Dynamical behaviors of a stochastic SIS epidemic model with mean-reverting inhomogeneous geometric brownian motion

Dynamical b eha viors of a sto c hastic S I S epidemic mo del with mean-rev erting inhomogeneous geometric bro wnian motion Lahcen Khammic h, Driss Kiouach L2MASI Lab oratory , F aculty of sciences Dhar El Mahraz, Sidi Mohamed Ben Abdellah Universit y , F ez, Morocco. lahcen.khammich@usmba.ac.ma d.kiouach@uiz.ac.ma Marc h 26, 2026 Abstract The main purp ose of this pap er is to study the Dynamical b eha viors of a sto c hastic S I S epidemic mo del using mean-rev erting inhomogeneous geometric bro wnian motion pro cess. First w e demonstrate the existence of a global-in-time solution and establish that is unique and remains p ositiv e. Then we derive a sufficient condition for exp onen tial extinction of infectious diseases and we show that our extinction threshold in the stochastic case coincides with that of the deterministic case. Finaly , w e define an appropriate theoretical framew ork to guarantee the existence of an ergo dic stationary distribution. Keyw ords: Sto c hastic S I S epidemic mo del, mean-reverting inhomogeneous geometric brownian motion pro cess , extinction, ergo dic stationary distribution. 1 In tro duction The Susceptible-Infected-Susceptible ( S I S ) mo del is a fundamental mathematical framework in epidemiology used to study the transmission dynamics of infectious diseases within a p opulation. In this model, individuals transition b etw een t wo compartments: susceptible S and infected I , with the p ossibility of b ecoming reinfected after recov ery . The evolution of the epidemic is typically describ ed through differen tial equations, whic h facilitate the analysis of critical epidemiological metrics, suc h as the basic reproduction num b er R 0 , and the long-term b eha vior of the disease. A wide range of differen tial equation form ulations-deterministic and sto c hastic hav e b een emplo yed to inv estigate the SIS mo del under v arious assumptions and scenarios [14, 13, 3]. One of the most in teresting works in this vein is the recen t pap er of Zhidong T enga, Lei W angb [11]. In this latter, the authors hav e prop osed the following SIS epidemic mo dels with nonlinear incidence rate : ( d S ( t ) = (Λ − χ ( t ) g ( I ( t ) S ( t ) + γ I ( t ) − µ S ( t )) d t, d I ( t ) = ( χ ( t ) g ( I ( t ) S ( t ) − ( γ α + µ ))d t. (1) Where the parameters app earing in this system are describ ed as: • Λ represents the recruitment rate. • χ is the transmission co efficient b et ween susceptible and infected individuals. • µ the natural mortality rate. 1 S I χS g ( I ) Λ µ S γ I µ I α I Figure 1: Schematic diagram for SIS mo del • α is the disease-related mortalit y rate of infected individuals. • γ is the recov ery rate. And g ( I ) satisfies the conditions: ( i ) g (0) = 0 , ( ii ) g ′ ( I ) > 0 , ( iii ) − p ≤ ( g ( I ) I ) ′ ≤ 0 with p is a p ositiv e constant. By the analysis in [10] , w e see that Γ :=  ( S ( t ) , I ( t )) ∈ R 4 +     Λ µ = S 0 > S + I  is the p ositiv ely inv arian t set of 1, and the repro duction num b er is R 0 = Λ g ′ (0) χ µ ( µ + γ + α ) for more detail ab out asymptotically analysis and disease equilibrium (see [10]). Recen t researc h suggests that environmen tal sounds cause many principal parameters in epidemic mo dels to swing around an a verage v alue. Considering this effect while mo deling contagious illnesses impro ves our understanding of their spread b eha vior. Decision-makers can mitigate the spread of these illnesses by implemen ting effective con trol measures. Man y academics and scholars are in ter- ested in studying stochastic epidemic mo dels[5, 1, 11], particularly those with randomized disease transmission parameters using the Ornstein-Uhlen b eck pro cess [12, 10, 7]. The Orstein-Uhlembeck pro cess incorp orates aleatory effects due to its b ounded v ariance for short time p eriods [0 , t ] . This aligns with the constan t disturbance prop ert y of sto c hastic noise. The Ornstein-Uhlenbeck process is typically stated using the follo wing sto c hastic differential equation: d χ ( t ) = r ( ¯ χ − χ ( t ))d t + σ d B ( t ) In this context, ¯ χ > 0 and r > 0 denote, resp ectivel y , the long-term mean of the pro cess and the sp eed of mean reversion, while σ > 0 represen ts the instantaneous volatilit y of random fluctuations, mo deled b y a standard Brownian motion B ( t ) defined on a complete probability space (Ω , F , P ) with a filtration {F t } t ≥ 0 satisfying the usual conditions (i.e., it is right-con tin uous and F 0 con tains all P -n ull sets). Since the transmission rate χ must remain p ositive, an y sto chastic mo deling of χ should preserv e this property . The classical Ornstein-Uhlenbeck process do es not guaran tee p ositivit y , making it unsuitable for this purpose. T o address this, we propose a p erturbation strategy inspired by the Ornstein-Uhlen b ec k pro cess but adapted to ensure that χ remains p ositiv e at all times. W e will use the mean-reverting inhomogeneous geometric brownian motion (IGBM) approac h to p erturb the parameter χ [1, 9, 2] .this latter pro cess is often characterized using the follo wing sto chastic differential equation: d χ ( t ) = r ( ¯ χ − χ ( t ))d t + σ χ ( t )d B ( t ) 2 A ccording to [1, 8, 6], the sto c hastic pro cess has an ergodic stationary distribution that follo ws the in verse-gamma density with shap e σ 2 +2 r σ 2 and scal 2 r ¯ χ σ 2 , and for any π -integral function φ w e hav e: Z ∞ 0 φ ( x ) π dx = lim t →∞ 1 t Z t 0 φ ( χ ( s )) ds In the light of what precedes, the system 1 can b e rewriting as follo w:      d S ( t ) = (Λ − χ ( t ) g ( I ( t ) S + γ I ( t ) − µ S ( t )) d t, d I ( t ) = ( χ ( t ) g ( I ( t ) S ( t ) − ( α + γ + µ ))d t, d χ ( t ) = r ( ¯ χ − χ ( t ))d t + σ χ ( t )d B ( t ) (2) The remainder of this pap er is structured as follows: Section 2 fo cuses on demonstrating the well- p osedness of mo del 2, sho wing that it has a unique, global-in-time, and p ositiv e solution. In Section 3 we present the sufficien t conditions leading to extinction. In In Section4, w e presen t the necessary conditions for the existence of a stationary distribution . Finally , the main conclusions of the pap er are discussed in Section 5. 2 Existence and uniqueness of the global p ositiv e solution Theorem 2.1. F or any initial value ( S (0) , I (0) , χ (0)) ∈ Γ wher e Γ :=  ( S ( t ) , I ( t ) , χ ( t )) ∈ R 3 +     0 < S + I < Λ µ  it c orr esp onds a unique solution ( S ( t ) , I ( t ) , χ ( t )) to the sto chastic system 2 on R + . Pr o of. Let us consider the C 2 -real v alued function Ψ : R 3 + → R + defined by Ψ( S, I , χ ) = [ S − 1 − ln S ] + [ I − 1 − ln I ] + N g ′ (0) r [ χ − 1 − ln χ ] By using the renowned Ito’s form ula, one obtains that L Φ =  1 − 1 S  [Λ − χS g ( I ) + γ I − µ S ] +  1 − 1 I  [ χ S g ( I ) − ( µ + γ + α ) I ] + N g ′ (0) r  1 − 1 β  h r ( ¯ β − β ) i + σ 2 r g ′ (0) N Therefore, we get L Φ ⩽ Λ + 2 µ + γ + α + g ′ (0) N ¯ β + σ 2 r g ′ (0) N | {z } := C , where N = max ( S (0) + I (0) , Λ µ ) and C is a p ositive constant that is not dep ending on the initial v alues S (0) , V (0) , χ (0) . The rest of the pro of is similar to the pro of of Theorem 3.1 in [4] So we skip it here. 3 3 Extinction In this section, w e fo cus on outlining the sufficient conditions required for the extinction of the disease. Theorem 3.1. F or any initial value ( S (0) , I (0) , χ (0)) ∈ Γ , If R e 0 = R 0 = Λ ¯ χg ′ (0) µ ( µ + γ + α ) < 1 we have lim T → + ∞ I ( t ) = 0 a.s. Pr o of. By applying Itô’s form ula we ha ve L (ln I ) = χ ( t ) S g ( I ) I − ( µ + γ + α ) In tegrating from 0 to t and then dividing by t on both sides, and using the ergo dicité of β ( t ) , we get lim sup t →∞ ln ( I ( t )) t ≤ Λ µ g ′ (0) lim t →∞ 1 t Z t 0 β ( s )d s − ( µ + γ + α ) ≤ Λ µ g ′ (0) Z + ∞ 0 xπ ( x )d x − ( µ + γ + α ) ≤ Λ µ g ′ (0) ¯ χ − ( µ + γ + α ) =( µ + γ + α )  g ′ (0) ¯ χ Λ µ ( µ + γ + α ) − 1  Then we hav e lim sup t →∞ ln ( I ( t )) t ≤ ( µ + γ + α ) ( R e 0 − 1) Therefore, if R e 0 < 1 , then the disease will go to extinction. this completes the proof. Remark 1. In this p articular c ase, and c ontr ary to what is typic al ly observe d, the extinction thr eshold of the sto chastic mo del c oincides pr e cisely with that of the deterministic mo del. 4 Stationary distribution This section in vestigates the existence of a stationary distribution in the sto c hastic mo del, as it serv es as an indicator of the long-term p ersistence of infectious diseases. Lemma 4.1. L et H ⊂ R d b e a b ounde d, close d domain with a r e gular b oundary L 0 . A ssume that for any initial value X (0) ∈ R d , the fol lowing c ondition holds: lim inf t → + ∞ 1 t Z t 0 P ( s, X (0) , H )d s > 0 , a.s., wher e P ( s, X ( s ) , H ) r epr esents the tr ansition pr ob ability of X ( t ) . Then, the system p ossesses a solution with the F el ler pr op erty, and system 2 admits at le ast one stationary distribution η ( · ) on R d . 4 Define R s 0 = Λ ¯ χg ′ (0) µ ( µ + γ + α + 2 D S 0 g ′ (0) + D ( S 0 ) 2 p ) , where D = 1 2 Z + ∞ 0 | ¯ χ − x | π ( x ) dx. and π ( x ) is the inv ariant density of in verse-gamma with shap e σ 2 +2 r σ 2 and scale 2 r ¯ χ σ 2 Theorem 4.2. If R s 0 > 1 , the sto chastic system describ e d by e quation 2 admits at le ast one er go dic stationary distribution, denote d by η ( . ) , within the domain Γ . Pr o of. Let defining the follo wing C 2 -real v alued functions: W 1 = − ln I − ¯ χg ′ (0) µ ( S 0 − S ) W 2 = − ln S − ln( Λ µ − S − I ) W 3 = χ − ln χ W = M W 1 + W 2 + N W 3 Applying Ito’s formula for the functions ab o v e,and letting h ( I ) = g ( I ) I w e hav e L ( − ln I ) = − χ S h ( I ) + µ + γ + α = − χ S 0 h (0) + µ + γ + α + χ S 0 h (0) − χ S 0 h ( I ) + χS 0 h ( I ) − χ S h ( I ) = − χ S 0 h (0) + µ + γ + α + χS 0 ( h (0) − h ( I )) + χh ( I )( S 0 − S ) (3) Note that h (0) = lim x → 0 g ( I ) I = lim x → 0 g ( I ) − g (0) I = g ′ (0) and − p ≤  g ( I ) I  ′ = g ( I ) I − lim x → 0 g ( I ) I I = g ( I ) I − g ′ (0) I ≤ 0 Then we hav e h (0) − h ( I ) = g ′ (0) − g ( I ) I ≤ p I and h ( I ) = g ( I ) I ≤ g ′ (0) By substituting we get 5 L ( − ln I ) ≤ − χ S 0 g ′ (0) + µ + γ + α + χ S 0 p I + χg ′ (0)( S 0 − S ) = − ¯ χ S 0 g ′ (0) + µ + γ + α + ( ¯ χ − χ ) S 0 g ′ (0) + ( χ − ¯ χ ) S 0 p I + ¯ χ S 0 p I + ( χ − ¯ χ ) g ′ (0)( S 0 − S ) + ¯ χg ′ (0)( S 0 − S ) ≤ − ¯ χ S 0 g ′ (0) + µ + γ + α + ( ¯ χ − χ ) + S 0 g ′ (0) + ( χ − ¯ χ ) + S 0 p I + ¯ χ S 0 p I + ( χ − ¯ χ ) + g ′ (0)( S 0 − S ) + ¯ χg ′ (0)( S 0 − S ) ≤ − ¯ χ S 0 g ′ (0) + µ + γ + α + ( ¯ χ − χ ) + S 0 g ′ (0) + ( χ − ¯ χ ) + ( S 0 ) 2 p + ¯ χ S 0 p I + ( χ − ¯ χ ) + g ′ (0) S 0 + ¯ χg ′ (0)( S 0 − S ) ≤ − ¯ χ S 0 g ′ (0) + µ + γ + α + 2 D g ′ (0) S 0 + D p ( S 0 ) 2 + ¯ χg ′ (0)( S 0 − S ) + ¯ χ S 0 p I + (( ¯ χ − χ ) + − D ) S 0 g ′ (0) + (( χ − ¯ χ ) + − D )( S 0 ) 2 p + (( χ − ¯ χ ) + − D ) g ′ (0) S 0 (4) Then we hav e: L ( − ln I ) ≤ − ( µ + γ + α + 2 D S 0 g ′ (0) + D ( S 0 ) 2 p )  Λ ¯ χg ′ (0) µ ( µ + γ + α + 2 D S 0 g ′ (0) + D ( S 0 ) 2 p ) − 1  + ¯ χg ′ (0)( S 0 − S ) + ¯ χ S 0 p I + F 1 ( χ ) + F 2 ( χ ) + F 3 ( χ ) ≤ − ( µ + γ + α + 2 D S 0 g ′ (0) + D ( S 0 ) 2 p ) ( R s 0 − 1) + ¯ χg ′ (0)( S 0 − S ) + ¯ χ S 0 p I + F 1 ( χ ) + F 2 ( χ ) + F 3 ( χ ) (5) Where R s 0 = Λ ¯ χg ′ (0) µ ( µ + γ + α + 2 D S 0 g ′ (0) + D ( S 0 ) 2 p ) And F 1 ( χ ) = (( ¯ χ − χ ) + − D ) S 0 g ′ (0) , F 2 ( χ ) = (( χ − ¯ χ ) + − D )( S 0 ) 2 p, F 3 ( χ ) = (( χ − ¯ χ ) + − D ) g ′ (0) S 0 . Applying Ito formula to − S + I µ w e hav e L − ( S + I µ ) = − ( S 0 − S ) − µ + α µ I (6) Then we hav e for W 1 L W 1 ≤ − ( µ + γ + α + 2 D S 0 g ′ (0) + D ( S 0 ) 2 p ) ( R s 0 − 1) + ( ¯ χS 0 p + ¯ χg ′ (0)( µ + α ) µ ) I + F 1 ( χ ) + F 2 ( χ ) + F 3 ( χ ) (7) F or W 2 w e hav e: L W 2 = − Λ S + Λ − µ ( S + I ) − α I Λ µ − ( S + I ) + χg ′ (0) I + µ ≤ − Λ S − α I Λ µ − S − I + χg ′ (0) S 0 + 2 µ (8) F or W 3 w e hav e L W 3 =  1 − 1 χ  ( r ( ¯ χ − χ )) + σ 2 2 6 L W ≤ − λM + A + M ( ¯ χ S 0 p + ¯ χg ′ (0)( µ + α ) µ ) I − Λ S − α I Λ µ − S − I + ( g ′ (0) S 0 − N r ) χ − N r ¯ χ χ + M F 1 ( χ ) + M F 2 ( χ ) + M F 3 ( χ ) (9) Where ( λ = ( µ + γ + α + 2 D S 0 g ′ (0) + D ( S 0 ) 2 p ) ( R s 0 − 1) A = 2 µ + N σ 2 2 + N r + N r ¯ χ T aking G ( S , I , χ ) = − λM + A + M ( ¯ χ S 0 p + ¯ χg ′ (0)( µ + α ) µ ) I − Λ S − α I Λ µ − S − I + ( g ′ (0) S 0 − N r ) χ − N r ¯ χ χ Then we hav e L W ≤ : G ( S , I , χ ) + M F 1 ( χ ) + M F 2 ( χ ) + M F 3 ( χ ) Let now construct a compact set H =  ( S , I , χ ) ∈ Γ | ϵ ≤ S , ϵ ≤ I , , S + I ≤ Λ µ − ϵ 2 , ϵ ≤ χ ≤ 1 ϵ  suc h that G ( S , I , χ ) ≤ − 1 for any ( S , I , χ ) ∈ Γ \ H := H c let H c = S 5 i =1 H c i , where H c 1 = { ( S , I , χ ) ∈ Γ | I ∈ (0 , ϵ ) } , H c 2 = { ( S , I , χ ) ∈ Γ | S ∈ (0 , ϵ ) } , H c 3 =  ( S , I , χ ) ∈ Γ | S + I ∈ ( Λ µ − ϵ 2 , ∞ ) , I ∈ [ ϵ, + ∞ )  , H c 4 = { ( S , I , χ ) ∈ Γ | χ ∈ (0 , ϵ ) } , H c 5 =  ( S , I , χ ) ∈ Γ | χ ∈ ( 1 ϵ , ∞ )  , By denoting K = M ( ¯ χ S 0 p + ¯ χg ′ (0)( µ + α ) µ ) Λ µ − 2 and choosing ϵ = 1 M 2 with N = M 3 and M is large enough to make the following inequalities true:            − λM ≤ − 2 , − 2 + M ( ¯ χ S 0 p + ¯ χg ′ (0)( µ + α ) µ ) ϵ ≤ − 1 , − min(Λ ,α,N r ¯ χ ) ϵ + K ≤ − 1 ( g ′ (0) Λ µ − N r ) ϵ + K ≤ − 1 7 1 st case : If ( S , I , χ ) ∈ H c 1 , than we hav e G ( S , I , χ ) ≤ − 2 + M ( ¯ χ S 0 p + ¯ χg ′ (0)( µ + α ) µ ) I ≤ − 2 + M ( ¯ χS 0 p + ¯ χg ′ (0)( µ + α ) µ ) ϵ ≤ − 1 . (10) 2 nd case : If ( S , I , χ ) ∈ H c 2 , than we hav e G ( S, I χ ) ≤ − 2 + M ( ¯ χ S 0 p + ¯ χg ′ (0)( µ + α ) µ ) I − Λ S ≤ K − Λ ϵ ≤ − min(Λ , α, N r ¯ χ ) ϵ + K ≤ − 1 (11) 3 st case : If ( S , I , χ ) ∈ H c 3 , than we hav e G ( S , I I , χ ) ≤ − α I Λ µ − S − I − 2 + M ( ¯ χS 0 p + ¯ χg ′ (0)( µ + α ) µ ) I ≤ − α ϵ + K ≤ − min(Λ , α, N r ¯ χ ) ϵ + K ≤ − 1 (12) 4 th case : If ( S , I , χ ) ∈ H c 4 , than we hav e G ( S , I , χ ) ≤ − N r ¯ χ χ − 2 + M ( ¯ χS 0 p + ¯ χg ′ (0)( µ + α ) µ ) I ≤ − N r ¯ χ ϵ + K ≤ − min(Λ , α, N r ¯ χ ) ϵ + K ≤ − 1 (13) 5 th case : If ( S , I χ ) ∈ H c 5 ,than we hav e G ( S , I , χ ) ≤ ( g ′ (0) Λ µ − N r ) χ − 2 + M ( ¯ χS 0 p + ¯ χg ′ (0)( µ + α ) µ ) I ≤ ( g ′ (0) Λ µ − N r ) ϵ + K ≤ − 1 (14) Summarizing the Five cases depicted ab o ve, one can deduce that G ( S , I , χ ) ≤ − 1 for all ( S , I , χ ) ∈ H c . Alternativ ely , since W tends to ∞ as ∥ ( S , I , χ ) ∥ → ∞ or approach the b oundary of Γ , we can ensure the existence of a p oin t  ˜ S , ˜ I , ˜ χ  in the interior of Γ where W  ˜ S , ˜ I , ˜ χ  attains its minimum., So we can construct a non-negativ e C 2 -function Ψ = W − W  ˜ S , ˜ I , ˜ χ  ,Then Applying Itô’s formula to this function gives us: L Ψ ≤ G ( S , I , χ ) + M F 1 ( χ ) + M F 2 ( χ ) + M F 3 ( χ ) 8 By taking the exp ectation and integrating we get: E Ψ ( S ( t ) , I ( t ) , χ ( t )) t ≥ 0 = E Ψ ( S (0) , I (0) , χ (0)) t + 1 t Z t 0 E ( L Ψ ( S ( τ ) , I ( τ ) , χ ( τ ))) d τ ≤ E Ψ (( S (0) , I (0) , χ (0)) t + 1 t Z t 0 E ( G ( S ( τ ) , I ( τ ) , χ ( τ ))) d τ M E ( F 1 ( χ )) + M E ( F 2 ( χ )) + M E ( F 3 ( χ )) . (15) Using the ergo dicité of χ we hav e: lim t → + ∞ E ( F 1 ( χ )) = 0 a.s. lim t → + ∞ E ( F 2 ( χ )) = 0 a.s. and lim t → + ∞ E ( F 3 ( χ )) = 0 a.s. On the other hand let B 0 = sup ( S , I ,χ ) ∈ Γ G ( S , I , χ ) and B = max ( B 0 , − 1) + 1 then : G ( S , I , χ ) ≤ B , ∀ ( S , I , χ ) ∈ Γ lim inf t → + ∞ 1 t Z t 0 E ( G ( S ( τ ) , I ( τ ) , χ ( τ ))) d τ = lim inf t → + ∞ 1 t Z t 0 E ( G ( S ( τ ) , I ( τ ) , χ ( τ ))) 1 { ( S ( τ ) , I ( τ ) ,χ ( τ )) ∈ H } d τ + lim inf t → + ∞ 1 t Z t 0 E ( G ( S ( τ ) , I ( τ ) , χ ( τ ))) 1 { ( S ( τ ) , I ( τ ) ,χ ( τ )) ∈ H c } d τ ≤ B lim inf t → + ∞ 1 t Z t 0 1 { ( S ( τ ) , I ( τ ) ,χ ( τ )) ∈ H } d τ − lim inf t → + ∞ 1 t Z t 0 1 { ( S ( τ ) , I ( τ ) ,χ ( τ )) ∈ H c } d τ ≤ − 1 + ( B + 1) lim inf t → + ∞ 1 t Z t 0 1 { ( S ( τ ) , I ( τ ) ,χ ( τ )) ∈ H } d τ . (16) Therefore, we hav e lim inf t → + ∞ 1 t Z t 0 1 { ( S ( τ ) , I ( τ ) ,χ ( τ )) ∈ H } dτ ≥ 1 B + 1 > 0 ..as Let P ( t, S ( τ ) , I ( τ ) , χ ( τ ) , Ω) as the transition probabilit y of ( S ( τ ) , I ( τ ) , χ ( τ )) b elongs to the set ω . Making the use of F atou’s lemma, we hav e lim inf t → + ∞ 1 t Z t 0 P ( τ , S ( τ ) , I ( τ ) , χ ( τ ) , L ) dτ ≥ 1 B + 1 > 0 , , as.. Witc h completes the pro of. 9 5 Conclusion In this w ork, a stochastic SIS epidemic model with nonlinear incidence rate is in v estigated to analyze its spreading behavior. In contrast to the approac h considered by Zhenfeng Shi and Daqing Jiang in [10], the p ositivit y of the transmission parameter χ is accoun ted for by p erturbing it with Mean-reverting inhomogeneous geometric Brownian motion process instead of the logarithmic Ornstein–Uhlen b ec k pro cess. Under this consideration, the existence of a unique and global p ositiv e solution to the system is established. In addition, a suitable C 2 -function is emplo yed to sho w that, under certain assumptions, R s 0 = Λ ¯ χg ′ (0) µ ( µ + γ + α + 2 D S 0 g ′ (0) + D ( S 0 ) 2 p ) > 1 the model admits at least one stationary distribution. 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