Co-jumps and Markov counting systems in random environments

Co-jumps and Marko v counting systems in random en vironments Carles Bret ´ o 1 Departament o de Estadistic a and Insti tuto Flores de Lemus, Univer sidad Carlos III de Madrid, C / M adrid 126, Getafe, 28903, Madrid, Spain Abstract W e provide transition rates for M arkov counting systems su bject to correlated environmental noises motivated by multi-strain disease mo dels. Such no ises induce simultaneo us counts, which can help model infinitesimal cou nt correlation (regardless of whether such correlation is due to correlated noises). K eywor ds: Continuou s-time Markov chains, Infinitesimal moments, Compartmen tal models, Inf ectious disease models, En vironm ental stochasticity 1. Introduction Continuou s-time stochastic processes ha ve proved to be useful fo r research in many a reas of scien ce. Such pro - cesses are naturally d efined by infinitesimal para meter fu nctions, like tran sition rates in the case of Markov cha ins. Assuming that such p arameter functio ns are subjec t to extern al no ise has been re ferred to as a random e n vironm ent and has proved to be a useful ap proach to data analysis. A r ecent example of such applied work is th at in Shrestha et al. (2011). This w ork studied the dynam ics governing inter actions a mong multiple infectiou s path ogens; it found prom is- ing results based on Markov countin g systems for which m ultiple transition r ates are subject to a single, c ommon external noise. Such comm on noise results in correlation between these transition rates. Su bjecting transition rates to such noise is k nown to re sult in a new Markov counting system defined by new rates. However , these new rates hav e been derived in closed form only for the case of independen t no ises. This lack of closed-fo rm rates for the case of cor- related noises c reates uncertainty about the co rrelated s ystem properties and might make applied r esearchers reluctant to take advantage of the promising approach presented by Shrestha et al. (2011). T o pre vent such reticence, this paper considers intro ducing co rrelated external noises to the rates of Markov countin g systems an d pr ovides closed- form expressions for new rates that capture the e ff e ct of correlated no ise. These new rates are th e main co ntribution of the paper and are b ased on n ovel closed-form expressions for th e infinitesimal covariances of Markov counting systems. In addition, this unusual focus on system covariances pr ovides an alternati ve in terpretation of corr elated no ises in terms of simultaneous transitions in non-ra ndom en viron ments. The stud y o f pro perties of countin g processes ha s ben efited among others th e fields of e pidemiolo gy an d ecol- ogy , which hav e re lied on Mar kov c ounting sy stems both in deterministic and stoch astic e n vironm ents ( the latter sometimes bein g fav o ured b y empir ical evidence). Research in these two disciplin es has taken advantage of theoret- ical investigation of c ounting system s bo th historically ( Bartlett, 1 956; Kermack and McK endrick, 19 27) and m ore recently (Cauch emez and Fer guson, 200 8; He et al., 2010). Such counting systems can o ften be seen as systems of in- teracting Markov counting pro cesses or Markov counting systems (Bret ´ o and Ion ides, 201 1), which include networks of queues (Br ´ emaud, 1999) and com partmental models (Jacq uez, 1996; Matis and Ki ff e, 2000). Ma rkov cou nting sys- tems are Markov chains and are hence naturally defined by transition rates. Noisy transition rates are often referred to as environmental stocha sticity in epidem iology a nd ecolog y (Enge n et al., 1998). Th e role of such stoch asticity has been extensiv ely studied, including in the con text of determ inistic ODE skeletons dri ven by di ff usions (Dureau et al., 2013; Hu and W ang, 2011; Ionides et al., 2006; King et al., 2008) and driven by L ´ evy processes (Bhad ra et al., 2011; Email addr ess: carles.bret o@uc3m.es (Carles Bret ´ o) 1 T el. + 34916245855; Fax: + 3491624 9848 Prep rint submitted to Elsevi er J uly 31, 2018 S      ✒ λ 1 ξ 1 ❅ ❅ ❅ ❅ ❅ ❘ λ 2 ξ 2 I 1 ✲ r I 2 ✲ r S 2 ✲ (1 − γ ) λ 2 ξ 2 S 1 ✲ (1 − γ ) λ 1 ξ 1 I ∗ 2 ❅ ❅ ❅ ❅ ❅ ❘ r I ∗ 1      ✒ r R Figure 1: Multi-strain SIR-type compartmen tal model of Bret ´ o et al. (2009). This m odel w ill be used in Section 5 to illustra te our results. Each indi vidual fal ls in one compartment : S , suscepti ble to both strains ; I 1 , infecte d with strain 1; I 2 , infecte d with strain 2; S 1 , susceptibl e to strain 1 (but imm une to strai n 2); S 2 , susceptible to strain 2 (b ut immune to strain 1); I ∗ 1 , infe cted with strain 1 (but imm une to strain 2); I ∗ 2 , infe cted with strain 2 (but immune to strain 1); and R , immune to both strains. Regarding demography , births enter S from compartment B (not plotted), at rate b ( t ) driv en by birth data (which is treat ed as a co va riate), and all indivi duals hav e a common mortality rate m at which the y leav e each compartment in the diagram into D (not plotte d). Rega rding disease dynamics, r is the recove ry rate from infection; γ measures the strength of cross-immunit y between strains; and λ i is the per-c apita infect ion rate of strain i with ξ i being the stochast ic noise on this rate. Moreov er , λ i = β ( t )( I i ( t ) + I ∗ i ( t )) α / P ( t ) + ω , where 0 ≤ β ( t ) is parameteriz ed with a trend and a smoot h seasonal component, 0 ≤ ω models infections from an en vironmental reservoir and 0 ≤ α ≤ 1 captures inhomogeneous m ixing of the popul ation. Laneri et al., 2010). The r ole of stochastic environments has also be en studied in the con text of Markov co unting systems, b oth payin g attention to the system pr obabilistic prop erties (e.g ., Bret ´ o et al., 20 09; Ma rion and Renshaw, 2000; V arughe se and Fatti, 2 008) an d focusing o n the b iological im plications for application s (e.g., Shrestha et al., 2011). Epidemio logical applicatio ns have come to conside r multip le in teracting path ogens and to study th em based on counting systems subject to genu inely correlated environmental noise, fitting in the f ramework provid ed by Bret ´ o et al. (2009) who formalize the transition rates of the system subject to noise. Pathogen inter action has recei ved atten - tion for som e time n ow (Fenton and Pedersen, 2005; Kamo an d Sasaki, 200 2), both without considerin g the role of external no ises ( Aguiar et al., 2 011; Buckee et al., 20 11; Reich et al., 2013) and considerin g it (Bret ´ o et al., 200 9; Shrestha et al., 20 11). In particular, Shre stha et al. (201 1) consider a Markov coun ting system co rrespond ing to a co m- partmental model of the susceptible-infec tious-recovered type (a simpler version o f which is consid ered by Bret ´ o et al. (2009) and reprodu ced in Figure 1). I n Shrestha et al. (2011), two pathogens co -exist but th ere are m ore than two pos- sible di ff e rent types of infection (depend ing on the history of past infections of individuals). The rate at which th ese di ff erent types of infectio n occur are assumed to be subject to a single (commo n to all inf ection types) external white noise, making all infectio n rates correlated. Such rate correlatio n has been formalized by Bret ´ o et al. (2009), sho wing that the system sub ject to noises is a new Markov countin g system n ot on ly when the noises are ind ependen t (in which case they ev en provide closed-fo rm rates) but als o when the noises are correlated, on which this paper focuses. The problem we take up in this paper is providin g closed-f orm tran sition r ates that define Markov counting systems accountin g for correlated noises and is made di ffi cult b y the lack of closed-form transition probabilities of general sys- tems, w orkin g against the inclusion of biologically genuine noise correlation. Closed-form transition prob abilities are readily av ailable for basic systems, e.g., those of a Poisson process correspond to a Poisson distrib ution and those of a linear pu re death process to a b inomial distribution ( Bharucha- Reid , 19 60). Howe ver , they are not available fo r ge neral compartm ental models, including the s ystem of interacting birth-death processes conside red by Shrestha et al. (201 1 ) or that represen ted in Figu re 1. If such closed-f orm gen eral system transition pro babilities were available, then the desired closed-form transition rates might be p ursued by direct integration of the noise fro m those (now rando mized) probab ilities. Such d irect app roach is feasible in basic cases (like the biv ariate death pr ocess of Bret ´ o and Ionides, 2011) but n ot in mor e sop histicated mode ls, where the lack of closed -form ra tes casts a shadow over the appeal of correlated noises in applications of a realistic degree of complexity . A k ey downside of lacking closed-f orm rates is that the prom ising r esults of Sh restha et al. (2011) and the genuin e biological rationale beh ind corr elated noises may b e ou tweighted by uncertain ty abou t the p roper ties of th e model sub- ject to noise and about the interpretation of empirical results, which we seek to prev ent with this paper . Shrestha et al. (2011) show th at it is fe asible to a rrive at corr ect and precise biolog ical conclu sions regarding patho gen intera ction based on their Markov coun ting systems with co rrelated n oises. In additio n, a heuristic biolo gical justification for correlated no ises could be as fo llows: wh ile localize d environmental v ariations n eed not a ff ect all types of infection , 2 changes at a larger scale in the en vironm ent shou ld be expected to, like heat or cold w aves. Ho wev er, unless the proper ties of the mod el after subjecting it to correlated noise are clear and appealing, such correlations might be con- sidered a nuisanc e or something foreign and hence av oided in actu al applicatio ns, wher e empir ical findings need to be interpre ted (which migh t be d one more confidently in the co ntext of simpler mo dels). Pro viding a tool to help in such interpretatio n is the ultimate goal of this paper . The main contribution of this paper is to provide closed-form transition rates fo r Markov counting systems subject to correlated noises based on the system infinitesimal covariances and to provide an illustratio n in the context of biological analy sis of multi-strain pathoge n dynamics. The p rovided closed-form expressions apply to a broad ra nge of cases considered in the app lied literature. They reduce th e u ncertainty abo ut the m odel p roperties by gi ving a precise definition of the system as formalized in Section 2. In addition, they are moti vated by the system infinitesimal covariances derived in Sections 3 a nd 4, which allo w circu mventing th e above mentioned pro blem of u nav ailable transition pr obabilities from which to directly in tegrate out the noise. Our foc us on infinitesimal covariances is u nusual in the context of Markov counting systems (althou gh as natural as in th e context of multiv ariate di ff usions) and leads to the novel closed-form expressions for them provided in Th eorem 2. The se expressions sho w that correlated noises induce simultaneous counts and that these in turn induce stronger correla tions within the system. Hence, if additional correlation is demand ed by data, it could b e m odelled with random environmen ts. In this case, th ese environmen ts could be interpreted as devices that generate the needed correlation in a non-rando m environment, instead of as actual random chan ges in parameters (very much like par ameter rand omization can be interpr eted as a device to gene rate over -dispersion ). This is illustrated in Section 5, wher e the rates and interpretatio n of the role of corr elated noises for Figure 1 are given. 2. Markov counting systems without external noise Markov counting systems a re defin ed as M arkov c hains driven by a collection of interactin g countin g pr ocesses that fully c haracterizes th e transition r ates of th e system (Bret ´ o et al., 20 09) and suc h definition can often be formalize d in a diagr am (similar to tha t in Figure 1). Before fo rmally defining Markov counting systems, we intr oduce their key aspects. Fir st, consider a p opulation whose m embers ar e at any point in tim e in one (and only one) o f C possible stages (or compartm ents) of t heir lives, with s tages belonging to fi nite collection C . Next, let the number o f population members that are at stage c at time t defin e integer-v alued rand om variables X c ( t ), which make up the system { X ( t ) } ≡ { X c ( t ) : c ∈ C} . Then, let the nu mber of population memb ers th at have tran sitioned from stage i to stage j by time t define non- decreasing , integer-v alued rand om variables N i j ( t ), which in tur n, fo r all pairs ( i , j ) belongin g to a collection of allowed transitions T , define the collection of cou nting processes { N ( t ) } ≡ { N i j ( t ) : ( i , j ) ∈ T } . Next, let the collection { N ( t ) } drive the dy namics of the system { X ( t ) } via the “conservation of mass” identity X c ( t ) = X c (0) + X ( i , c ) ∈T N ic ( t ) − X ( c , j ) ∈T N c j ( t ) , (1) so that ch anges in { X ( t ) } are the resu lt of ch anges in { N ( t ) } . Mass con servation identity (1) restricts the tran sitions that can occur in { X ( t ) } as follows. Let N 0 ( N ) be th e natural nu mbers including (excludin g) zero and consider initial counts n ∈ N T 0 and initial system cond itions x ∈ N C 0 . For any given increments of the c ollection of cou nts ℓ ≡ { ℓ i j : ( i , j ) ∈ T } ∈ N T , the system { X ( t ) } mu st m ake transition s u ≡ { u c : c ∈ C } ∈ Z C with u c = P ( i , c ) ∈T ℓ ic − P ( c , j ) ∈T ℓ c j . Finally , let the following t ransition rates define the Markov chain { X ( t ) , N ( t ) } q ( x , ℓ ) ≡ lim h ↓ 0 P  N ( t + h ) = n + ℓ , X ( t + h ) = x + u | N ( t ) = n , X ( t ) = x  h . (2) Since the left hand side o f (2) only depen ds on x (and not n ), { X ( t ) } is itself a contin uous-time Markov chain and we call it a Markov countin g system 2 , which we illustrate with the following e xamp le. 2 The transiti on rates in (2) are time homogeneous, s ince its left hand side does not depend on t . This homogene ity adds clarity to the concepts, results and proofs bu t can readily be relaxed . 3 Figure 1 defines a Markov cou nting system by relying on the concep ts of marginal transition rates and of pairwise transition rates, wh ich are necessary for its interpr etation and wh ich are also key to stu dy th e e ff ect of correlated external noise. Consider th e rate at which k po pulation memb ers simultaneously u ndergo a transition of the i j -type (regardless o f whether o ther mem bers u ndergo o ther tran sitions), which can b e d efined as q i j ( x , k ) ≡ P ℓ : ℓ i j = k q ( x , ℓ ) for k ∈ N and which we call the ( i , j ) marginal transition rate. Margina l transition r ates o f size one q i j ( x , 1) are the labe ls on the a rrows in Fig ure 1 . Marginal r ates o f sizes greater tha n on e d o no t app ear in Figure 1 because they are assumed to be zero. Another assumptio n needed to in terpret Figure 1 is that there are no co-jumps of di ff erent typ es. T o fo rmalize this seco nd assum ption, con sider the rate at which k = ( k i j , k i ′ j ′ ) populatio n member s simultaneou sly undergo transitions o f the ( i , j ) and ( i ′ , j ′ ) typ es (regar dless of wheth er other members u ndergo other transitions), which can be defined as q i j , i ′ j ′ ( x , k ) ≡ P ℓ : ℓ i j = k i j ,ℓ i ′ j ′ = k i ′ j ′ q ( x , ℓ ) fo r k ∈ n N 2 0 − (0 , 0 ) o and which we call the ( i , j ) − ( i ′ , j ′ ) pairwise transition rate. Requiring a ll p airwise tra nsition rates to satisfy q i j , i ′ j ′ ( x , (1 , 0)) = q i j ( x , 1) and q i j , i ′ j ′ ( x , (0 , 1)) = q i ′ j ′ ( x , 1) guarantees no co-jum ps and allows interpr eting figures such as Figure 1 as Markov chains (see Ande rson and May, 1991; Br ´ ema ud, 199 9; Jacquez, 1 996). Such interpr etation also assumes that all non-zer o rates q i j ( x , 1) are deterministic functions of the chain s tate x and not subject to external noise. 3. Correlated external noise in bivariate death Marko v counting systems Introd ucing correlated noise to the rates is easier if one consider s two indep endent death pro cesses and results in both co-ju mps an d in infinitesimal covariance as stated in Pro position 1 belo w , which conside rs a comm on multiplica- ti ve g amma external noise, a co mmon death rate, and wh ich will later be usefu l wh en con sidering ge neral Markov counting systems. P ropo sition 1 was pr oved in Bret ´ o and Ion ides (201 1) and we state it here to make the paper self- contained . W e state it in ter ms o f our no tation fo r Markov cou nting sy stems, af ter in troducin g our notation f or the noise. The noise a ff ecting the individual d eath rates is assum ed to be c ontinuo us-time white noise ob tained f rom a gamma process, which is also the choice of noise in Bret ´ o et al. (2009) an d Shrestha et al. (2011). Gam ma wh ite noise is defined as { ξ ( t ) } ≡ { d Γ ( t ) / d t } with Γ ( t ) ∼ Gamma ( t /τ , τ ), E [ Γ ( t )] = t , and V [ Γ ( t )] = τ t , so that τ parameter izes the magnitud e of the noise. Proposition 1 ( Proposition 7 o f Bret ´ o a nd Ionides, 201 1) . Consider the biva riate Markov coun ting system { Y ( t ) } ≡  Y 1 ( t ) , Y 2 ( t )  defined by counting pr oc esses  N Y 1 D ( t ) , N Y 2 D ( t )  thr ou gh mass conservation equations Y i ( t ) = Y i (0) − N Y i D ( t ) , and by transition r a tes q Y i D ( y i , 1 ) = δ y i I { 0 < y i } i. e., two independen t linea r death pr oce sses having equal individual death rate δ ∈ R + and in itial po pulation sizes Y i (0) . Consider sub jecting bo th q Y i D ( y i , 1 ) to a commo n gamma white noise, w hich defines countin g pr ocesses { N ˜ Y i ˜ D ( t ) } = { N Y i D  Γ ( t )  } and the co rr esp onding Markov counting system { ˜ Y ( t ) } , i.e., two death pr ocesses each having stochastic rate δ ξ ( t ) . Then, the transition rates of { ˜ Y ( t ) } co rr e spond to pairwise tr ansition rates, letting k i = k ˜ Y i ˜ D ∈  N 2 0 − (0 , 0) : k i ≤ ˜ y i  , q ˜ Y 1 ˜ D , ˜ Y 2 ˜ D  ( ˜ y 1 , ˜ y 2 ) , ( k 1 , k 2 )  = ˜ y 1 k 1 ! ˜ y 2 k 2 ! k 1 + k 2 X j = 0 k 1 + k 2 j ! ( − 1) k 1 + k 2 − j + 1 τ − 1 ln  1 + δτ ( ˜ y 1 + ˜ y 2 − j )  . Furthermore , the infinitesimal covariance between { N ˜ Y 1 ˜ D ( t ) } and { N ˜ Y 2 ˜ D ( t ) } is lim h ↓ 0 h − 1 C ov " N ˜ Y 1 ˜ D ( t + h ) − N ˜ Y 1 ˜ D ( t ) , N ˜ Y 2 ˜ D ( t + h ) − N ˜ Y 2 ˜ D ( t )      ˜ Y ( t ) = ˜ y # = ˜ y 1 ˜ y 2 τ − 1 ln (1 + δτ ) 2 1 + 2 δ τ ! > 0 . The commo n rate assumption of Propo sition 1 can be relaxed at the cost of more complex clo sed-form expressions for the covariance and for the pairwise r ates. Although di ff erent dea th rates are assumed in the interacting dea th processes of Figure 1, they will be a ssumed to b e equal fo r the sake of simplicity w hen we illustrate in Section 5 our results for general Markov countin g systems (which hold re gard less of whether individual death rates are equa l). 4 4. Correlated external noise in genera l M arkov counting systems Consider generalizing Proposition 1 to general Marko v counting s ystems, which will l ead us to defining infinitesi- mal covariances of such general systems. Consider a general Markov cou nting system as defined by tr ansition rates (2) that satisfies the standard assumption s to interpret Figure 1 o f neith er m ultiple jumps no r co-jump s and denote such system by { W ( t ) } . Consider now subjecting some (or all) transition rates o f { W ( t ) } to a collection of ( possibly cor- related, not necessarily gamma) white n oises derived from { Γ ( t ) } (analo gously to Section 3) and call the r esulting process { ˜ W ( t ) } . The transitions rates of { ˜ W ( t ) } could be obta ined by integrating out th e n oises { Γ ( t ) } fr om th e (now random ized) tra nsition probabilities app earing in side the limit in (2) as follows. Con sider th e collec tion o f ra ndomized time inc rements H ≡ { H i j : ( i , j ) ∈ T } , necessarily with E [ H i j ] = h . The natur e of H i j depend s on whether q i j ( w , 1 ) is sub ject to noise: if y es, then H i j is the corr esponding noise rando m variable with d ensity f H i j ; if not, th en it is the degenerate random v ariable H i j = h . Then, provided they e xist, the transition rates of { ˜ W ( t ) } a re q ( ˜ w , ℓ ) = lim h ↓ 0 Z P  N ( t + s ) = n + ℓ , W ( t + s ) = w + u | N ( t ) = n , W ( t ) = w  s f H ( s ) d s . (3) While such direct integration of the noise in equation (3) was straightfor ward for the biv ariate death process of Prop o- sition 1 , it is not so straig htforward for mo re so phisticated m odels, like the one in Figure 1 (Bret ´ o et al., 200 9) o r similar models (Shrestha et al., 2011). Hence, instead of obtain ing the new transition rates by direct integration , we propo se constructin g such new rates by directly specifying transition rates that produce the s ame infinitesimal cov ari- ance as that produced by introducin g noise to appropriate biv ariate systems. In the case of Figure 1 , noises ξ i ( t ) a ff ect the rate of death pro cesses { N S I i } an d { N S i I ∗ i } . In this case, o ur pro posal amoun ts to specifyin g a n ew set of tra nsition rates for the system represented by Fi gure 1 such that the infinitesimal covariance between { N S I i } and { N S i I ∗ i } matches the cov ariance gi ven by Proposition 1 between { N Y 1 D } and { N Y 2 D } . T o do this, we first der i ve closed-form e xpression s for the infinitesimal cov ariances between two coun ting processes in volved in a Ma rkov counting system. 4.1. Infin itesimal covariance of Markov counting syst ems Define the in finitesimal covariances o f a gen eral Markov coun ting system { X ( t ) } as defined in Sectio n 2 as the collection { σ d X ( x ) } ≡ n σ i j , i ′ j ′ d X ( x ) : ( i , j ) , ( i ′ , j ′ ) ∈ T o of infinitesimal cov ariances between counting processes { N i j ( t ) } and { N i ′ j ′ ( t ) } : σ i j , i ′ j ′ d X ( x ) ≡ lim h ↓ 0 h − 1 C ov h N i j ( t + h ) − N i j ( t ) , N i ′ j ′ ( t + h ) − N i ′ j ′ ( t )    X ( t ) = x i . (4) Our closed-for m expressions below require one moment existence condition. Similar conditions were required by Theorem 1 of Bret ´ o and Ionides ( 2011) to provid e closed- form expression s fo r the infinitesimal me an and variance of Markov cou nting pr ocesses. Since Bret ´ o and Ionid es (2 011) ca ll their cond ition fo r the mean ( P 1 ⋆ ) and th at fo r the variance ( P 2 ⋆ ), we shall call o ur cond ition for covariances ( P 3 ⋆ ). ( P 3 ⋆ ) is related to the numb er of tra nsitions occurrin g in th e M arkov coun ting system over a time interval. This num ber of tran sitions is related n ot on ly to th e sizes o f th e incr ements of each cou nting proc ess { N i j ( t ) } but also to the overall rate at which these incr ements occu r , which we call the rate function of the Markov coun ting s ystem and define as λ ( x ) ≡ lim h ↓ 0 1 − P  N i j ( t + h ) − N i j ( t ) = 0 f or all ( i , j ) ∈ T    X ( t ) = x  h . (5) This quantity is also kno w as the intensity of the process in the point process literature (Daley and V ere-Jones, 2003). If the rate func tion satisfies that λ ( x ) = P ℓ q ( x , ℓ ) < ∞ fo r all x , then the process i s said to be stable and conservati ve. Consider stochastically bounding th e rate f unction and the incre ment o f each pair o f c ounting processes ov er [ t , t + ¯ h ] by: ¯ Λ ( t ) ≡ sup t ≤ s ≤ t + ¯ h λ  X ( s )  , ¯ Z i j , i ′ j ′ ( t ) ≡ sup n sup t ≤ s ≤ t + ¯ h d N i j ( s ) , sup t ≤ s ≤ t + ¯ h d N i ′ j ′ ( s ) o . (6) 5 A combination of these two bounds gi ves the following property: P3 ⋆ . For each t , x and ( i , j ) , ( i ′ , j ′ ) there is some ¯ h > 0 such that E h ¯ Z 2 i j , i ′ , j ′ ( t ) ¯ Λ ( t ) | X ( t ) = x i < ∞ . Property ( P 3 ⋆ ) r equires th at the Markov cou nting system do es no t h av e an explo si ve behaviour and holds, for ex- ample, for SIR-typ e com partmental models like that of Figure 1, as sh own in Section 5. It su ffi ces to guara ntee that infinitesimal covariances exist and that are given by the expression in Theorem 2 below . Theorem 2 (Infinitesimal covariances of a Mar kov c ounting sy stem) . Let { X ( t ) } be a time homogeneous Markov counting system defin ed b y counting pr o cesses { N ( t ) } an d by transition rates q ( x , ℓ ) as in (2) th at is stable an d co n- servative. Su pposing (P3 ⋆ ), the infinitesima l co variance between { N i j ( t ) } and { N i ′ j ′ ( t ) } is σ i j , i ′ j ′ d X ( x ) = X k k i j k i ′ j ′ q i j , i ′ j ′ ( x , k ) . Theorem 2 generalize s Theo rem 1 of Bret ´ o and Io nides (2011) to cov ariances and is proved in App endix A. 5. T ransition rates of SIR-type models subject to external correlated noises Theorem 2 can b e used to show that, after minimal simp lifications to ad d clarity to o ur contr ibution, the system represented in Figure 1 can be defin ed by transition r ates that reprod uce the e ff ects (identified in Propo sition 1) o f correlated noises. First, consider the Markov ch ain { Z ( t ) } ≡  S ( t ) , I 1 ( t ) , I 2 ( t ) , S 1 ( t ) , S 2 ( t ) , I ∗ 1 ( t ) , I ∗ 2 ( t ) , R ( t )  specified by Figur e 1. The standard interpretatio n of Figure 1 g i ves the tr ansition rates for { Z ( t ) } in T able 1 (e.g., letting th e ξ i be dete rministic con stants). Next, before con sidering adding noise to { Z ( t ) } , we make the following simplifications to { Z ( t ) } so that the con tributions in this p aper can be presented mor e clearly : ( i) instead of a time-inh omogen eous birth rate b ( t ), birth s c ompensate deaths so that the total p opulation size remains constan t and is equ al to P < ∞ , as in Shresth a et al. (201 1); (ii) instead of a time- inhomo geneou s infection rate β ( t ) with in λ i , this rate is constant a nd equal to β ; a nd (iii) in stead of d i ff erent transition rates fr om S to I i and fro m S i to I ∗ i , these rates are both λ i ξ i ( t ), i.e., γ = 0. Simp lifications (i) and ( ii) im pose time-ho mogene ity , which allows for a simpler notation in th e rest o f the paper . Simplification (iii) allows for simpler transition rates and cov ariance closed-form expressions. Now , let stable, conservati ve Mar kov ch ain { ˜ Z ( t ) } be defined by th e rates of T able 1 modified accor ding to (i)–(iii) an d by non -zero pairwise rates of transitions in volving the ξ i equal to those in Propo sition 1 as fo llows: q ˜ S ˜ I i , ˜ S i ˜ I ∗ i ( ˜ z , k ) = ˜ s k 1 ! ˜ s i k 2 ! k 1 + k 2 X j = 0 k 1 + k 2 j ! ( − 1) k 1 + k 2 − j + 1 τ − 1 ln  1 + λ i τ ( ˜ s + ˜ s i − j )  . System { ˜ Z ( t ) } d efined by these rates satisfies ( P 3 ⋆ ), since letting the fixed population size be ˜ P , λ  ˜ Z ( t )  = m  ˜ P − ˜ S ( t )  + r  ˜ I 1 ( t ) + ˜ I 2 ( t ) + ˜ I ∗ 1 ( t ) + ˜ I ∗ 2 ( t )  + X k q ˜ S ˜ I 1 , ˜ S 1 ˜ I ∗ 1  ˜ Z ( t ) , k  + X k q ˜ S ˜ I 2 , ˜ S 2 ˜ I ∗ 2  ˜ Z ( t ) , k  ≤ m + r + λ 1 + λ 2 ! ˜ P (7) where the inequality follows by substituting all comp artments by ˜ P and because X k q ˜ S ˜ I i , ˜ S i ˜ I ∗ i  ˜ Z ( t ) , k  = τ − 1 ln  1 + τλ i  ˜ S ( t ) + ˜ S i   ≤ λ i ˜ P (as follows from the pro perties of th e bino mial gamm a p rocess of Bret ´ o and Ionides, 20 11). Sin ce ( 7) is not time- varying, it a lso b ounds ¯ Λ ( t ) in volved in ( P 3 ⋆ ). Similarly , ˜ P is an uppe r bound for the incr ements: ¯ Z i j , i ′ j ′ ( t ) ≤ ˜ P , so that E h ¯ Z 2 i j , i ′ j ′ ( t ) ¯ Λ ( t ) | ˜ Z ( t ) = ˜ z i ≤ m + r + λ 1 + λ 2 ! ˜ P 3 6 ( P 3 ⋆ ) can be analog ously verified for the b i variate system { ˜ Y ( t ) } of Pr oposition 1 by assuming, f or example, that the initial population sizes are deterministic, i.e., for fixed y i (0) = ˜ y i (0) λ  ˜ Y ( t )  = X k q ˜ Y 1 ˜ D , ˜ Y 2 ˜ D  ˜ Y ( t ) , k  = τ − 1 ln  1 + τδ  ˜ Y 1 ( t ) + ˜ Y 2 ( t )   ≤ δ  ˜ y 1 (0) + ˜ y 2 (0)  . Hence, it follows directly from Theore m 2 that σ ˜ S ˜ I i , ˜ S i ˜ I ∗ i d ˜ Z ( ˜ z ) = ˜ s ˜ s i τ − 1 ln (1 + λ i τ ) 2 1 + 2 λ i τ ! > 0 . (8) Equation (8) can be inter preted as follows. First, th e e ff ects of intr oducing cor related noises in the biv ariate system of Prop osition 1 can be repr oduced in mo re gener al systems. Second, it per mits an a lternative interp retation of correlated noises ξ i ( t ) in a non-ran dom environment co ntext. These n oises have e ff ectively b een integrated out in { ˜ Z ( t ) } , wh ich can be seen as a regular Markov chain in a non-ra ndom en viro nment, wit h the caveat that it now allows for simultaneous co-transitions that driv e the ne w infinitesimal correlations. T able 1: Transit ion rates acc ording th e s tandard interpret ation of Figure 1 as a contin uous-time Marko v chain with rate functio n λ Z ( z ) ≡ P ( i , j ) ∈T q i j ( z , 1) and with all margi nal rates q i j ( z , k ) for k > 1 and all pairwise transition rates q i j , i ′ j ′ ( z , k ) assumed to be zero. 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Then lim h ↓ 0 h − 1 C ov h ∆ N i j ( t ) , ∆ N i ′ j ′ ( t )    X ( t ) = x i = lim h ↓ 0 h − 1 ( E h ∆ N i j ( t ) ∆ N i ′ j ′ ( t )    X ( t ) = x i − o ( h ) ) = lim h ↓ 0 h − 1 E h ∆ N i j ( t ) ∆ N i ′ j ′ ( t )    X ( t ) = x i . The rest of this pro of follows closely the pro of of Theorem 1 in Bret ´ o (20 12). Wh ile that theor em provided expressions for the infinitesimal mean and variance of { X ( t ) } , th is one provides them for the covariances. All pro babilities and exp ectations in this proof ar e condition al on X ( t ) = x (in add ition to other con ditioning, where app ropriate) . Define the following: (i) let { ¯ N i j ( t ) } be a p rocess su ch th at, conditional on ¯ Λ ( t ) and ¯ Z ( t ) ≡ ¯ Z i j , i ′ j ′ ( t ), realizations of { ¯ N i j ( t ) } are those of a compou nd Poisson pro cess (Cox and Isham, 198 0) with Poisson e vent rate ¯ Λ ( t ) and degenerate jump or batch size distribution (Daley and V ere-Jones, 2003) with mass one at ¯ Z ( t ), i.e., a process with jumps arriving accord ing to the Poisson pro cess and fo r which the size of the jumps is ¯ Z ( t ); and (ii) let S be the event that there i s exactly one transition time occurring in the interv al [ t , t + h ] in the Markov coun ting system { X ( t ) } . T hen, E h ∆ N i j ( t ) ∆ N i ′ j ′ ( t ) i = E h ∆ N i j ( t ) ∆ N i ′ j ′ ( t ) I { S } i + E h ∆ N i j ( t ) ∆ N i ′ j ′ ( t ) I { S c } i . (A.1) Consider the first term o n the rig ht hand side of ( A.1). Let S i j , i ′ j ′ ⊂ S be the event that there is exactly one transition time occurrin g in the interval [ t , t + h ] in the MCS { X ( t ) } and that this transition incr eases bo th the { N i j ( t ) } and the { N i ′ j ′ ( t ) } pr ocesses (and possibly other processes). Then, in (A.1), letting k ≡ ( k i j , k i ′ j ′ ) E [ ∆ N i j ( t ) ∆ N i ′ j ′ ( t ) I { S } ] = E [ ∆ N i j ( t ) ∆ N i ′ j ′ ( t ) | S i j , i ′ j ′ ] × P ( S i j , i ′ j ′ | S ) × P ( S ) = X k k i j k i ′ j ′ q i j , i ′ j ′ ( x , k ) P k q i j , i ′ j ′ ( x , k ) × P k q i j , i ′ j ′ ( x , k ) λ ( x ) × h h λ ( x ) + o ( h ) i (A.2) = h X k k i j k i ′ j ′ q i j , i ′ j ′ ( x , k ) + o ( h ) (A.3) 8 where P ( S ) in (A.2) follows by a standard result on Markov chains (see for example: R oss, 199 6, page 492). T o finish the pro of, we sho w that the second term on the righ t hand side of (A.1) disappear s infinitesimally . Let ¯ S be the event that there is e xactly one transition time occurring in the in terval [ t , t + h ] in the comp ound Poisson proce ss { ¯ N i j ( t ) } . Since the random variable ∆ N i j ( t ) ∆ N i ′ j ′ ( t ) is stoch astically smaller than  ∆ ¯ N i j ( t )  2 , E h ∆ N i j ( t ) ∆ N i ′ j ′ ( t ) I { S c } i ≤ E   ∆ ¯ N i j ( t )  2 I { ¯ S c }  = E " E h ∆ ¯ N i j ( t )  2    ¯ Λ ( t ) , ¯ Z ( t ) i # − E " E h ∆ ¯ N i j ( t )  2 I { ¯ S }    ¯ Λ ( t ) , ¯ Z ( t ) i # (A.4) = E h ¯ Z 2 ( t ) ¯ Λ ( t ) h i + E h ¯ Z 2 ( t ) ¯ Λ 2 ( t ) i h 2 | {z } = o ( h ) ! − E h ¯ Z 2 ( t ) ¯ Λ ( t ) h exp {− h ¯ Λ ( t ) } i (A.5) = E  ¯ Z 2 ( t ) ¯ Λ ( t ) h  1 − exp  − h ¯ Λ ( t )    + o ( h ) where (A.4) follows as in (A.1), an d ( A.5) follows by the pro perties of the co mpoun d Poisson d istribution. Since ¯ z 2 ¯ λ  1 − exp  − h ¯ λ   ≤ ¯ z 2 ¯ λ and E h ¯ Z 2 ( t ) ¯ Λ ( t ) i is assumed finite (note that the distrib ution of ¯ Z 2 ( t ) ¯ Λ ( t ) depe nds on ¯ h and not h ), it follows by dom inated con vergen ce that lim h ↓ 0 E  ¯ Z 2 ( t ) ¯ Λ ( t ) h  1 − exp  − h ¯ Λ ( t )    h = E  lim h ↓ 0 ¯ Z 2 ( t ) ¯ Λ ( t )  1 − exp  − h ¯ Λ ( t )    = 0 . 9