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Research article Special Issues

Pathogens stabilize or destabilize depending on host stage structure

  • Received: 01 August 2023 Revised: 31 October 2023 Accepted: 31 October 2023 Published: 10 November 2023
  • A common assumption is that pathogens more readily destabilize their host populations, leading to an elevated risk of driving both the host and pathogen to extinction. This logic underlies many strategies in conservation biology and pest and disease management. Yet, the interplay between pathogens and population stability likely varies across contexts, depending on the environment and traits of both the hosts and pathogens. This context-dependence may be particularly important in natural consumer-host populations where size- and stage-structured competition for resources strongly modulates population stability. Few studies, however, have examined how the interplay between size and stage structure and infectious disease shapes the stability of host populations. Here, we extend previously developed size-dependent theory for consumer-resource interactions to examine how pathogens influence the stability of host populations across a range of contexts. Specifically, we integrate a size- and stage-structured consumer-resource model and a standard epidemiological model of a directly transmitted pathogen. The model reveals surprisingly rich dynamics, including sustained oscillations, multiple steady states, biomass overcompensation, and hydra effects. Moreover, these results highlight how the stage structure and density of host populations interact to either enhance or constrain disease outbreaks. Our results suggest that accounting for these cross-scale and bidirectional feedbacks can provide key insight into the structuring role of pathogens in natural ecosystems while also improving our ability to understand how interventions targeting one may impact the other.

    Citation: Jessica L. Hite, André M. de Roos. Pathogens stabilize or destabilize depending on host stage structure[J]. Mathematical Biosciences and Engineering, 2023, 20(12): 20378-20404. doi: 10.3934/mbe.2023901

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  • A common assumption is that pathogens more readily destabilize their host populations, leading to an elevated risk of driving both the host and pathogen to extinction. This logic underlies many strategies in conservation biology and pest and disease management. Yet, the interplay between pathogens and population stability likely varies across contexts, depending on the environment and traits of both the hosts and pathogens. This context-dependence may be particularly important in natural consumer-host populations where size- and stage-structured competition for resources strongly modulates population stability. Few studies, however, have examined how the interplay between size and stage structure and infectious disease shapes the stability of host populations. Here, we extend previously developed size-dependent theory for consumer-resource interactions to examine how pathogens influence the stability of host populations across a range of contexts. Specifically, we integrate a size- and stage-structured consumer-resource model and a standard epidemiological model of a directly transmitted pathogen. The model reveals surprisingly rich dynamics, including sustained oscillations, multiple steady states, biomass overcompensation, and hydra effects. Moreover, these results highlight how the stage structure and density of host populations interact to either enhance or constrain disease outbreaks. Our results suggest that accounting for these cross-scale and bidirectional feedbacks can provide key insight into the structuring role of pathogens in natural ecosystems while also improving our ability to understand how interventions targeting one may impact the other.



    Recently, [1] introduced the process SH,K={SH,Kt,t0} on the probability space (Ω,F,P) with indices H(0,1) and K(0,1], named the sub-bifractional Brownian motion (sbfBm) and defined as follows:

    SH,Kt=12(2K)/2    (BH,Kt+BH,Kt),

    where {BH,Kt,tR} is a bifractional Brownian motion (bfBm) with indices H(0,1) and K(0,1], namely, {BH,Kt,tR} is a centered Gaussian process, starting from zero, with covariance

    E[BH,KtBH,Ks]=12K[(|t|2H+|s|2H)K|ts|2HK],

    with H(0,1) and K(0,1].

    Clearly, the sbfBm is a centered Gaussian process such that SH,K0=0, with probability 1, and Var(SH,Kt)=(2K22HK1)t2HK. Since (2H1)K1<K10, it follows that 2HK1<K. We can easily verify that SH,K is self-similar with index HK. When K=1, SH,1 is the sub-fractional Brownian motion (sfBm). For more on sub-fractional Brownian motion, we can see [2,3,4,5] and so on. The following computations show that for all s,t0,

    RH,K(t,s)=E(SH,KtSH,Ks)=(t2H+s2H)K12(t+s)2HK12|ts|2HK (1.1)

    and

    C1|ts|2HKE[(SH,KtSH,Ks)2]C2|ts|2HK, (1.2)

    where

    C1=min{2K1,2K22HK1},    C2=max{1,222HK1}. (1.3)

    (See [1]). [6] investigated the collision local time of two independent sub-bifractional Brownian motions. [7] obtained Berry-Esséen bounds and proved the almost sure central limit theorem for the quadratic variation of the sub-bifractional Brownian motion. For more on sbfBm, we can see [8,9,10].

    Reference [11] studied the limits of bifractional Brownian noises. [12] obtained limit results of sub-fractional Brownian and weighted fractional Brownian noises. Motivated by all these studies, in this paper, we will study the increment process {SH,Kh+tSH,Kh,t0} of SH,K and the noise generated by SH,K and see how close this process is to a process with stationary increments. In principle, since the sub-bifractional Brownian motion is not a process with stationary increments, its increment process depends on h.

    We have organized our paper as follows: In Section 2 we prove our main result that the increment process of SH,K converges to the fractional Brownian motion BHK. Section 3 is devoted to a different view of this main result and we analyze the noise generated by the sub-bifractional Brownian motion and study its asymptotic behavior. In Section 4 we prove limit theorems to the sub-bifractional Brownian motion from a correlated non-stationary Gaussian sequence. Finally, Section 5 describes the behavior of the tangent process of sbfBm.

    In this section, we prove the following main result which says that the increment process of the sub-bifractional Brownian motion SH,K converges to the fractional Brownian motion with Hurst index HK.

    Theorem 2.1. Let K(0,1). Then, as h,

    {SH,Kh+tSH,Kh,t0}d{BHKt,t0},

    where d means convergence of all finite dimensional distributions and BHK is the fractional Brownian motion with Hurst index HK.

    In order to prove Theorem 2.1, we first show a decomposition of the sub-bifractional Brownian motion with parameters H and K into the sum of a sub-fractional Brownian motion with Hurst parameter HK plus a stochastic process with absolutely continuous trajectories. Some similar results were obtained in [13] for the bifractional Brownian motion and in [14] for the sub-fractional Brownian motion. Such a decomposition is useful in order to derive easier proofs for different properties of sbfBm (like variation, strong variation and Chung's LIL).

    We consider the following decomposition of the covariance function of the sub-bifractional Brownian motion:

    RH,K(t,s)=E(SH,KtSH,Ks)=(t2H+s2H)K12(t+s)2HK12|ts|2HK                 
        =[(t2H+s2H)Kt2HKs2HK]
                                  +[t2HK+s2HK12(t+s)2HK12|ts|2HK]. (2.1)

    The second summand in (2.1) is the covariance of a sub-fractional Brownian motion with Hurst parameter HK. The first summand turns out to be a non-positive definite and with a change of sign it will be the covariance of a Gaussian process. Let {Wt,t0} a standard Brownian motion, for any 0<K<1, define the process XK={XKt,t0} by

    XKt=0(1eθt)θ1+K2dWθ. (2.2)

    Then, XK is a centered Gaussian process with covariance:

    E(XKtXKs)=0(1eθt)(1eθs)θ1Kdθ                                                    
    =0(1eθt)θ1Kdθ0(1eθt)eθsθ1Kdθ          
    =0(t0θeθudu)θ1Kdθ0(t0θeθudu)eθsθ1Kdθ
    =t0(0θKeθudθ)dut0(0θKeθ(u+s)dθ)du          
    =Γ(1K)K[tK+sK(t+s)K],                                     (2.3)

    where Γ(α)=0xα1exdx.

    Therefore we obtain the following result:

    Lemma 2.1. Let SH,K be a sub-bifractional Brownian motion, K(0,1) and assume that {Wt,t0} is a standard Brownian motion independent of SH,K. Let XK be the process defined by (2.2). Then the processes {KΓ(1K)XKt2H+SH,Kt,t0} and {SHKt,t0} have the same distribution, where {SHKt,t0} is a sub-fractional Brownian motion with Hurst parameter HK.

    Proof. Let Yt=KΓ(1K)XKt2H+SH,Kt. Then, from (2.1) and (2.3), we have, for s,t0,

    E(YsYt)=KΓ(1K)E(XKs2HXKt2H)+E(SH,KsSH,Kt)               
    =t2HK+s2HK(t2H+s2H)K                    
        +(t2H+s2H)K12(t+s)2HK12|ts|2HK
    =t2HK+s2HK12(t+s)2HK12|ts|2HK,

    which completes the proof.

    Lemma 2.1 implies that

    {SH,Kt,t0}d={SHKtKΓ(1K)XKt2H,t0} (2.4)

    where d= means equality of all finite-dimensional distributions.

    By Theorem 2 in [13], the process XK has a version with trajectories that are infinitely differentiable trajectories on (0,) and absolutely continuous on [0,).

    Reference [15] presented a decomposition of the sub-fractional Brownian motion into the sum of a fractional Brownian motion plus a stochastic process with absolutely continuous trajectories. Namely, we have the following lemma.

    Lemma 2.2. Let BH be a fractional Brownian motion with Hurst parameter H, SH be a sub-fractional Brownian motion with Hurst parameter H and B={Bt,t0} is a standard Brownian motion. Let

    YHt=0(1eθt)θ1+2H2dBθ. (2.5)

    (1) If 0<H<12 and suppose that BH and B are independent, then the processes

    {HΓ(12H)YHt+BHt,t0} and {SHt,t0} have the same distribution.

    (2) If 12<H<1 and suppose that SH and B are independent, then the processes

    {H(2H1)Γ(22H)YHt+SHt,t0} and {BHt,t0} have the same distribution.

    Proof. See the proof of Theorem 2.2 in [15] or the proof of Theorem 3.5 in [14].

    By (2.4) and Lemma 2.2, we get, as 0<HK<12,

    {SH,Kt,t0}d={BHKt+HKΓ(12HK)YHKtKΓ(1K)XKt2H,t0} (2.6)

    and as 12<HK<1,

    {SH,Kt,t0}d={BHKtHK(2HK1)Γ(22HK)YHKtKΓ(1K)XKt2H,t0}. (2.7)

    The following Lemma 2.3 comes from Proposition 2.2 in [11].

    Lemma 2.3. Let XKt be defined by (2.2). Then, as h,

    E[(XK(h+t)2HXKh2H)2]=Γ(1K)K2KH2K(1K)t2h2(HK1)(1+o(1)).

    Therefore, as h,

    {XK(h+t)2HXKh2H,t0}d{Xt0,t0}.

    Lemma 2.4. Let YHt be defined by (2.5). Then, as h,

    E[(YHKh+tYHKh)2]=22HK2Γ(22HK)t2h2(HK1)(1+o(1)).

    Therefore, as h,

    {YHKh+tYHKh,t0}d{Yt0,t0}.

    Proof. By Proposition 2.1 in [15], we have

    E(YHtYHs)={Γ(12H)2H[t2H+s2H(t+s)2H],if  0<H<12;Γ(22H)2H(2H1)[(t+s)2Ht2Hs2H],if  12<H<1.

    When 0<HK<12, we get

    E(YHKtYHKs)=Γ(12HK)2HK[t2HK+s2HK(t+s)2HK].

    In particular, for every t0,

    E[(YHKt)2]=Γ(12HK)2HK(222HK)t2HK.

    Hence, we obtain

    E[(YHKh+tYHKh)2]=Γ(12HK)2HK22HK[(h+t)2HK+h2HK]+Γ(12HK)2HK2(2h+t)2HK.

    Then, for every large h>0, by using Taylor's expansion, we have

    I:=2HKΓ(12HK)E[(YHKh+tYHKh)2]                                                                 
    =22HK[(h+t)2HK+h2HK]+2(2h+t)2HK                                               
    =22HKh2HK[(1+th1)2HK+1]+2h2HK(2+th1)2HK                              
    =22HKh2HK[2+2HKth1+HK(2HK1)t2h2(1+o(1))]                        
    +2h2HK[22HK+22HK12HKth1+22HK2HK(2HK1)t2h2(1+o(1))]
    =22HK1HK(12HK)t2h2(HK1)(1+o(1)).                                               

    Thus,

    E[(YHKh+tYHKh)2]=22HK2(12HK)Γ(12HK)t2h2(HK1)(1+o(1))              
    =22HK2Γ(22HK)t2h2(HK1)(1+o(1)).

    Similarly, we can prove the case 12<HK<1. Therefore we finished the proof of Lemma 2.4.

    Proof of Theorem 2.1. It is obvious that Theorem 2.1 is the consequence of (2.6), (2.7), Lemma 2.3 and Lemma 2.4.

    In this section, we can understand Theorem 2.1 by considering the sub-bifractional Brownian noise, which is increments of sub-bifractional Brownian motion. For every integer n0, the sub-bifractional Brownian noise is defined by

    Yn:=SH,Kn+1SH,Kn.

    Denote

    R(a,a+n):=E(YaYa+n)=E[(SH,Ka+1SH,Ka)(SH,Ka+n+1SH,Ka+n)]. (3.1)

    We obtain

    R(a,a+n)=fa(n)+g(n)g(2a+n+1), (3.2)

    where

    fa(n)=[(a+1)2H+(a+n+1)2H]K[(a+1)2H+(a+n)2H]K
    [a2H+(a+n+1)2H]K+[a2H+(a+n)2H]K

    and

    g(n)=12[(n+1)2HK+(n1)2HK2n2HK].

    We know that the function g is the covariance function of the fractional Brownian noise with Hurst index HK. Thus we need to analyze the function fa to understand "how far" the sub-bifractional Brownian noise is from the fractional Brownian noise. In other words, how far is the sub-bifractional Brownian motion from a process with stationary increments?

    The sub-bifractional Brownian noise is not stationary. However, the meaning of the following theorem is that it converges to a stationary sequence.

    Theorem 3.1. For each n, as a, we have

    fa(n)=2H2K(K1)a2(HK1)(1+o(1)) (3.3)

    and

    g(2a+n+1)=22HK2HK(2HK1)a2(HK1)(1+o(1)). (3.4)

    Therefore limafa(n)=0 and limag(2a+n+1)=0 for each n.

    Proof. (3.3) is obtained by Theorem 3.3 in Maejima and Tudor. For (3.4), we have

    g(2a+n+1)=12[(2a+n+2)2HK+(2a+n)2HK2(2a+n+1)2HK]                                    
                    =22HK1a2HK[(1+n+22a1)2HK+(1+n2a1)2HK2(1+n+12a1)2HK]
            =22HK1a2HK[1+2HKn+22a1+HK(2HK1)(n+22)2a2(1+o(1))
                        +1+2HKn2a1+HK(2HK1)(n2)2a2(1+o(1))
                                            2(1+2HKn+12a1+HK(2HK1)(n+12)2a2(1+o(1)))]
    =22HK2HK(2HK1)a2(HK1)(1+o(1)).                                    

    Hence the proof of Theorem 3.1 is completed.

    We are now interested in the behavior of the sub-bifractional Brownian noise (3.1) with respect to n (as n). We have the following result.

    Theorem 3.2. For integers a,n0, let R(a,a+n) be given by (3.1). Then for large n,

    R(a,a+n)=HK(K1)[(a+1)2Ha2H]n2(HK1)+(12H)+o(n2(HK1)+(12H)).

    Proof. By (3.2), we have

    R(a,a+n)=fa(n)+g(n)g(2a+n+1).

    By the proof of Theorem 4.1 in [11], we get, for large n, the term fa(n) behaves as

    HK(K1)[(a+1)2Ha2H]n2(HK1)+(12H)+o(n2(HK1)+(12H)).

    We know that the term g(n) behaves as HK(2HK1)n2(HK1) for large n. For g(2a+n+1), it is similar to the computation for Theorem 3.1, we can obtain g(2a+n+1) also behaves as HK(2HK1)n2(HK1) for large n. Hence we have finished the proof of Theorem 3.2.

    It is easy to obtain the following corollary.

    Corollary 3.1. For integers a1 and n0, let R(a,a+n) be given by (3.1). Then, for every aN, we have

    n0R(a,a+n)<.

    Proof. By Theorem 3.2, we get that the main term of R(a,a+n) is n2HK2H1, and since 2HK2H1<1, the series is convergent.

    In this section, we prove two limit theorems to the sub-bifractional Brownian motion. Define a function g(t,s),t0,s0 by

    g(t,s)=2RH,K(t,s)ts=4H2K(K1)(t2H+s2H)K2(ts)2H1+HK(2HK1)|ts|2HK2
    HK(2HK1)(t+s)2HK2
    =:g1(t,s)+g2(t,s)g3(t,s),                          (4.1)

    for (t,s) with ts, t0, s0 and t+s0.

    Theorem 4.1. Assume that 2HK>1 and let {ξj,j=1,2,} be a sequence of standard normal random variables. g(t,s) is defined by (4.1). Suppose that E(ξiξj)=g(i,j). Then, as n,

    {nHK[nt]j=1ξj,t0}d{SH,Kt,t0}.

    Remark 1. Theorem 4.1 and 4.2 (below) are similar to the central limit theorem and can be used as a basis for many subsequent studies.

    In order to prove Theorem 4.1, we need the following lemma.

    Lemma 4.1. When 2HK>1, we have

    t0s0g(u,v)dudv=(t2H+s2H)K12(t+s)2HK12|ts|2HK.

    Proof. It follows from the fact that g(t,s)=2RH,K(t,s)ts for every t0,s0 and by using that 2HK>1.

    Proof of Theorem 4.1. It is enough to show that, as n,

    In:=E[(nHK[nt]i=1ξi)(nHK[ns]j=1ξj)]E(SH,KtSH,Ks).

    In fact, we have

    In=n2HK[nt]i=1[ns]j=1E(ξiξj)=n2HK[nt]i=1[ns]j=1g(i,j).

    Note that

    g(in,jn)=4H2K(K1)[(in)2H+(jn)2H]K2(ijn2)2H1                                    
    +HK(2HK1)|injn|2HK2HK(2HK1)(in+jn)2HK2
    =n2(1HK)g(i,j).                                                                     (4.2)

    Thus, as n,

    In=n2HK[nt]i=1[ns]j=1n2HK2g(in,jn)                  
    =n2[nt]i=1[ns]j=1g(in,jn)                             
    t0s0g(u,v)dudv                                
    =(t2H+s2H)K12(t+s)2HK12|ts|2HK
    =E(SH,KtSH,Ks).                                     

    Hence, we finished the proof of Theorem 4.1.

    We now consider more general sequence of nonlinear functional of standard normal random variables. Let f be a real valued function such that f(x) does not vanish on a set of positive measure, E[f(ξ1)]=0 and E[(f(ξ1))2]<. Let Hk denote the k-th Hermite polynomial with highest coefficient 1. We have

    f(x)=k=1ckHk(x),

    where k=1c2kk!< and ck=E[f(ξj)Hk(ξj)] (see e.g. [16]). Assume that c10. Let ηj=f(ξj),j=1,2,, where {ξj,j=1,2,} is the same sequence of standard normal random variables as before.

    Theorem 4.2. Assume that 2HK>32 and let {ξj,j=1,2,} be a sequence of standard normal random variables. g(t,s) is defined by (4.1). Suppose that E(ξiξj)=g(i,j). Then, as n,

    {nHK[nt]j=1ηj,t0}d{c1SH,Kt,t0}.

    Proof. Note that ηj=f(ξj)=c1ξj+k=2ckHk(ξj). We obtain

    nHK[nt]j=1ηj=c1nHK[nt]j=1ξj+nHK[nt]j=1k=2ckHk(ξj).

    Using Theorem 4.1, it is enough to show that, as n,

    E[(nHK[nt]j=1k=2ckHk(ξj))2]0. (4.3)

    In fact, we get

    Jn:=E[(nHK[nt]j=1k=2ckHk(ξj))2]                   
    =n2HK[nt]i=1[nt]j=1k=2l=2ckclE[Hk(ξi)Hl(ξj)].

    We know that, if ξ and η are two random variables with joint Gaussian distribution such that E(ξ)=E(η)=0, E(ξ2)=E(η2)=1 and E(ξη)=r, then

    E[Hk(ξ)Hl(η)]=δk,lrkk!,

    where

    δk,l={1,if  k=l;0,if  kl.

    Thus,

    Jn=n2HK[nt]i=1[nt]j=1k=2c2k(E(ξiξj))kk!                              
    =n2HK[nt]k=2c2kk!+n2HK[nt]i,j=1;ijk=2c2kk![g(i,j)]k.

    Since |g(i,j)|(E(ξ2i))12(E(ξ2j))12=1, we get, by (4.2),

    Jnn2HK[nt]k=2c2kk!+n2HK[nt]i,j=1;ijk=2c2kk![g(i,j)]2                       
    =n2HK[nt]k=2c2kk!+n2HKk=2c2kk![nt]i,j=1;ij[g(i,j)]2                    
    tn12HKk=2c2kk!+n2(HK1)(k=2c2kk!)n2[nt]i,j=1;ij[g(in,jn)]2. (4.4)

    On one hand, by k=2c2kk!< and 2HK>32>1, we get, as n,

    tn12HKk=2c2kk!0. (4.5)

    On the other hand, we have

    n2[nt]i,j=1;ij[g(in,jn)]2=n2[nt]i,j=1;ij[g1(in,jn)+g2(in,jn)g3(in,jn)]2                        
                                3n2[nt]i,j=1;ij{[g1(in,jn)]2+[g2(in,jn)]2+[g3(in,jn)]2}.

    Since |g1(u,v)|C(uv)HK1 and 2HK>32>1, we obtain

    n2[nt]i,j=1;ij[g1(in,jn)]2t0t0g21(u,v)dudvCt0t0(uv)2HK2dudv<. (4.6)

    We know that

    n2[nt]i,j=1;ij[g3(in,jn)]2t0t0g23(u,v)dudv                                                
                        =H2K2(2HK1)2t0t0(u+v)4HK4dudv
    <,                                 (4.7)

    since 2HK>32>1.

    We have also

    n2[nt]i,j=1;ij[g2(in,jn)]2t0t0g22(u,v)dudv                                                
                        =H2K2(2HK1)2t0t0(uv)4HK4dudv
    <,                                (4.8)

    since 2HK>32. Thus (4.3) holds from (4.4)–(4.8) and 2HK>32. The proof is completed.

    Remark 2. [11] pointed out, when 2HK>1, the convergence of

    n2(HK1)n2[nt]i,j=1;ij[g2(in,jn)]2

    had been already proved in [16]. But we can not find the details in [16]. Here we only give the proof when 2HK>32, because the holding condition for (4.8) is 2HK>32.

    In this section, we study an approximation in law of the fractional Brownian motion via the tangent process generated by the sbfBm SH,K.

    Theorem 5.1. Let H(0,1) and K(0,1). For every t0>0, as ϵ0, we have, the tangent process

    {SH,Kt0+ϵuSH,Kt0ϵHK,u0}d{BHKu,u0}, (5.1)

    where BHKu is the fractional Brownian motion with Hurst index HK.

    Proof. As 0<HK<12, by (2.6), we get

    {SH,Kt,t0}d={BHKt+HKΓ(12HK)YHKtKΓ(1K)XKt2H,t0}.

    By (2.5) in [12], there exists a constant C(H,K)>0 such that

    E[(XK(t0+ϵu)2HXK(t0)2HϵHK)2]=C(H,K)t2(HK1)0u2ϵ2(1HK)(1+o(1)),

    which tends to zero, as ϵ0, since 1HK>0.

    On the other hand, similar to the proof of Lemma 2.4, we obtain

    E[(YHKt0+ϵuYHKt0ϵHK)2]=22HK2Γ(22HK)t2(HK1)0u2ϵ2(1HK)(1+o(1)),

    which also tends to zero, as ϵ0. Therefore (5.1) holds. Similarly, (5.1) also holds for the case 12<HK<1. We finished the proof.

    In this paper, we prove that the increment process generated by the sub-bifractional Brownian motion converges to the fractional Brownian motion. Moreover, we study the behavior of the noise associated to the sbfBm and the behavior of the tangent process of the sbfBm. In the future, we will investigate limits of Gaussian noises.

    Nenghui Kuang was supported by the Natural Science Foundation of Hunan Province under Grant 2021JJ30233. The author wishes to thank anonymous referees for careful reading of the previous version of this paper and also their comments which improved the paper.

    The author declares there is no conflict of interest.



    [1] H. McCallum, A. Dobson, Detecting disease and parasite threats to endangered species and ecosystems, Trends Ecol. Evolut., 10 (1995), 190–194. https://doi.org/10.1016/S0169-5347(00)89050-3 doi: 10.1016/S0169-5347(00)89050-3
    [2] B. R. Forester, E. A. Beever, C. Darst, J. Szymanski, W. C. Funk, Linking evolutionary potential to extinction risk: applications and future directions, Front. Ecol. Environ., 20 (2022), 507–515. https://doi.org/10.1002/fee.2552 doi: 10.1002/fee.2552
    [3] J. Bosch, A. M.-C. de Alba, S. Marquínez, S. J. Price, B. Thumsová, J. Bielby, Long-term monitoring of amphibian populations of a National Park in northern Spain reveals negative persisting effects of Ranavirus, but not Batrachochytrium dendrobatidis, Front. Veter. sci., 8 (2021). https://doi.org/10.3389/fvets.2021.645491
    [4] M. C. Fisher, T. W. Garner, Chytrid fungi and global amphibian declines, Nat. Rev. Microbiol., 18 (2020), 332–343. https://doi.org/10.1038/s41579-020-0335-x doi: 10.1038/s41579-020-0335-x
    [5] C. X. Cunningham, S. Comte, H. McCallum, D. G. Hamilton, R. Hamede, A. Storfer, et al., Quantifying 25 years of disease-caused declines in Tasmanian devil populations: host density drives spatial pathogen spread, Ecol. Letters, 24 (2021), 958–969. https://doi.org/10.1111/ele.13703 doi: 10.1111/ele.13703
    [6] D. F. Jacobs, H. J. Dalgleish, C. D. Nelson, A conceptual framework for restoration of threatened plants: The effective model of American chestnut (Castanea dentata) reintroduction, New Phytolog., 197 (2013), 378–393. https://doi.org/10.1111/nph.12020 doi: 10.1111/nph.12020
    [7] F. De Castro, B. Bolker. Mechanisms of disease-induced extinction, Ecol. Letters, 8 (2005), 117–126. https://doi.org/10.1111/j.1461-0248.2004.00693.x doi: 10.1111/j.1461-0248.2004.00693.x
    [8] J. L. Hite, J. Bosch, S. Fernández-Beaskoetxea, D. Medina, S. R. Hall, Joint effects of habitat, zooplankton, host stage structure and diversity on amphibian chytrid, Proceed. Royal Soc. B Biol. Sci., 283 (2016), 20160832. https://doi.org/10.1098/rspb.2016.0832
    [9] C. G. Becker, S. E. Greenspan, R. A. Martins, M. L. Lyra, P. Prist, J. P. Metzger, et al., Habitat split as a driver of disease in amphibians, Biol. Rev., (2023). https://doi.org/10.1111/brv.12927
    [10] A. M. de Roos, J. A. J. Metz, L. Persson, Ontogenetic symmetry and asymmetry in energetics, J. Math. Biol., 66 (2013), 889–914. https://doi.org/10.1007/s00285-012-0583-0 doi: 10.1007/s00285-012-0583-0
    [11] D. L. Preston, E. L. Sauer, Infection pathology and competition mediate host biomass overcompensation from disease, 2020. https://doi.org/10.1002/ecy.3000
    [12] A. M. de Roos, L. Persson. Population and community ecology of ontogenetic development, volume 51, Princeton University Press, 2013. https://doi.org/10.23943/princeton/9780691137575.001.0001
    [13] P. A. Abrams, When does greater mortality increase population size? The long history and diverse mechanisms underlying the hydra effect, Ecol. letters, 12 (2009), 462–474. https://doi.org/10.1111/j.1461-0248.2009.01282.x doi: 10.1111/j.1461-0248.2009.01282.x
    [14] A. M. de Roos, The impact of population structure on population and community dynamics, Theor. Ecol., (2020), 53–73. https://doi.org/10.1093/oso/9780198824282.003.0005
    [15] R. Iritani, E. Visher, M. Boots, The evolution of stage-specific virulence: Differential selection of parasites in juveniles, Evolut. Letters, 3 (2019), 162–172. https://doi.org/10.1002/evl3.105 doi: 10.1002/evl3.105
    [16] M. W. Simon, M. Barfield, R. D. Holt. When growing pains and sick days collide: Infectious disease can stabilize host population oscillations caused by stage structure, Theoret. Ecol., 15 (2022), 285–309. https://doi.org/10.1007/s12080-022-00543-z doi: 10.1007/s12080-022-00543-z
    [17] S. Panter, D. A. Jones, Age-related resistance to plant pathogens, Adv. Botan. Res., 38 (2002), 251–280. https://doi.org/10.1016/S0065-2296(02)38032-7 doi: 10.1016/S0065-2296(02)38032-7
    [18] B. Ashby, E. Bruns, The evolution of juvenile susceptibility to infectious disease, Proceed. Royal Soc. B, 285 (2018), 20180844. https://doi.org/10.1098/rspb.2018.0844 doi: 10.1098/rspb.2018.0844
    [19] F. Ben-Ami, Host age effects in invertebrates: Epidemiological, ecological, and evolutionary implications, Trends Parasitol., 35 (2019), 466–480. https://doi.org/10.1016/j.pt.2019.03.008 doi: 10.1016/j.pt.2019.03.008
    [20] M. J. Keeling, P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008. https://doi.org/10.1515/9781400841035
    [21] S. Carran, M. Ferrari, T. Reluga, Unintended consequences and the paradox of control: Management of emerging pathogens with age-specific virulence, PLoS Neglected Trop. Diseases, 12 (2018), e0005997. https://doi.org/10.1371/journal.pntd.0005997 doi: 10.1371/journal.pntd.0005997
    [22] S. V. Cousineau, S. Alizon, Parasite evolution in response to sex-based host heterogeneity in resistance and tolerance, J. Evolut. Biol., 27 (2014), 2753–2766. https://doi.org/10.1111/jeb.12541 doi: 10.1111/jeb.12541
    [23] F. Magpantay, A. King, P. Rohani, Age-structure and transient dynamics in epidemiological systems, J. Royal Soc. Interface, 16 (2019), 2019. https://doi.org/10.1098/rsif.2019.0151 doi: 10.1098/rsif.2019.0151
    [24] A. Esteve, I. Permanyer, D. Boertien, J. W. Vaupel, National age and coresidence patterns shape COVID-19 vulnerability. Proceed. Nat. Aca. Sci., 117 (2020), 16118–16120. https://doi.org/10.1073/pnas.2008764117
    [25] C. E. Cressler, D. V. McLeod, C. Rozins, J. Van Den Hoogen, T. Day, The adaptive evolution of virulence: a review of theoretical predictions and empirical tests, Parasitology, 143 (2016), 915–930. https://doi.org/10.1017/S003118201500092X doi: 10.1017/S003118201500092X
    [26] J. L. Abbate, S. Kada, S. Lion, Beyond mortality: Sterility as a neglected component of parasite virulence. PLoS Pathogens, 11 (2015), e1005229. https://doi.org/10.1371/journal.ppat.1005229
    [27] P. J. Hurtado, S. R. Hall, S. P. Ellner, Infectious disease in consumer populations: Dynamic consequences of resource-mediated transmission and infectiousness, Theoret. Ecol., 7 (2014), 163–179. https://doi.org/10.1007/s12080-013-0208-2 doi: 10.1007/s12080-013-0208-2
    [28] V. H. Smith, R. D. Holt, M. S. Smith, Y. Niu, M. Barfield, Resources, mortality, and disease ecology: Importance of positive feedbacks between host growth rate and pathogen dynamics, Israel J. Ecol. Evolut., 61 (2015), 37–49. https://doi.org/10.1080/15659801.2015.1035508 doi: 10.1080/15659801.2015.1035508
    [29] J. L. Hite, C. E. Cressler. Resource-driven changes to host population stability alter the evolution of virulence and transmission, Philosophical Transactions of the Royal Society B: Biological Sciences, 373 (2018), 20170087. https://doi.org/10.1098/rstb.2017.0087 doi: 10.1098/rstb.2017.0087
    [30] C. J. Briggs, R. A. Knapp, V. T. Vredenburg, Enzootic and epizootic dynamics of the chytrid fungal pathogen of amphibians, Proceed. Nat. Aca. Sci., 107 (2010), 9695–9700. https://doi.org/10.1073/pnas.0912886107 doi: 10.1073/pnas.0912886107
    [31] J. L. Hite, R. M. Penczykowski, M. S. Shocket, A. T. Strauss, P. A. Orlando, M. A. Duffy, et al., Parasites destabilize host populations by shifting stage-structured interactions, Ecology, 97 (2016), 439–449. https://doi.org/10.1890/15-1065.1 doi: 10.1890/15-1065.1
    [32] L. Persson, K. Leonardsson, A. M. de Roos, M. Gyllenberg, B. Christensen, Ontogenetic scaling of foraging rates and the dynamics of a size-structured consumer-resource model, Theor. Popul. Biol., 54 (1998), 270–293. https://doi.org/10.1006/tpbi.1998.1380 doi: 10.1006/tpbi.1998.1380
    [33] K. Lika, R. M. Nisbet, A dynamic energy budget model based on partitioning of net production, J. Math. Biol., 41 (2000), 361–386. https://doi.org/10.1007/s002850000049 doi: 10.1007/s002850000049
    [34] A. M. de Roos, T. Schellekens, T. V. Kooten, K. E. V. D. Wolfshaar, D. Claessen, L. Persson, Simplifying a physiologically structured population model to a stage-structured biomass model, Theor. Popul. Biol., 73 (2008), 47–62. https://doi.org/10.1016/j.tpb.2007.09.004 doi: 10.1016/j.tpb.2007.09.004
    [35] J. L. Hite, A. C. Pfenning, C. E. Cressler, Starving the enemy? Feeding behavior shapes host-parasite interactions, Trends Ecol. Evolut., 35 (2020), 68–80. https://doi.org/10.1016/j.tree.2019.08.004 doi: 10.1016/j.tree.2019.08.004
    [36] O. Diekmann, M. Gyllenberg, J. A. J. Metz, Steady-state analysis of structured population models, Theor. Popul.n Biol., 63 (2003), 309–338. https://doi.org/10.1016/S0040-5809(02)00058-8 doi: 10.1016/S0040-5809(02)00058-8
    [37] R. M. Anderson, R. M. May, Infectious diseases of humans: Dynamics and control, Oxford university press, 1992. https://doi.org/10.1093/oso/9780198545996.001.0001
    [38] J. Heesterbeek, K. Dietz, The concept of Ro in epidemic theory, Statistica Neerlandica, 50 (1996), 89–110. https://doi.org/10.1111/j.1467-9574.1996.tb01482.x doi: 10.1111/j.1467-9574.1996.tb01482.x
    [39] A. M. d. Roos, Numerical methods for structured population models: The Escalator Boxcar Train, Numer. Methods Partial Differ. Equat., 4 (1988), 173–195. https://doi.org/10.1002/num.1690040303 doi: 10.1002/num.1690040303
    [40] A. M. d. Roos, O. Diekmann, J. A. J. Metz, Studying the dynamics of structured population models: A versatile technique and its application to Daphnia, Am. Natural., 139 (1992), 123–147. https://doi.org/10.1086/285316 doi: 10.1086/285316
    [41] P. A. Abrams, H. Matsuda, The effect of adaptive change in the prey on the dynamics of an exploited predator population, Canadian J. Fisher. Aquat. Sci., 62 (2005), 758–766. https://doi.org/10.1139/f05-051 doi: 10.1139/f05-051
    [42] R. M. Anderson, R. M. May, The invasion, persistence and spread of infectious diseases within animal and plant communities, Philosoph. Transact. Royal Soc. London. B Biol. Sci., 314 (1986), 533–570. https://doi.org/10.1098/rstb.1986.0072 doi: 10.1098/rstb.1986.0072
    [43] A. P. Dobson, The population biology of parasite-induced changes in host behavior, Quarterly Rev. Biol., 63 (1988), 139–165. https://doi.org/10.1086/415837 doi: 10.1086/415837
    [44] Y. Xiao, F. Van Den Bosch, The dynamics of an eco-epidemic model with biological control, Ecol. Model., 168 (2003), 203–214. https://doi.org/10.1016/S0304-3800(03)00197-2 doi: 10.1016/S0304-3800(03)00197-2
    [45] A. Fenton, S. Rands, The impact of parasite manipulation and predator foraging behavior on predator–prey communities, Ecology, 87 (2006), 2832–2841. https://doi.org/10.1890/0012-9658(2006)87[2832:TIOPMA]2.0.CO;2 doi: 10.1890/0012-9658(2006)87[2832:TIOPMA]2.0.CO;2
    [46] R. Anderson, R. M. May, Regulation and stability of host-parasite population interactions, J. Animal Ecol., 47 (1978), 219–247. https://doi.org/10.2307/3933 doi: 10.2307/3933
    [47] F. M. Hilker, K. Schmitz, Disease-induced stabilization of predator–prey oscillations, J. Theor. Biol., 255 (2008), 299–306. https://doi.org/10.1016/j.jtbi.2008.08.018 doi: 10.1016/j.jtbi.2008.08.018
    [48] P. Rohani, X. Zhong, A. A. King, Contact network structure explains the changing epidemiology of pertussis, Science, 330 (2010), 982–985. https://doi.org/10.1126/science.1194134 doi: 10.1126/science.1194134
    [49] C. J. E. Metcalf, J. Lessler, P. Klepac, A. Morice, B. T. Grenfell, O. Bjørnstad, Structured models of infectious disease: Inference with discrete data, Theor. Popul. Biol., 82 (2012), 275–282. https://doi.org/10.1016/j.tpb.2011.12.001 doi: 10.1016/j.tpb.2011.12.001
    [50] P. Klepac, H. Caswell, The stage-structured epidemic: Linking disease and demography with a multi-state matrix approach model, Theor. Ecol., 4 (2011), 301–319. https://doi.org/10.1007/s12080-010-0079-8 doi: 10.1007/s12080-010-0079-8
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