Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol168.Issue1.Nov2000:

Horizontal versus vertical transmission of parasites in a

stochastic spatial model

Rinaldo B. Schinazi

*

Department of Mathematics, University of Colorado, Colorado Springs, CO 80933-7150, USA

Received 3 December 1999; received in revised form 25 July 2000; accepted 16 August 2000

Abstract

A number of pathogens may be transmitted from parent to child at or before birth (vertically) or from one individual to another by contact (horizontally). A natural deterministic and non-spatial model, in-troduced by Lipsitch et al. [Proc. Roy. Soc. London Ser. B 260 (1995) 321] shows that an epidemic is possible if the vertical transmission or the horizontal transmission is high enough. In contrast, we introduce a stochastic spatial model that shows that, on a particular graph, if the vertical transmission is not high enough, then the infected individuals disappear even for very high horizontal transmission. This illustrates

the fact that introducing space may greatly change the qualitative behavior of a model. Ó _{2000 Elsevier}

Science Inc. All rights reserved.

MSC:60K35

Keywords:Vertical transmission; Horizontal transmission; Spatial stochastic model

1. Introduction and results

In this paper, we are interested in parasites that may be transmitted in two ways. The parasites may be transmitted from a parent to its o€spring at or before birth, in this case the transmission is said to be vertical. The parasites may also be transmitted by contact from an infected individual to an uninfected individual, in that case the transmission is said to be horizontal. A wide range of pathogens are transmitted by a combination of horizontal and vertical transmission, including HIV, hepatitis B and C, and rubella virus in man. Vertical transmission also exists in animals and plants. For instance, mammary tumor viruses in certain mouse strains, avian visceral

*_{Tel.: +1-719 262 3515; fax: +1-719 262 3605.}

E-mail address:schinazi@math.uccs.edu (R.B. Schinazi).

0025-5564/00/$ - see front matter Ó _{2000 Elsevier Science Inc. All rights reserved.}

lymphomatosis virus in chickens and mosaic viruses in plants are thought to be transmitted vertically, see Refs. [1,2]. Recently, there has been some interest in understanding the e€ect of vertical transmission on the evolution of virulence in pathogens with vertical and horizontal transmission, see Refs. [3,4]. Because of the number of pathogens that may be transmitted ver-tically and horizontally we feel it is important to understand the relative e€ects of vertical and horizontal transmissions in the propagation of a pathogen.

We start by formulating a particular case of a deterministic non-spatial model introduced in [5]. Let u1 and u2 denote the densities of susceptible individuals and of infected individuals, respec-tively.

u0₁ ˆk1u1‰1ÿ …u1‡u2†=KŠ ÿu1ÿbu1u2;

u0₂ ˆk2u2‰1ÿ …u1‡u2†=KŠ ÿu2‡bu1u2:

In words, healthy and infected individuals give birth at rates k1 and k2, respectively. Healthy individuals always give birth to healthy children and infected individuals always give birth to infected children. Healthy individuals get infected by infected individuals at rateb. Infected and susceptible individuals die at the same rate 1. The parameterKcorresponds to the limiting density of the population. Lipsitch et al. [5] observe that the parasite may invade an uninfected host population if

bK…1ÿ1=k1† ‡k2=k1 >1:

Note that bK…1ÿ1=k1† corresponds to horizontal transmission while k2=k1 corresponds to ver-tical transmission. So the condition above says that if the verver-tical or the horizontal transmission is high enough, then the parasite persists. We will now give an example of a spatial stochastic model for which this does not hold. For our example, if the vertical transmission is too low, then the parasite does not persist no matter how high the horizontal transmission is.

We formulate our spatial stochastic model. Each sitexon the latticeZd may be in one of three states: 0 denotes an empty site, 1 a healthy site and 2 is an infected site. If the model is in stateg, letn1…x;g†andn2…x;g† be the number of nearest neighbors ofx(among the 2d nearest neighbors ofx) that are in states 1 and 2, respectively. Assume that the model is in stateg. Then the state at a given site xevolves as

0!1 at rate k1n1…x;g† ‡cn2…x;g†;

0!2 at rate k2n2…x;g†;

1!2 at rate bn2…x;g†;

1!0 at rate 1;

2!0 at rate 1:

shows that the fraction of vertically infected children drops dramatically, see Ref. [7]. Note that there are known cases of very high fraction of vertical transmission. For instance, in [3] it is observed that 80% of the seeds of a Danthonia infected by the fungal pathogen Atkinsonella hypoxylonare infected.

To limit the number of parameters we are assuming that the death rates are the same for susceptible and infected individuals. The virulence of the parasite may be translated into a di-minished birth ratek2. Note thatk2 is the vertical transmission rate and thatb is the horizontal transmission rate. We make the biologically meaningful assumptions

k26k1 and c6k1:

Since there is no spontaneous appearance of particles, note that if we start with 1s only or 2s only, then the process evolves with only one type. Moreover, the process is then a basic contact process. That is, if we start with no 2s, for instance, then the process evolves as

0!1 at rate k1n1…x;g†;

1!0 at rate 1:

It is known that there is a critical parameterkc(that depends on the dimensiond) for the contact process such that ifk16kc then the 1s die out in the following two senses. If we start the contact process with a single 1 the 1s die out after a ®nite time with probability one. We also have that the only stationary distribution for the process withk16kcis the all 0s con®guration. For more on the contact process, see Refs. [8,9]. We need the critical parameter of the contact process to formulate the following results.

Theorem 1.

(a)Consider the spatial stochastic model onZwithk2 <kcandcˆ0.Then for anyb>0the in-fected individuals die out. That is,starting with a2at the origin and an arbitrary configuration of

0s and 1s elsewhere, the2s will die out after a finite time with probability one.

(b)Consider the spatial stochastic model on Zd_{, dP1,with k}

2ˆ0 andb>kc. Ifc is large

en-ough,then there is a positive probability that the 2s will survive at all times.

Theorem 1(a) shows that, unlike what happens for the deterministic model above, if the vertical transmission is too low, then the infected individuals die out even for very high horizontal transmission. Note that this result is proved only ind ˆ1 and with nearest neighbor interaction. In contrast, Theorem 1(b) shows that even with no vertical transmission (k2 ˆ0), if the infected individuals give birth to healthy individuals at a rate high enough, then there will be an epidemic, provided horizontal transmission is high enough. Note that Theorem 1(b) holds in anydP1. For

a proof of Theorem 1(b) see the proof of Theorem 1(b) in [10].

Theorem 2. Consider the spatial stochastic model onZd,dP1,starting with an infected individual at the origin and an arbitrary configuration of0s and 1s elsewhere.

(a)Ifmin…k2;b†>kc,for anycP0and in anydP1,there may be an epidemic. That is,there is a positive probability that there will be2s at all times.

(c)Ifk1 >k2,for anycP0and in anydP1,there is abc>0 (depending onk1 andk2)such that

if b<bc, then the infected individuals die out.

Theorem 2(a) is not surprising: it says that if vertical and horizontal transmissions are high, then an epidemic is possible. Theorem 2(b) is more interesting. Even ifk1is much larger thank2, if min…k2;b†>kc, then we may get 100% infection of the population. We believe that this result holds in anydP1 but our proof uses in a crucial waydˆ1. Theorem 2(c) tells us that even if the parasite has low virulence, i.e.,k2 is almost as large ask1, then the infected individuals die out if the horizontal transmission is not high enough.

2. Discussion

The main point of this paper is contained in Theorem 1. There, we give an example of a spatial stochastic model whose behavior is drastically di€erent from the corresponding deterministic model. For a similar result, see Ref. [11]. We show that if the vertical transmission is not high enough, then the infected individuals disappear from the population even for very high horizontal transmission. That is, high virulence of the virus (i.e., k2 <kc) cannot be compensated by high horizontal transmission. For the deterministic model high virulence may be compensated by high horizontal transmission. Our result suggests that whether an epidemic will occur may depend crucially on the spatial habitat of the population.

We think that the di€erence in behavior between our spatial stochastic model and the non-spatial deterministic model is due to the space aspect rather than to the stochastic aspect. To support our opinion we note that in [12] there is a simulation of our model in the particular case k2 ˆcˆ0 and there is evidence that an epidemic is possible ind ˆ2 even in this extreme case (no births from infected individuals). This is in sharp contrast with our result in dˆ1 and shows clearly that it is the spatial aspect that is determinant for this model. Theorem 1(b) also points towards the same conclusion. By allowing infected individuals to give birth to healthy individuals we provide some additional mixing of the infected and healthy populations and this allows the epidemic to spread (even in d ˆ1).

Spatial stochastic models for epidemics have been studied for at least 25 yr. In [13] the contact process was introduced. In [11] the spatial stochastic epidemic model with k1 ˆk2 ˆcˆ0 was studied and bounds on the velocity of a front for epidemics were computed. Kelly in [11, Discussion section] shows that starting with ®nitely many infected individuals in

d ˆ1 there can be no epidemic. In [14] it is proved that starting from any initial con®guration there can be no epidemic in dˆ1. The argument in Lemma 2.2 in [14] applies to a number of one-dimensional systems including the one in [11]. In [15] a class of models that includes the model in [11] is studied and bounds for the threshold of an epidemic in dP2 are computed. In

We think that our model is di€erent from the model in [5] because they are in some sense at two extreme opposites. In the deterministic non-spatial model there is somehow total mixing: every individual is in contact with all the other individuals. On the other extreme, in our spatial sto-chastic model individuals are just in contact with their nearest neighbors. A better model might be one with intermediate mixing, maybe a spatial model where individuals perform random walks at a certain rate on top of the current rules of the model. Note that it has been shown that if the rate of the random walk is fast enough, then, in many cases, there is convergence of the spatial sto-chastic model to the deterministic non-spatial model, see for instance Ref. [19] and the references therein. However, for intermediate rates for the random walks it is not clear how to approach the mathematical analysis of such a model.

Theorem 2(b) shows that even a highly virulent pathogen (i.e., k1 much larger than k2) may infect the whole population provided k2 and b are large enough. This is in agreement with the analysis of [5]. As always with interacting particle systems we do not get sharp conditions on the parameters. Such conditions are easy to get for the di€erential equations model. In contrast the reader will see in the proof of Theorem 2(b) that we get a pathwise analysis of how the 1s are driven out by the 2s that is not available for the di€erential equations model. This illustrates the fact that di€erential equations and spatial stochastic models complement each other by shedding light on di€erent aspects of a given question.

3. Proofs

Throughout this section, the processes considered are being generated by their graphical rep-resentations. That is to say, we are given appropriate families of Poisson processes which may be used to couple together the di€erent processes corresponding to di€erent initial conditions. Such constructions are standard, for an example, see Ref. [10].

Proof of Theorem 1(a).Letg_tdenote our epidemic process. It is easy to see thatg_t has less 2s than the basic contact processnt that evolves according to the rules

0!2 at rate max…k2;b†n2…x;n†;

2!0 at rate 1:

rightmost 2 is atrt, then births of 2s at `tÿ1 and atrt‡1 occur at rate b. At all the other sites particles give birth at rate k2. Particles die at rate 1. Note that the boundary contact process is a slight modi®cation of the basic contact process and that ifbˆk2, the two processes are identical. We use a coupling from [8, Chapter VI, Theorem 1.9]. We start the boundary contact process, f_t, and the epidemic process g_t with a single 2 at the origin. The other sites are empty for f_t and there is an arbitrary con®guration of 0s and 1s for g_t. At any given time the 2s in g_t and ft are paired in increasing order. In particular, the leftmost 2 off_tis paired to the leftmost 2 ofg_tand the rightmost 2 offt is paired to the rightmost 2 ofgt. For more details on this coupling, see Lemma 2.2 in [20].

Assuming that b>k2, the rightmost and leftmost 2s of ft give birth more often than the rightmost and leftmost 2s ofg_t. This is so because if there is a 1 atRt‡1, whereRtis the rightmost 2 ofg_t, then a 2 appears atRt‡1 at rateb, while if there is a 0 atRt‡1, then a 2 appears atRt‡1 at ratek2. Forft, the birth rate atrt‡1 is constantlyb>k2. Since at the boundariesftgives more births thang_t and that between the extreme 2s the birth rates are the same it is clear that at any given time there are more 2s inftthan there are ingt. However, in [20, Theorem 2] it is proved that the 2s die out f_t if k2 <kc. The same must hold forgt and this completes the proof of Theorem 1(a).

Proof of Theorem 2(a).We couple the epidemic process g_t to the following contact process n_t.

0!2 at rate min…k2;b†n2…x;n†;

2!0 at rate 1:

Startn_t with a single 2 at the origin and 0s everywhere else. Startg_t with a 2 at the origin and 0s and 1s everywhere else. This time it is easy to see thatg_thas more 2s thann_tat all times. This is so because the death rates are identical but the birth rates of 2s forg_tare larger than the birth rates for n_t. Since we are assuming that min…k2;b†>kc we know that the 2s in nt have a positive probability of not dying out. The same must be true for g_t and the proof of Theorem 2(a) is complete.

Proof of Theorem 2(b). Let n_t be the contact process de®ned in the proof of Theorem 2(a). As-suming that min…k2;b†>kc we know that the event A that 2s in nt will survive forever has a positive probability. Let`tandrt be the leftmost and rightmost 2s ofnt, respectively. Conditioned on Awe have that

lim t!1

`t

t ˆ ÿa and tlim!1

rt

t ˆa;

wherea>0 (see Ref. [8]). LetLtandRtthe leftmost and rightmost 2s ofgt, respectively. As noted in the proof of Theorem 2(a), the processesn_t andg_t may be constructed in the same probability space in such a way thatg_thas more 2s thanntat all times. In particular, we have for alltP0 that

Lt6`t and RtPrt:

cˆ0). Conditioned onA, we have a linearly growing region of 2s and 0s that drives away the 1s. This completes the proof of Theorem 2(b).

Proof of Theorem 2(c).We use ideas to be found in [21, Theorem 2]. LetAbe a ®nite subset ofZd. We will show that the following holds for smallb. There exists an a.s. ®nite (random) timeTAsuch that the space±time regionA ‰TA;1†(Zd ‰0;1†) contains no 2s. This statement implies the claim of the theorem. We prove this under the assumption that dˆ2. We de®ne two nested space±time regions as

A_{ˆ ‰ÿ2L;2LŠ}2 _‰0;2TŠ; B_{ˆ ‰ÿL;LŠ}2 _‰T;2TŠ;

whereL and T are integers to be chosen later. We will compare the process g_t to a certain de-pendent percolation process on the set L_ˆZ2_{Z‡, whereZ‡ˆ f0;1;2;}

. . .g. We say that the

site…k;m;n†inL_{iswetif there exist no 2s in the box…kL;mL;nT† ‡}B_{. Moreover, we want these} events to occur for the process restricted to the ®nite box‰ÿML;MLŠ2 ‰0;TŠ ‡ …kL;mL;nT†. Note that the eventf…k;m;n† is wetgdepends only on the existence (or not) of Poisson marks within a ®nite box. This is to ensure that the percolation process in L_{, though dependent, has an} inter-action with only ®nite range. Sites which are not wet are calleddry.

We start by considering the processg_t withbˆ0 and cˆ0. This process has been introduced in [22] under the name multitype contact process. In [23] the following lemma is proved for the multitype contact process.

Lemma (Durrett±Neuhauser).Assume thatk1>k2,T ˆL2_{.For anyd>0,there are L and M such}

that there exist no2s in the box…kL;mL;nT† ‡B_{,for the multitype contact process restricted to the} finite box‰ÿML;MLŠ2 ‰0;TŠ ‡ …kL;mL;nT†, with probability at least 1ÿd=2.

See Proposition 3.2 and Lemma 3.7 in [23]. Using this lemma, we get that

if bˆ0; then P……k;m;n† is wet†P1ÿd=2:

It is easy to check that allowingc>0 does not change the lemma above. Ifbˆ0, then a multitype contact withc>0 has fewer 2s (and more 1s) than a multitype with cˆ0. That is, if the system with cˆ0 has a 1 at some site then the system with c>0 has also a 1 at the same site. If the system withc>0 has a 2 at some site, then the system withcˆ0 has also a 2 at the same site. One can check that no transition can break this property, assumingc<k1.

Since the space time box…kL;mL;nT† ‡B_{is ®nite there isb}

c>0 such that ifb<bcthere are no attempted infections (2s infecting 1s) in …kL;mL;nT† ‡B _{with probability at least 1ÿd=2.} Therefore, for anycP0,

if b<bc; thenP……k;m;n† is wet†P1ÿd:

timet. This can happen only if there is a chain of 2s that starts somewhere at time 0 and goes to the origin at some time larger thant. This corresponds with a path of dry sites of length at least

t=T for the percolation model. By takingdsmall enough, it is easy to see that the probability of a path of length t=T decays exponentially fast with t. Thus, T0 is almost surely ®nite and this completes the proof of Theorem 2(c). For more details see the proof of Theorem 4.4 in [18].

Acknowledgements

We thank two anonymous referees and the editor for their many suggestions that helped to improve this paper.

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