Analysis of a stochastic coronavirus (COVID-19) Lévy jump model with protective measures

Analysis of a stochastic coronavirus (COVID-19) L´evy jump model with protective measures Tom´as Caraballoa,∗, Mohamed El Fatinib, Mohamed El Khalifib, Anandaraman Rathinasamyc,d aDpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Universidad de Sevilla, Tarfia, s/n, 41012, Sevilla, Spain carabal[email protected] bStochastic Modelling and Statistics group, Lab of PDE’s, ASG, Ibn Tofail University, FSK, BP 133, K´enitra, Morocco [email protected]om ; [email protected]om cDepartment of Mathematics, Anna University, Chennai – 600025, India dDepartment of Mathematics, MIT Campus, Anna University, Chennai – 600044, India a [email protected] Abstract This paper studied a stochastic epidemic model of the spread of the novel coronavirus (COVID- 19). Severe factors impacting the disease transmission are presented by white noise and compensated poisson noise with possibly infinite characteristic measure. Large time estimates are established based on Kunita’s inequality rather than Burkholder-Davis-Gundy inequality for countinuous diffusions. The effect of stochasticity is taken into account in the formulation of sufficient conditions for the extinction of COVID-19 and its persistence. Our results prove that environmental fluctuations can be privileged in controlling the pandemic behaviour. Based on real parameter values, numerical results are presented to illustrate obtained results concerning the extinction and the persistence in mean of the disease. Keywords: Stochastic differential equation; L´evy noise; COVID-19; extinction; persistence in mean; Kunita’s inequality. 1. Introduction Since its first emergence on December 2019, coronavirus disease 2019 (COVID-19), the respiratory illness caused by Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) is ∗Corresponding author Email address: [email protected] (Tom´as Caraballo) Preprint submitted to Journal of L A T EX Templates September 30, 2021 continuing its spread across the world taking more than one million and one-third of a million lives [1] with more than 50 millions infected cases by the end of November 2020 [2]. This large pandemic was not the first of the twenty-first century, there were already the SARS CoV-1 and MERS in 2002-2003 and 2012 respectively [3, 4]. Coughs and sneezes and talking expel respiratory droplets that can be breathed by others allowing the spread of COVID-19 virus. An individual could be able to catch the infection through face-touching right after touching contaminated surfaces and objects. Collective efforts of public, health authorities and all societies actors are main strategies to stay safe and reduce the impact of the current sanitary crisis while waiting an effective COVID-19 vaccine. Following guidelines of World Health Organisation, measures such as, social distancing, wearing face masks, regular hands cleaning, isolate infected individuals, among others can limit the spread of the disease [5, 6]. Besides medical researches to assess infectious diseases transmission understanding, mathematics has long been used in biology and disease modelling. Mainly, Kermack and McKendrick were the first to elaborate a compartmental ordinary differential equation model where the total population is divided into susceptible, infected and removed classes (SIR) [7]. Thenceforth, mathematical models become progressively more crucial as tool of diseases dynamics analysis. Beyond deterministic systems, environmental fluctuations are introduced through difussion processes theory to formulate stochastic differential equation (SDE) based models for epidemiology [8, 9, 10]. SDEs and related tools have been extensively employed for real world diseases investigation. Indeed, [11] proposed a stochastic SIR model with vaccination for hepatitis B and analysed its persistence and extinction. [12] studied a stochastic SIRD epidemic model of Ebola and explored the existence of a unique stationary distribution. Respecting measles transmission dynamics, [13] developed a stochastic SIR-type model incorporating double dose vaccination and studied the stability of disease-free and endemic equilibria. For other diseases modelling using SDEs, we refer, for instance, to [9, 14, 15, 16] and references cited therin. A great amount of literature has been produced in the field of mathematical epidemiology for COVID-19 dynamics analysis [17, 18, 19]. For instance, [20] studied a stochastic SIR model for novel coronavirus spreading forecast in Kuwait. Author in [21] has proposed a stochastic model for COVID-19 transmission and prediction, taking into account the effect of asymptomatic infectives. Mandal et al. [22] have examined a deterministic model of COVID-19 fitting and 2 short-term forecasting for some Indian states coronavirus data. Moreover, they formulated an optimal control problem to reduce infected population size. Their ODE model is illustrated by the diagram in figure 1 and given by the following equation. Susceptibles Exposed Infectives Quarantined Recovered Figure 1: Transition diagram in the epidemic model (1). Dashed arrows stand for births, black arrows for deaths and red arrows represent different transition flows between the states                      dSt= [A−β(1 −ρ1)(1 −ρ2)StEt+b1Qt−µSt]dt, dEt= [β(1 −ρ1)(1 −ρ2)StEt−(b2+α+ν+µ)Et]dt, dQt= [b2Et−(b1+c+µ)Qt]dt, dIt= [αEt+cQt−(η+µ+δ)It]dt, dRt= [ηIt+νEt−µRt]dt, (1) where the explanation of parameters is given in Table 1 and the labels S, E, Q, I and Rstand for susceptible, exposed, quarantined, hospitalized infectives and recovered individuals respectively. In system (1), hospitalized infectives are not responsible of the spread of COVID-19 as they are completely isolated until they recover or die. Moreover, individuals of the class Eare only the spreaders as well as this class contains asymptomatic as well as symptomatic but unreported infected population. Henceforth, an infection may occurs only when a portion (1 −ρ1) of susceptibles comes into contact with the portion (1 −ρ2)of exposed persons who do not take safety measures. It is shown in [22] that system (1) always has a disease free equilibrium E0 which is locally asymptotically stable under the condition R0=Aβ(1 −ρ1)(1 −ρ2) µ(b2+α+ν+µ)<1.When R0>1,besides E0that is unstable, there is a unique endemic equilibrium E∗that is locally asymptotically stable. 3 Table 1: Model parameters description. Parameter Description Athe constant recruitment rate. βthe disease transmission rate. ρ1the portion of susceptibles maintaining proper precaution measure (0< ρ1<1). ρ2the portion of exposed persons maintaining proper precaution measure (0< ρ2<1). b1the portion of quarantine class moving to susceptible class after quarantine period. b2the portion of the exposed class going to quarantine class. αthe portion of the exposed class going to infected class. cthe portion of quarantine class moving to infective class. ηthe recovery rate of hospitalized infected population. νthe recovery rate of exposed population. δthe disease induced death rate. µthe natural mortality rate. Latterly, Boukanjime et al. [23] introduced environmental fluctuations into system (1) by considering both white noise and telegraph noise and a generalized incidence function. In their work, the impact of factors like humidity, rainfall fluctuations, meteorological factors, etc., are taken into account in COVID-19 spreading modelling. They showed the extinction of COVID-19 under R0 s<1and its persistence when R0 s>1,where the expression of R0 sdepends on white noise intensity and regimes states [23]. There are many important sources of stochasticity affecting the infection rate of diseases including, for instance, susceptibility that differs from person to person according to their health states and uncertain number of contacts per infective individual. The infection rate of sexually transmitted diseases experiences different perturbations induced gender, age, and sexual preference and number of partners per person. In this paper, we introduce another type of noises to the system (1) to present serious environmental perturbations. This is the L´evy noise that is widely exploited to reflect the discontinuity of epidemic models solutions due to sudden environmental shocks. Actually, natural disasters like earthquakes and weather-related disasters can dramatically expand the number of exposed 4 individuals to diseases and amplify the deaths level, which cannot mathematically represented by continuous models and necessitates the introduction of discontinuous noise sources as L´evy one. See [24, 11, 25, 26, 27] and references cited therein. At this stage, we let the L´evy noise act on the COVID-19 transmission rate as follows β→β+σ˙ B(t) + ˙ J(t),(2) where B(t)is a standard Brownian motion defined in a complete probability space (Ω,Ft,P) satisfying the usual conditions, σ > 0is the intensity of white noise which is independent of the L´evy jumps J(t) = Rt 0RR∗C(z)˜ N(ds, dz),where ˜ Nis the compensator of a Poisson counting measure Nwith characteristic measure λon R∗=R\ {0}, C :R∗×Ω→Ris bounded and continuous with respect to λand is B(R∗)× Ft-measurable, where B(R∗)is a σ-algebra with respect to the set R∗.All processes are defined on (Ω,Ft,P).Under these hypotheses, we propose the following system                                  dSt= [A−β(1 −ρ1)(1 −ρ2)StEt+b1Qt−µSt]dt −σStEtdBt−RR∗St−Et−C(z)˜ N(ds, dz), dEt= [β(1 −ρ1)(1 −ρ2)StEt−(b2+α+ν+µ)Et]dt +σStEtdBt+RR∗St−Et−C(z)˜ N(ds, dz), dQt= [b2Et−(b1+c+µ)Qt]dt, dIt= [αEt+cQt−(η+µ+δ)It]dt, dRt= [ηIt+νEt−µRt]dt, (3) To analyse the model (3), we need to impose the following assumption: Assumption (H1).C(z)is bounded, −1< C(z),kCk∞<µ A,and RR∗C2(z)λ(dz)<∞. Notice that we do not require the condition λ(R∗)to be finite. Hence one can see that previous works, including [11, 28, 29, 30, 26] etc., do not include the case we consider in this work. This paper is structured as follows. Section 2 is devoted to state preliminaries on the existence of a unique global positive solution to system (3). Moreover, some useful estimates are established based on discountinuous processes theory. Sufficient conditions to the extinction and persistence in mean of COVID-19 are presented in sections 3 and 4 respectively. Based on real data parameter values, our analytical results are illustrated through numerical simulations in Section 4. 5 2. Preliminaries After formulating the SDE system (3) for COVID-19 dynamics, it is necessary to verify if it admits a unique global positive solution. Following the same steps of the proofs of Theorem 2.1 and Theorem 1 in [27] and [26] respectively, we can easily arrive at the hereunder results. Theorem 2.1. Assume that (H1)holds, then for any given initial value (S(0), E(0), Q(0), I(0), R(0)) ∈ R5 +, there exists a unique solution (S(t), E(t), Q(t), I(t), R(t)) ∈R5 +of system (3) on t⩾0and the solution will remain in R5 +with probability 1. Remark 1. The following domain ∆ = ((x1,· · · , x5)∈R5 +: 5 X i=1 xi≤A µ)(4) is a positively invariant set of system (3) for any initial condition in ∆. By virtue of Kunita’s first inequality (Theorem 4.4.23 of [31]), we can prove the following estimates. Lemma 1. Let (S(t), E(t), Q(t), I(t), R(t)) be the solution to system (3) for an initial value in ∆.If Assumption (H1)holds, then i) lim t→∞ 1 tZt 0ZR∗ ln (1 + Sr−C(z)) ˜ N(dr, dz)=0, a.s., lim t→∞ 1 tZt 0ZR∗ Sr−Er−C(z)˜ N(dr, dz)=0, a.s., ii) lim t→∞ 1 tZt 0 SrdBr= 0, a.s., lim t→∞ 1 tZt 0 SrErdBr= 0, a.s. Proof. Denote Zt=1 tZt 0ZR∗ ln (1 + Sr−C(z)) ˜ N(dr, dz). By virtue of Kunita’s first inequality, for any p≥2,there is a positive real number cpsuch that Esup 0<r≤t |Zr|p≤cpEZt 0ZR∗ |ln (1 + Sr−C(z)) |pλ(dz)dr +cpE"Zt 0ZR∗ |ln (1 + Sr−C(z)) |2λ(dz)drp/2#. On the other hand, one can easily show that |ln (1 + Sr−C(z)) |≤ A µ  1 + AkCk∞ 2µ1−A µkCk∞2  |C(z)|. 6 Therefore, there is a positive Cpsuch that Esup 0<r≤t |Zr|p≤CptZR∗ |C(z)|pν(dz) + Cptp/2ZR∗ C2(z)λ(dz)p/2 .(5) For any ε > 0,Doob’s martingale inequality implies that Psup k≤t≤k+1 |Zt|p> k1+ε+p/2≤1 k1+ε+p/2Esup k≤t≤k+1 |Zt|p ≤Cp(k+ 1) k1+ε+p/2ZR∗ |C(z)|pν(dz) + Cp(k+ 1)p/2 k1+ε+p/2ZR∗ C2(z)λ(dz)p/2 . As a result of the Borel-Cantelli lemma, we have for almost all ω∈Ω, sup k≤t≤k+1 |Zt|p≤k1+ε+p/2, holds for all but finitely many k. Hence, for almost all ω∈Ω,there is a k0=k0(ω)such that for all k≥k0 ln |Zt| ln t≤(1 + ε+p/2) ln k pln t≤1 2+1 + ε p, for any ε > 0and k≤t≤k+ 1.Letting ε→0yields lim t→∞ ln |Zt| ln t≤1 2+1 p. Then, for all ˜ε∈(0,1/2−1/p),there is a finite random time ¯ T=¯ T(ω)such that ln |Zt|≤ 1 2+1 p+ ˜εln t, t ≥¯ T, and so lim sup t→∞ |Zt| t≤lim sup t→∞ t˜ε−1/2+1/p = 0, a.s. This gives that lim t→∞ |Zt| t= 0, a.s. Following the same steps as above, one can obtain lim t→∞ 1 tZt 0ZR∗ Sr−Er−C(z)˜ N(dr, dz)=0, a.s. The proof of the second assertion is similar to one of Lemma 2.2 of [30] and we omit it here. This concludes the proof. 7 3. Extinction In the next theorem, we will establish a condition under which the COVID-19 dies out from the population. For the convenience of the reader, we define the following quantities ˜ β=β(1 −ρ1)(1 −ρ2),˜σ=1 2     σ2+1 1 + A µkCk∞2ZR∗ C2(z)λ(dz)     ,(6) Re j= ˜ βA µ b2+α+ν+µ+ ˜σA µ2.(7) We have the next theorem. Theorem 3.1. Let (S(t), E(t), Q(t), I(t), R(t)) be the solution of system (3) with an initial value (S(0), E(0), Q(0), I(0), R(0)) ∈∆.If the following conditions Re j<1and 2˜σA µ≤˜ βhold, then the pandemic goes out of the population exponentially with probability one. That is to say lim sup t→∞ ln Et t<0a.s., lim sup t→∞ ln Qt t<0a.s., lim sup t→∞ ln It t<0a.s., Proof. Applying the Itˆo formula with jumps (Theorem 4.4.7 of [31]) to the function Et7→ ln Et, we get d ln Et=˜ βSt−(b2+α+ν+µ)−σ2 2S2 t+ZR∗ ln (1 + StC(z)) −StC(z)λ(dz)dt +σStdBt+ZR∗ ln (1 + St−C(z)) ˜ N(dt, dz).(8) Using the Taylor’s formula, we have for some θ∈(0,1) that ln (1 + StC(z)) −StC(z) = −(StC(z))2 2 (1 + θStC(z))2≤ − C2(z) 21 + A µkCk∞2S2 t.(9) Hence, ln Et t≤˜ βhSit−(b2+α+ν+µ)−˜σhS2it+M1(t) t+M2(t) t,(10) where M1(t) = ln E0+σZt 0 SrdBrand M2(t) = Zt 0ZR∗ ln (1 + Sr−C(z)) ˜ N(dr, dz).(11) 8 From the system (3), one can obtain dSt+ dEt+ dQt=A−µSt−(α+ν+µ)Et−(c+µ)Qt, which leads to hSit=A µ−α+ν+µ µhEit+c+µ µhQit+φ1(t),(12) where φ1(t) = S0+E0+Q0 µt −St+Et+Qt µt .Then, using the Cauchy-Schwarz inequality and direct computation, we get hS2it≥ hSi2 t=A µ−α+ν+µ µhEit+c+µ µhQit+φ1(t)2 ≥A µ2 −2A µ+φ1(t)α+ν+µ µhEit+c+µ µhQit. Moreover, from the SDE (3), we have dQt+ dIt+ dRt= (b2+α+ν)Et−(b1+µ)Qt−(µ+δ)It−µRt. Therefore, −hQit≥ −b2+α+ν b1+µhEit+φ2(t), where φ2(t) = Qt+It+Rt (b1+µ)t−Q0+I0+R0 (b1+µ)t.Hence, hS2it≥A µ2 −2A µα+ν+µ µ+(c+µ)(b2+α+ν) µ(b+µ)hEit−φ3(t),(13) where φ3(t) = 2φ1(t)α+ν+µ µ+(c+µ)(b2+α+ν) µ(b+µ)hEit−2φ2(t)c+µ µA µ+φ1(t). Injecting (12) and (13) into (10), we obtain ln Et t≤ − ˜ β−2˜σA µα+ν+µ µ+ 2˜σA µ (c+µ)(b2+α+ν) µ(b+µ)hEit +˜ βA µ−(b2+α+ν+µ)−˜σA µ2 +φ4(t) + M1(t) t+M2(t) t,(14) 9 Despite the interest arising from the studied model, some limitations can be listed up for further investigations. For instance, public health efforts are on reducing transmission and disease fatality rates and raising up the recovery rate. Consequently, time-independent parameter models are not able to represent this property and then still restrictives. Other compartments and transitions can be also taken into account to formulate more reasonable COVID-19 models. Demographic noise could be also considered as an internal source of stochasticity as in [33]. 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