a stochastic advection-reaction-diffusion model Cyclic fluctuations, climatic changes and role of noise in planktonic foraminifera in the mediterranean sea Noise-induced effects in population dynamics Role of the noise on the transient dynamics of an ecosystem of interacting species Phase dynamics in graphene-based josephson junctions in the presence of thermal and correlated fluctuations Stabilization effects of dichotomous noise on the lifetime of the superconducting state in a long josephson junction Geometry of quantum phase transitions Spike train statistics for consonant and dissonant musical accords in a simple auditory sensory model Measurement of energies of oxygen ion diffusion in yttria stabilized zirconia by flicker noise spectroscopy Neurohybrid memristive CMOS-integrated systems for biosensors and neuroprosthetics Uhlmann curvature in dissipative phase transitions Competitive exclusion in a general multi-species Chemostat model with stochastic perturbations Dynamics of a stochastic predator-prey model with habitat complexity and prey aggregation Noise-induced transitions in a non-smooth SIS epidemic model with media alert Critical value in a SIR network model with heterogeneous infectiousness and susceptibility Stability of stochastic differential equations with distributed and state-dependent delays Improving stability conditions for equilibria of SIR epidemic model with delay under stochastic perturbations Stability analysis of delayed tumor-antigen-activated immune response in combined BCG and IL-2 immunotherapy of bladder cancer Global dynamics of an SEIR model with two age structures and a nonlinear incidence Ergodic stationary distribution and extinction of a J o u r n a l P r e -p r o o f Journal Pre-proof stochastic sirs epidemicmodel with logistic growth and nonlinear incidence Progressive dynamics of a stochastic epidemic model with logistic growth and saturated treatment Exploring the stochastic host-pathogen tuberculosis model with adaptive immune response Stability and Complexity in Model Ecosystems The behavior of an SIR epidemic model with stochastic perturbation Threshold behaviour of a stochastic SIR model Asymptotic behavior of global positive solution to a stochastic SIR model Long-time behavior of a stochastic SIR model Modeling the impact of unreported cases of the COVID-19 in the north african countries Age-structured modeling of COVID-19 epidemic in the USA, UAE and Algeria Stochastic dynamical probes in a triple delayed SICR model with general incidence rate and immunization strategies Dynamic threshold probe of stochastic SIR model with saturated incidence rate and saturated treatment function A stochastic SACR epidemic model for J o u r n a l P r e -p r o o f HBV transmission Stochastic permanence of an epidemic model with a saturated incidence rate Sharp conditions for the existence of a stationary distribution in one classical stochastic chemostat Dynamic behavior of a stochastic SIQS epidemic model with Levy jumps Stability in a system subject to noise with regulated periodicity A simple noise model with memory for biological systems Stochastic SIR model with jumps Threshold of a stochastic SIR epidemic model with Levy jumps Stochastic dynamics of the delayed chemostat with Levy noises New results on the asymptotic behavior of an SIS epidemiological model with quarantine strategy, stochastic transmission, and Levy disturbance Developing new techniques for obtaining the threshold of a stochastic SIR epidemic model with 3-dimensional Levy process The long-time behaviour of a stochastic SIR epidemic model with distributed delay and multidimensional Levy jumps Dynamic characterization of a stochastic sir infectious disease model with dual perturbation Ergodic stationary distribution of a stochastic hepatitis B epidemic model with interval-valued parameters and compensated poisson process Stochastic Differential Equations A stochastic model of HIV infection incorporating combined therapy of haart driven by Levy jumps A dynamics stochastic model with HIV infection of CD4 T cells driven by Levy noise Threshold behavior of a stochastic Lotka Volterra food chain chemostat model with jumps Persistence and extinction of a stochastic sis epidemic model with regime switching and Levy jumps Stochastic viral infection model with lytic and nonlytic immune responses driven by Levy noise Asymptotic behaviors of stochastic epidemic models with jump-diffusion Stochastic sir Levy jump model with heavy tailed increments Effects of Levy noise on the dynamics of sine-gordon solitons in long josephson junctions Anomalous transport effects on switching currents of graphene-based josephson junctions Periodic solution and stationary distribution of stochastic SIR epidemic models with higher order perturbation Higher order stochastically perturbed SIRS epidemic model with relapse and media impact Stationary distribution and extinction of a stochastic SIR model with nonlinear perturbation Stationary distribution and extinction of a stochastic predator-prey model with additional food and nonlinear perturbation Extinction and stationary distribution of an impulsive stochastic chemostat model with nonlinear perturbation Survival and stationary distribution of a SIR epidemic model with stochastic perturbations Stationary distribution and persistence of a stochastic predator-prey model with a functional response Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations Classification of asymptotic behavior in a stochastic SIR model Dynamical behavior of a stochastic predator-prey model with general functional response and nonlinear jump-diffusion Statistical inference for ergodic diffusion processes Asymptotic properties of jump-diffusion processes with state-dependent switching Stochastic Differential Equations and Applications Stochastic stability of differential equations On the existence and uniqueness of invariant measure for continuous-time markov processes The stationary distribution of the facultative population model with a degenerate noise Mathematical analysis and simulation of a stochastic COVID-19 levy jump model with isolation strategy Strong approximations of stochastic differential equations with jumps Stationary solution, extinction and density function for a high-dimensional stochastic SEI epidemic model with general distributed delay Stationary distribution and extinction of a multi-stage HIV model with nonlinear stochastic perturbation Dynamics of a stochastic sica epidemic model for HIV transmission with higher-order perturbation The authors express their gratitude to the editor and expert reviewers for their comments and suggestions. The first author warmly thanks Professor Sanling Yuan (Shanghai, China) for his clarifications. This work was supported by the DST -Science and Engineering Research Board of India (EEQ/2021/001003). The real and theoretical data used to support the findings of this study are already included in the article. On behalf of all authors, the corresponding author states that there is no conflict of interest. For this purpose, this study is dedicated to presenting a new general setting and to answer the above questions. Accurately, 1. We have mentioned (in the introduction) that in the majority of real and concrete situations, external disturbances are not continuous. For this, we have used the stochastic model with Lévy jumps.2. Using the ergodic characteristic of the auxiliary system (2.1), the probabilistic comparison result, and the Lyapunov function approach, we have provided the sufficient and necessary criterion for the extinction and ergodicity of the distributed system (1.4) . We indicate that the critical case 0 =0 is still an open question that we will treat in the future. It is interesting to highlight that the state of 0 =0 is an absorbed state and that the conditions for extinction and persistence of contagion are not very clear from a physical point of view 3. In this study, we have given the exact value of 0 . It is obvious that the linear noise intensities 21  and 21 () u  have a passive influence on its value, and the quadratic noise quantities have no effect on it.4. To prove the ergodicity, we have presented a novel technique that joins the Lyapunov method with the analysis used in [53] .To illustrate the sharpness of our results, we have performed some numerical simulations and we have confirmed that the impact of quadratic jumps on the threshold value is negligible. However, the non-linearity hypothesis has a positive effect on the disease in the permanence case.Generally speaking, we point out that this paper extends the study presented in [71] to the case of quadratic Lévy jumps and delivers some new insights for understanding the propagation of diseases with complex fluctuations. In other words, the proposed approach leaves many research paths to be explored in future works.