Horizontal versus vertical transmission of parasites in a stochastic spatial model

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1 Mathematical Biosciences 168 (2000) 1±8 Horizontal versus vertical transmission of parasites in a stochastic spatial model Rinaldo B. Schinazi * Department of Mathematics, University of Colorado, Colorado Springs, CO , 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, introduced 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.: ; fax: address: schinazi@math.uccs.edu (R.B. Schinazi) /00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S (00)

2 2 R.B. Schinazi / Mathematical Biosciences 168 (2000) 1±8 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 vertically 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 u 1 and u 2 denote the densities of susceptible individuals and of infected individuals, respectively. u 0 1 ˆ k 1u 1 1 u 1 u 2 =KŠ u 1 bu 1 u 2 ; u 0 2 ˆ k 2u 2 1 u 1 u 2 =KŠ u 2 bu 1 u 2 : In words, healthy and infected individuals give birth at rates k 1 and k 2, 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 rate b. Infected and susceptible individuals die at the same rate 1. The parameter K corresponds to the limiting density of the population. Lipsitch et al. [5] observe that the parasite may invade an uninfected host population if bk 1 1=k 1 k 2 =k 1 > 1: Note that bk 1 1=k 1 corresponds to horizontal transmission while k 2 =k 1 corresponds to vertical transmission. So the condition above says that if the vertical 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 site x on the lattice Z d 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 state g, let n 1 x; g and n 2 x; g be the number of nearest neighbors of x (among the 2d nearest neighbors of x) that are in states 1 and 2, respectively. Assume that the model is in state g. Then the state at a given site x evolves as 0! 1 at rate k 1 n 1 x; g cn 2 x; g ; 0! 2 at rate k 2 n 2 x; g ; 1! 2 at rate bn 2 x; g ; 1! 0 at rate 1; 2! 0 at rate 1: In words, healthy individuals give birth to healthy children at rate k 1. Infected individuals give birth to healthy children at rate c or to infected children at rate k 2. Infected individuals infect healthy individuals at rate b. Compared to the deterministic model we introduced above we have an additional parameter c to take into account the fact that in many cases a fraction of the children of infected individuals are healthy. For instance 20% of the children of women infected with HIV are infected in the womb, see Ref. [6]. However, by treating the mother recent research

3 R.B. Schinazi / Mathematical Biosciences 168 (2000) 1±8 3 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 hypoxylon are 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 diminished birth rate k 2. Note that k 2 is the vertical transmission rate and that b is the horizontal transmission rate. We make the biologically meaningful assumptions k 2 6 k 1 and c 6 k 1 : 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 k 1 n 1 x; g ; 1! 0 at rate 1: It is known that there is a critical parameter k c (that depends on the dimension d) for the contact process such that if k 1 6 k c 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 with k 1 6 k c is 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 on Z with k 2 < k c and c ˆ 0. Then for any b > 0 the infected individuals die out. That is, starting with a 2 at the origin and an arbitrary configuration of 0s and 1s elsewhere, the 2s will die out after a finite time with probability one. (b) Consider the spatial stochastic model on Z d, d P 1, with k 2 ˆ 0 and b > k c. If c is large enough, 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 in d ˆ 1 and with nearest neighbor interaction. In contrast, Theorem 1(b) shows that even with no vertical transmission (k 2 ˆ 0), 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 any d P 1. For a proof of Theorem 1(b) see the proof of Theorem 1(b) in [10]. Theorem 2. Consider the spatial stochastic model on Z d, d P 1, starting with an infected individual at the origin and an arbitrary configuration of 0s and 1s elsewhere. (a) If min k 2 ; b > k c, for any c P 0 and in any d P 1, there may be an epidemic. That is, there is a positive probability that there will be 2s at all times. (b) If min k 2 ; b > k c, c ˆ 0 and d ˆ 1, then the 1s may die out. That is, there is a positive probability that for any site x there is a finite time after which there will never be a 1 at x.

4 4 R.B. Schinazi / Mathematical Biosciences 168 (2000) 1±8 (c) If k 1 > k 2, for any c P 0 and in any d P 1, there is a b c > 0(depending on k 1 and k 2 ) such that if b < b c, 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 if k 1 is much larger than k 2,if min k 2 ; b > k c, then we may get 100% infection of the population. We believe that this result holds in any d P 1 but our proof uses in a crucial way d ˆ 1. Theorem 2(c) tells us that even if the parasite has low virulence, i.e., k 2 is almost as large as k 1, 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., k 2 < k c ) 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 nonspatial 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 k 2 ˆ c ˆ 0 and there is evidence that an epidemic is possible in d ˆ 2 even in this extreme case (no births from infected individuals). This is in sharp contrast with our result in d ˆ 1 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 ˆ 1). 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 k 1 ˆ k 2 ˆ c ˆ 0 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 ˆ 1 there can be no epidemic. In [14] it is proved that starting from any initial con guration there can be no epidemic in d ˆ 1. 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 d P 2 are computed. In [16] a Shape Theorem for the case k 1 ˆ k 2 ˆ c ˆ 0 is proved in d ˆ 2. In [17] a spatial epidemic in dimension 2 for which births of healthy individuals occur spontaneously is introduced. It is proved there that an epidemic may occur provided that the infection rate is high enough. In [18] the phase diagram of the model in [17] is studied. In [10] the model with c was introduced. The present paper seems to be the rst one for which a spatial model is used to analyze vertical transmission.

5 R.B. Schinazi / Mathematical Biosciences 168 (2000) 1±8 5 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 stochastic 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 stochastic 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., k 1 much larger than k 2 ) may infect the whole population provided k 2 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 representations. 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). Let g t denote our epidemic process. It is easy to see that g t has less 2s than the basic contact process n t that evolves according to the rules 0! 2 at rate max k 2 ; b n 2 x; n ; 2! 0 at rate 1: We start n t and g t with a single 2 at the origin, the other sites of n 0 are empty and the other sites of g 0 are lled with 0s and 1s in an arbitrary way. Note that if a 2 in g t has a nearest neighbor which is a 1, then this 1 becomes a 2 at rate b. Ifa2ing t has a nearest neighbor which is a 0, then this 0 becomes a 2 at rate k 2. For n t, a 2 always gives birth at rate max k 2 ; b. Since the birth rate of 2s is larger for n t than for g t and that the death rates are the same there are more 2s in n t than in g t. That is, we may construct both processes in the same probability space in such a way that if there isa2atx for g t, then there is a 2 at x for n t.ifb 6 k 2, since we are also assuming that k 2 < k c,we have that the 2s in n t die out. So the 2s in g t die out as well and this proves Theorem 1(a) if b 6 k 2. We now turn to the more interesting case b > k 2. We will couple our epidemic process, g t, to the boundary contact process introduced in [20]. The boundary contact process is a process on Z, starting with a single 2 at the origin and 0s everywhere else. If the leftmost 2 is at `t and the

6 6 R.B. Schinazi / Mathematical Biosciences 168 (2000) 1±8 rightmost 2 is at r t, then births of 2s at `t 1 and at r t 1 occur at rate b. At all the other sites particles give birth at rate k 2. Particles die at rate 1. Note that the boundary contact process is a slight modi cation of the basic contact process and that if b ˆ k 2, 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 f t are paired in increasing order. In particular, the leftmost 2 of f t is paired to the leftmost 2 of g t and the rightmost 2 of f t is paired to the rightmost 2 of g t. For more details on this coupling, see Lemma 2.2 in [20]. Assuming that b > k 2, the rightmost and leftmost 2s of f t give birth more often than the rightmost and leftmost 2s of g t. This is so because if there is a 1 at R t 1, where R t is the rightmost 2ofg t, then a 2 appears at R t 1 at rate b, while if there is a 0 at R t 1, then a 2 appears at R t 1 at rate k 2. For f t, the birth rate at r t 1 is constantly b > k 2. Since at the boundaries f t gives more births than g 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 in f t than there are in g t. However, in [20, Theorem 2] it is proved that the 2s die out f t if k 2 < k c. The same must hold for g t 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 k 2 ; b n 2 x; n ; 2! 0 at rate 1: Start n t with a single 2 at the origin and 0s everywhere else. Start g t with a 2 at the origin and 0s and 1s everywhere else. This time it is easy to see that g t has more 2s than n t at all times. This is so because the death rates are identical but the birth rates of 2s for g t are larger than the birth rates for n t. Since we are assuming that min k 2 ; b > k c we know that the 2s in n t 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). Assuming that min k 2 ; b > k c we know that the event A that 2s in n t will survive forever has a positive probability. Let `t and r t be the leftmost and rightmost 2s of n t, respectively. Conditioned on A we have that lim t!1 `t t r t ˆ a and lim t!1 t ˆ a; where a > 0 (see Ref. [8]). Let L t and R t the leftmost and rightmost 2s of g t, respectively. As noted in the proof of Theorem 2(a), the processes n t and g t may be constructed in the same probability space in such a way that g t has more 2s than n t at all times. In particular, we have for all t P 0 that L t 6 `t and R t P r t : So L t and R t increase at least linearly. Moreover, it is easy to check that there are only 2s and 0s in L t ; R t Š (it is at this point that we make essential use of d ˆ 1, nearest neighbor interactions and

7 R.B. Schinazi / Mathematical Biosciences 168 (2000) 1±8 7 c ˆ 0). Conditioned on A, 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]. Let A be a nite subset of Z d. We will show that the following holds for small b. There exists an a.s. nite (random) time T A such that the space±time region A T A ; 1 ( Z d 0; 1 ) contains no 2s. This statement implies the claim of the theorem. We prove this under the assumption that d ˆ 2. We de ne two nested space±time regions as A ˆ 2L; 2LŠ 2 0; 2T Š; B ˆ L; LŠ 2 T ; 2T Š; where L and T are integers to be chosen later. We will compare the process g t to a certain dependent percolation process on the set L ˆ Z 2 Z, where Z ˆf0; 1; 2;...g. We say that the site k; m; n in L is wet if 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 ML; MLŠ 2 0; T Š kl; ml; nt. Note that the event f k; m; n is wetg depends 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 interaction with only nite range. Sites which are not wet are called dry. We start by considering the process g t with b ˆ 0 and c ˆ 0. 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 that k 1 > k 2, T ˆ L 2. For any d > 0, there are L and M such that there exist no 2s in the box kl; ml; nt B, for the multitype contact process restricted to the finite box ML; MLŠ 2 0; T Š kl; ml; nt, with probability at least 1 d=2. See Proposition 3.2 and Lemma 3.7 in [23]. Using this lemma, we get that if b ˆ 0; then P k; m; n is wet P 1 d=2: It is easy to check that allowing c > 0 does not change the lemma above. If b ˆ 0, then a multitype contact with c > 0 has fewer 2s (and more 1s) than a multitype with c ˆ 0. That is, if the system with c ˆ 0 has a 1 at some site then the system with c > 0 has also a 1 at the same site. If the system with c > 0 has a 2 at some site, then the system with c ˆ 0 has also a 2 at the same site. One can check that no transition can break this property, assuming c < k 1. Since the space time box kl; ml; nt B is nite there is b c > 0 such that if b < b c there are no attempted infections (2s infecting 1s) in kl; ml; nt B with probability at least 1 d=2. Therefore, for any c P 0, if b < b c ; then P k; m; n is wet P 1 d: We now position oriented edges between sites in L in order to obtain a percolation model. Let A k; m; n ˆ kl; ml; nt A. For each pair k; m; n ; x; y; z 2L, we draw an oriented edge from k; m; n to x; y; z if and only if n 6 z and A k; m; n \A x; y; z 6ˆ;. The wet sites on the ensuing directed graph constitute a (dependent) percolation model. Let T 0 be the supremum of all times t such that there is a 2 at the origin at time t.ift 0 > t, then the origin is occupied by a 2 after

8 8 R.B. Schinazi / Mathematical Biosciences 168 (2000) 1±8 time t. 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 than t. This corresponds with a path of dry sites of length at least t=t for the percolation model. By taking d small enough, it is easy to see that the probability of a path of length t=t decays exponentially fast with t. Thus, T 0 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. References [1] L. Gross, Pathogenic properties and vertical transmission of the mouse leukemia agent, Proc. Soc. Exp. Biol. Med. 78 (1951) 342. [2] P. Fine, E. Sylvester, Calculation of vertical transmission rates of infection illustrated with data on an aphid-borne virus, Am. Nat. 112 (1978) 781. [3] P. Kover, K. Clay, Trade-o between virulence and vertical transmission and the maintenance of a virulent plant pathogen, Am. Nat. 152 (1998) 165. [4] M. Lipsitch, S. Siller, M. Nowak, The evolution of virulence in pathogens with vertical and horizontal transmission, Evolution 50 (1996) [5] M. Lipsitch, M. Nowak, D. Ebert, R. May, The population dynamics of vertically and horizontally transmitted parasites, Proc. Roy. Soc. London Ser. B 260 (1995) 321. [6] J. Driscoll, Human Immunode ciency Virus Infections and Acquired Immune De ciency Syndrome, third ed., The Columbia University College of Physicians and Surgeons Complete Home Medical Guide, Crown Publishers, [7] M. McCarthy, Perinatal AIDS decreasing rapidly in USA, Lancet 354 (1999) 573. [8] T. Liggett, Interacting Particle Systems, Springer, New York, [9] C. Bezuidenhout, G. Grimmett, The critical contact process dies out, Ann. Probab. 18 (1990) [10] R. Schinazi, On an interacting particle system modeling an epidemic, J. Math. Biol. 34 (1996) 915. [11] D. Mollison, Spatial contact models for ecological and epidemic spread, J. Roy. Statist. Soc. Ser. B 39 (1977) 283. [12] K. Sato, H. Matsuda, A. Sasaki, Pathogen invasion and host extinction in lattice structured populations, J. Math. Biol. 32 (1994) 251. [13] T. Harris, Contact interactions on a lattice, Ann. Probab. 2 (1974) 969. [14] E. Andjel, R. Schinazi, A complete convergence theorem for an epidemic model, J. Appl. Probab. 33 (1996) 741. [15] K. Kuulasmaa, The spatial general epidemic and locally dependent random graphs, J. Appl. Probab. 19 (1982) 745. [16] J.T. Cox, R. Durrett, Limit theorems for the spread of epidemics and forest res, Stochastic Processes Appl. 30 (1988) 171. [17] R. Durrett, C. Neuhauser, Epidemics with recovery in D ˆ 2, Ann. Appl. Probab. 1 (1991) 189. [18] J. van den Berg, G. Grimmett, R. Schinazi, Dependent random graphs and spatial epidemics, Ann. Appl. Probab. 8 (1998) 317. [19] R. Durrett, Mutual invadability implies coexistence in spatial models, Preprint, [20] R. Durrett, R. Schinazi, Boundary modi ed contact processes, J. Theoret. Probab., 1999, to appear. [21] R. Durrett, R. Schinazi, Asymptotic critical value for a competition model, Ann. Appl. Probab. 3 (1993) [22] C. Neuhauser, Ergodic theorems for the multi-type contact process, Probab. Theory Related Fields 91 (1992) 467. [23] R. Durrett, C. Neuhauser, Coexistence results for some competition models, Ann. Appl. Probab. 7 (1997) 10.