STOCHASTIC SIS AND SIR MULTIHOST EPIDEMIC MODELS
ROBERT K. MCCORMACK AND LINDA J. S. ALLEN
Pathogens that infect multiple hosts are common. Zoonotic diseases, such as Lyme dis- ease, hantavirus pulmonary syndrome, and rabies, by their very definition are animal dis- eases transmitted to humans. In this investigation, we develop stochastic epidemic models for a disease that can infect multiple hosts. Based on a system of deterministic epidemic models with multiple hosts, we formulate a system of Itostochasticdiˆ fferential equations. Through numerical simulations, we compare the dynamics of the deterministic and the stochastic models. Even though the deterministic models predict disease emergence, this is not always the case for the stochastic models.
Copyright © 2006 R. K. McCormack and L. J. S. Allen. This is an open access article dis- tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop- erly cited.
1. Introduction Most pathogens are capable of infecting more than one host. Often these hosts, in turn, transmit the pathogen to other hosts. Approximately sixty percent of human pathogens are zoonotic including diseases such as Lyme disease, influenza, sleeping sickness, rabies, and hantavirus pulmonary syndrome [18]. Generally, there is only a few species (often only one species) considered reservoir species for a pathogen. Other species, infected by the pathogen, are secondary or spillover species, where the disease does not persist. For example, domestic dogs and jackals in Africa may both serve as reservoirs for the ra- bies virus [7, 12]. Humans and other wild carnivores are secondary hosts. Hantavirus, a zoonotic disease carried by wild rodents, is generally associated with a single reser- voir host [2, 11, 13]. Spillover infection occurs in other rodent species. Human infection results in either hantavirus pulmonary syndrome or hemorrhagic fever with renal syn- drome [13]. To study the role played by multiple reservoirs and secondary hosts, in previous re- search, we developed deterministic epidemic models with multiple hosts and showed that the disease is more likely to emerge with multiple hosts [10]. In this research, we
Hindawi Publishing Corporation Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 775–785 776 Stochastic SIS and SIR multihost epidemic models extend the deterministic models to stochastic models and compare the stochastic and the deterministic dynamics.
2. Deterministic models We describe the deterministic multihost epidemic models developed in previous research and summarize their dynamics [10]. In the first model, known as an SIS epidemic model, individuals in the host population are either susceptible, S, or infected (and infectious), I. When individuals recover, they do not develop immunity but become susceptible again. In the multihost SIS epidemic model, let Sj and Ij denote the total number of susceptible and infected hosts of species j, respectively, j = 1,2,...,n. Then the SIS model is given by the following system of equations:
n dSj Ik = Njbj − Sj dj Nj − Sj βjk Nk + γjIj , dt Nk k=1 (2.1) n dIj Ik =−Ijdj Nj + Sj βjk Nk − γj + αj Ij , dt Nk k=1 where Sj(0) > 0, Ij(0) ≥ 0, and Nj = Sj + Ij for j = 1,2,...,n.Theparameterbj is the birth rate, γj is the recovery rate, and αj is the disease-related death rate. All parameters are positive. The contact rate between an infected individual of species j and a suscep- tible individual of species k is dependent on the population size of species k: βjk(Nk). We assume two forms for βjk(Nk): standard incidence, where βjk(Nk) ≡ λjk,andmass action incidence, where βjk(Nk) ≡ λjkNk. The density-dependent natural death rate also depends on the population size Nj, and satisfies the following assumptions: (i) dj ∈ C1[0,∞); (ii) 0
n dSj Ik = Nj bj − Sjdj Nj − Sj βjk Nk , dt Nk k=1 n dIj Ik =−Ijdj Nj + Sj βjk Nk − γj + αj Ij , (2.2) dt Nk k=1
dRj =−R d N γ I dt j j j + j j, where Sj(0) > 0, Ij(0) ≥ 0, Rj(0) ≥ 0, and Nj = Sj + Ij + Rj for j = 1,2,...,n.Allpa- rameters are interpreted as in the SIS model except that recovered individuals develop R. K. McCormack and L. J. S. Allen 777 immunity. It should be noted that the only interaction between the hosts is through con- tact and spread of disease. Competition and predator-prey interactions are not considered [5, 6, 8, 15–17]. The basic reproduction number 0 for the multihost SIS and SIR epidemic mod- els can be defined using the next generation approach of Diekmann et al. [3]andvan den Driessche and Watmough [14]. The multihost SIS and SIR epidemic models have a unique disease-free equilibrium (DFE), where Ij ≡ 0 ≡ Rj and Sj = Kj . The next gener- ation matrix for a multihost SIS or SIR epidemic model with n hosts is an n × n matrix M = n n ( jk)j,k=1,where
Kj βjk Kk jk = (2.3) Kk γk + αk + bk is the jk entry in the matrix Mn, j,k = 1,2,...,n. Hence, the basic reproduction number for the epidemic models is the spectral radius of Mn,
0 = ρ Mn . (2.4)
The DFE of the multihost SIS and SIR epidemic models is locally asymptotically stable if 0 < 1 and unstable if 0 > 1[14]. In addition, it can be shown that as the number of hosts increases, so does the basic reproduction number. In particular, if one more host is added to the system and the original parameters do not change, then
ρ Mn ≤ ρ Mn+1 . (2.5)
This result holds because Mn and Mn+1 are nonnegative matrices and Mn is the leading principal submatrix of Mn+1 of order n [10]. It also holds for more complex epidemic models such as a multihost SEIR epidemic model. As a result, multiple reservoirs and secondary species that become infected and transmit the disease can contribute to persis- tence of the disease in a multihost system by increasing 0.
3. Stochastic models We formulate new stochastic differential equation (SDE) models for the multihost SIS and SIR epidemic models described in the previous section. Variability in the stochastic models is due to births, deaths, and infections. Let j , Ᏽj,andj denote continuous random variables for the susceptible, infected, and immune states, respectively. Random variables are denoted by calligraphic letters. The random variables for the SIS model satisfy j ,Ᏽj ∈ [0,∞), where j + Ᏽj = ᏺj and ᏺj ∈ [0,∞). Let the random vector
T ᐄ(t) = 1(t),2(t),...,n(t),Ᏽ1(t),Ᏽ2(t),...,Ᏽn(t) . (3.1) 778 Stochastic SIS and SIR multihost epidemic models
The SDE formulation is based on a Markov chain model with a small time step Δt,where Δᐄ(t) = ᐄ(t + Δt) − ᐄ(t) is approximately normally distributed [1, 9]. First, we make assumptions regarding the probability of a change in state during Δt. We assume there can be a change of at most one unit, ±1, during Δt.LetΔ(t) = Δj (t + Δt) − j (t)andΔᏵ(t) = Ᏽ(t + Δt) − Ᏽ(t). Then
Prob Δj = 1 | ᐄ(t) = bj ᏺjΔt + o(Δt), Prob Δj =−1 | ᐄ(t) = j dj ᏺj Δt + o(Δt), Prob ΔᏵj =−1 | ᐄ(t) = Ᏽjdj ᏺj + αj Ᏽj Δt + o(Δt), (3.2) Prob Δj = 1, ΔᏵj =−1 | ᐄ(t) = γj IjΔt + o(Δt),
Ᏽk Prob Δj =−1, ΔᏵk = 1 | ᐄ(t) = j βjk ᏺk Δt + o(Δt). ᏺk
Applying these transition probabilities, the expected rate of change E(Δᐄ(t)) satisfies
⎛ n ⎞ Ᏽk ⎜ ᏺ1b1 − 1d1 ᏺ1 − 1 β1k ᏺk + γ1Ᏽ1 ⎟ ⎜ ᏺk ⎟ ⎜ k= ⎟ ⎜ 1 ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ n ⎟ ⎜ Ᏽ ⎟ ⎜ k ⎟ ⎜ ᏺnbn − ndn ᏺn − n βnk ᏺk + γnᏵn ⎟ ⎜ ᏺk ⎟ ⎜ k=1 ⎟ Δt . ⎜ n ⎟ + o( ) (3.3) ⎜ Ᏽk ⎟ ⎜ −Ᏽ1d1 ᏺ1 + 1 β1k ᏺk − γ1 + α1 Ᏽ1 ⎟ ⎜ ᏺk ⎟ ⎜ k=1 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ n ⎟ ⎝ Ᏽk ⎠ −Ᏽndn ᏺn + n βnk ᏺk − γn + αn Ᏽn ᏺk k=1
Denote this 2n vector as μ(ᐄ(t))Δt + o(Δt). The stochastic variability for the system comes from the covariance for the rate of change in the state variables. The 2n × 2n co- variance matrix C(Δᐄ(t)) to order Δt is
⎛ ⎞ C11 C12 C Δᐄ(t) = ⎝ ⎠Δt, (3.4) C21 C22 R. K. McCormack and L. J. S. Allen 779 where n Ᏽk C11 = diag ᏺjbj + j dj ᏺj + j βjk ᏺk + γjᏵj , ᏺk k=1 (3.5) n Ᏽk C22 = diag Ᏽj dj ᏺj + j βjk ᏺk + γj + αj Ᏽj ᏺk k=1 are n × n submatrices. Matrix C(ᐄ(t)) is symmetric, so that the n × n submatrices C12 C C = CT = c and 21 satisfy 12 21 ( jk), where
⎧ ⎪ Ᏽk ⎪− β ᏺ − γ Ᏽ j = k ⎨ j jk k ᏺ j j if , c = K jk ⎪ (3.6) ⎪ Ᏽk ⎩−jβjk ᏺk if j = k. ᏺK
To order Δt, it follows that
Δᐄ(t) = E Δᐄ(t) + C Δᐄ(t) . (3.7)
Let C11 C12 B ᐄ(t) = , (3.8) C21 C22 where the Cij are defined by (3.5)and(3.6). Matrix B is the unique positive definite square root. Taking the limit as Δt → 0of(3.7), we obtain a system of Itoˆ SDEs [1, 4, 9],
dᐄ t dᐃ t ( ) = μ ᐄ t B ᐄ t ( ) dt ( ) + ( ) dt , (3.9)
T where ᐃ(t) = (ᐃ1(t),...,ᐃ2n(t)) , and each ᐃj(t), j = 1,...,2n, is an independent Wiener process. A system of SDEs for the multihost SIR epidemic model can be formulated in a similar manner. Let ᐄ(t) denote the random vector
T 1(t),2(t),...,n(t),Ᏽ1(t),Ᏽ2(t),...,Ᏽn(t),1(t),2(t),...,n(t) . (3.10)
The transition probabilities for the multihost SIR model are similar in form to the mul- tihost SIS model given in (3.2).TheexpectedrateofchangeE(Δᐄ(t)) for the SIR model 780 Stochastic SIS and SIR multihost epidemic models satisfies ⎛ ⎞ n Ᏽk ⎜ ᏺ b − d ᏺ − β ᏺ ⎟ ⎜ 1 1 1 1 1 1 1k k ᏺ ⎟ ⎜ k= k ⎟ ⎜ 1 ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ n Ᏽ ⎟ ⎜ k ⎟ ⎜ ᏺnbn − ndn ᏺn − n βnk ᏺk ⎟ ⎜ ᏺk ⎟ ⎜ k=1 ⎟ ⎜ n ⎟ ⎜ Ᏽ ⎟ ⎜ −Ᏽ d ᏺ β ᏺ k − γ α Ᏽ ⎟ ⎜ 1 1 1 + 1 1k k 1 + 1 1 ⎟ ⎜ ᏺk ⎟ Δt . ⎜ k=1 ⎟ + o( ) (3.11) ⎜ . ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ n ⎟ ⎜ Ᏽk ⎟ ⎜−Ᏽndn ᏺn + n βnk ᏺk − γn + αn Ᏽn⎟ ⎜ ᏺk ⎟ ⎜ k=1 ⎟ ⎜ ⎟ ⎜ − d ᏺ γ Ᏽ ⎟ ⎜ 1 1 1 + 1 1 ⎟ ⎜ ⎟ ⎜ . ⎟ ⎝ . ⎠ −ndn ᏺn + γnᏵn
Denote this 3n vector as μ(ᐄ(t))Δt + o(Δt). The covariance matrix for Δᐄ(t)isa3n × 3n matrix satisfying, to order Δt, ⎛ ⎞ C C 0 ⎜ 11 12 ⎟ ⎜ ⎟ C ᐄ(t) = ⎝C21 C22 C23⎠Δt, (3.12)
0 C32 C33 where the n × n submatrices are n Ᏽk C11 = diag ᏺjbj + jdj ᏺj + j βjk ᏺk , ᏺk k=1 n Ᏽk C22 = diag Ᏽj dj ᏺj + j βjk ᏺk + γj + αj Ᏽj , ᏺk (3.13) k=1 C33 = diag j dj ᏺj + γjᏵj , C23 = diag − γj Ᏽj = C32.
C C C = CT = c Submatrices 12 and 21 satisfy 12 21 ( jk), where
Ᏽk cjk =−j βjk ᏺk . (3.14) ᏺk Let B(ᐄ(t)) = (Cij). It follows that the system of Itoˆ SDEs for the SIR model is given by (3.9). R. K. McCormack and L. J. S. Allen 781
4. Numerical examples Two numerical examples illustrate the dynamics of the stochastic and deterministic mod- els. For both examples, there are three hosts in an SIS epidemic model, one reservoir species and two spillover species. Standard incidence is used in the first example, βjk(Nk)= λjk. Mass action incidence is used in the second example, βjk(Nk) = λjkNk. It is reasonable to assume that the reservoir species (j = 1) has a greater contact rate and longer period of infectivity than the spillover species. Therefore, we assume λ11 >λjk, λj1 >λjk,forj = 1 and k = 1, and γ1 <γj , j = 2,3 [10]. In the first example, we let λ11 = 3.5, λ1k = 0.3 = λkk, k = 2,3, λ21 = 0.6 = λ31,and λ23 = 0 = λ32 [10]. The disease is not spread between the spillover species. Hantavirus in rodents results in very few, if any, disease-related deaths. Therefore, we let αj = 0.01, j = 1,2,3. All species have the same birth rate, bj = 3 = b, j = 1,2,3, but different carrying capacities, Kj . The recovery rate for the reservoir species is γ1 = 0.55, and for the spillover species γj = 1, j = 2,3. The natural death rate for each species is given by dj (Nj) = a + (b − a)Nj/Kj , j = 1,2,3, where a = 0.5, K1 = 500, and Kj = 250, j = 2,3. The basic reproduction number corresponding to this first model is slightly greater than one, 0 = 1.0101. There exists a unique locally stable endemic equilibrium,
(S1,I1,S2,I2,S3,I3) ≈ (495,5,249.6,0.4,249.6,0.4). (4.1)
The simulation is initiated with a small number of infected individuals: S1(0) = 200, S2(0) = 100 = S3(0), I1(0) = 5, I2(0) = 1, and I3(0) = 0. The deterministic solution, one stochastic sample path, the mean of 1000 sample paths, and the frequency distribution at t = 50 for the first example are graphed in Fig- ures 4.1(a), 4.1(b), 4.1(c),and4.1(d), respectively. The deterministic solution approaches the endemic equilibrium given by (4.1). However, the sample paths and the frequency distribution show that this is not the case for the stochastic model. The means of the μ = . μ = . μ = . μ = . sample paths are close to the DFE: S1 498 9, S2 248 4, S3 248 3, I1 0 0023, μ = . μ = . t = I2 0 0001, and I3 0 0004 (at 50). The disease cannot persist even in the reservoir species. This is due to the variance inherent in the stochastic model and the relatively low endemic equilibrium values for the infected hosts. In addition, the basic reproduction number for the reservoir species is less than one (in the absence of the spillover species), 1 = . 0 0 983. In the second example, all of the parameter values and initial conditions are the same as in the first example, with the exception of the transmission rates and the carrying capacities. The carrying capacities are K1 = 1000 and K2 = 500 = K3. The values of the transmission rates at the carrying capacities satisfy β11(K1) = 3.5, β1k(Kk) = 0.3= βkk(Kk), k = 2,3, βj1(K1) = 2(β1j(Kj)), j = 2,3, and β23(K3) = 0 = β32(K2), so that they agree with the preceding example [10]. The basic reproduction number is 0 = 2.0202. There exists a locally stable endemic equilibrium given by S1,I1,S2,I2,S3,I3 ≈ (483.7,514.3,424.8,74.9,424.8,74.9). (4.2)
The deterministic solution, one stochastic sample path, the mean of 1000 sample paths, and the frequency distribution at t = 50 for the second example are graphed in 782 Stochastic SIS and SIR multihost epidemic models
S1 S2 S3 I1 I2 I3 600 600 600 10 10 10 400 400 400 5 5 5 200 200 200
0 0 0 0 0 0 050050050050050050 tttttt (a)
S1 S2 S3 I1 I2 I3 600 600 600 60 10 10 400 400 400 40 5 5 200 200 200 20
0 0 0 0 0 0 050050050050050050 tttttt (b)
S1 S2 S3 I1 I2 I3 600 600 600 10 10 10 400 400 400 5 5 5 200 200 200
0 0 0 0 0 0 050tttttt050050050050050 (c)
S1 S2 S3 I1 I2 I3 40 40 30 1000 1000 1000 30 30 20 20 20 500 500 500 10 10 10 0 0 0 0 0 0 400 500 600 0 200 400 0 200 400 −10 0 10 −10 0 10 −10 0 10 (d)
Figure 4.1. Solutions of the three-host deterministic and stochastic SIS epidemic model with standard incidence, 0 = 1.0101: (a) solution of the deterministic model; (b) one sample path of the stochastic model; (c) mean of 1000 sample paths of the stochastic model; (d) frequency distribution of the stochastic model at t = 50 based on 1000 sample paths. R. K. McCormack and L. J. S. Allen 783
S1 S2 S3 I1 I2 I3 1000 1000 1000 1000 1000 1000 800 600 500 500 500 500 500 400 200 0 0 0 0 0 050050050050050050 ttttt t (a)
S1 S2 S3 I1 I2 I3 1000 1000 1000 800 1000 1000 600 500 500 500 400 500 500 200 0 0 0 0 0 0 050050050050050050 ttttt t (b)
S1 S2 S3 I1 I2 I3 1000 1000 1000 1000 1000 1000
500 500 500 500 500 500
0 0 0 0 0 0 050050050050050050 ttt tt t (c)
S1 S2 S3 I1 I2 I3 100 60 40 80 60 60 30 60 40 40 40 50 20 40 20 20 20 10 20 0 0 0 0 0 0 0 1000 2000 200 400 600 200 400 600 0 500 1000 0 100 200 0 100 200 (d)
Figure 4.2. Solutions of the three-host deterministic and stochastic SIS epidemic model with mass action incidence, 0 = 2.0202: (a) solution of the deterministic model; (b) one sample path of the stochastic model; (c) mean of 1000 sample paths of the stochastic model; (d) frequency distribution of the stochastic model at t = 50 based on 1000 sample paths.
Figures 4.2(a), 4.2(b), 4.2(c),and4.2(d), respectively. Because of the large reproduction number and high prevalence of disease, there is less chance for disease extinction than in the previous example. The means of the sample paths are close to the deterministic 784 Stochastic SIS and SIR multihost epidemic models
μ = . μ = . μ = . μ = . μ = . μ = . solution; S1 508 0, S2 427 3, S3 428 0, I1 489 3, I2 71 2, and I3 71 2(at t = 50). Because in a few of the sample paths there is disease extinction, the frequency distribution is bimodal, with one mode at the DFE and another at the endemic equilib- rium. In summary, for multihost epidemic models, the deterministic models show that dis- ease persistence can be enhanced by the presence of secondary or spillover species. How- ever, if the level of prevalence in these spillover species is relatively low and the reproduc- tion number in the reservoir host is less than one, the stochastic models show that the disease does not persist in the multihost system.
Acknowledgment
This research was supported by Grant R01TW006986-02 from the Fogarty International Center under the NIH NSF Ecology of Infectious Diseases Initiative.
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Robert K. McCormack: Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA E-mail address: [email protected]
Linda J. S. Allen: Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, USA E-mail address: [email protected]