Infectious disease in consumer populations: dynamic consequences of resource-mediated transmission and infectiousness

Abstract

Nonhost species can strongly affect the timing and progression of epidemics. One central interaction—between hosts, their resources, and parasites—remains surprisingly underdeveloped from a theoretical perspective. Furthermore, key epidemiological traits that govern disease spread are known to depend on resource density. We tackle both issues here using models that fuse consumer–resource and epidemiological theory. Motivated by recent studies of a phytoplankton–zooplankton–fungus system, we derive and analyze a family of dynamic models for parasite spread among consumers in which transmission depends on consumer (host) and resource densities. These models yield four key insights. First, host–resource cycling can lower mean host density and inhibit parasite invasion. Second, host–resource cycling can create Allee effects (bistability) if parasites increase mean host density by reducing the amplitude of host–resource cycles. Third, parasites can stabilize host–resource cycles; however, host–resource cycling can also cause disease cycling. Fourth, resource dependence of epidemiological traits helps to govern the relative dominance of these different behaviors. However, these resource dependencies largely have quantitative rather than qualitative effects on these three-species dynamics. Given the extent of these results, host–resource–parasite interactions should become more fundamental components of the burgeoning theory for the community ecology of infectious diseases.

Appendices

Appendix A: Bifurcation summary

The dynamics illustrated in Fig. 4, particularly the bistability in regions ③ and ④, can be summarized as the generic dynamics near a generalized Hopf (Bautin) bifurcation with a nearby transcritical bifurcation of limit cycles BPC that correspond to the loss of parasites from the three-species cycles LC₂. A generalized Hopf bifurcation has three branches (see Fig. 4): a supercritical Hopf H ₋ (to the right of GH), a subcritical Hopf H ₊ (to the left of GH), and a saddle-node bifurcation of limit cycles LPC (for limit point cycle). The nature of the bistability that arises between LPC and H ₊, and the basins of attraction for each outcome, can be more clearly understood by considering the dynamics near these two bifurcation curves (also see Fig. 3). The subcritical Hopf H ₊ gives rise to an unstable periodic orbit LC _u which exists for k values above H ₊ in regions ③ and ④ and forms part of the separatrix that separates the basins of attraction between the three-species steady-state EQ₂ and the stable limit cycle in regions ③ (LC₂, parasites present) and ④ ( LC₁, no parasites). Ignoring BPC for the moment, traversing ③ and ④ from H ₊towards LPC, the amplitude of the unstable cycle LC _u increases as it approaches the stable limit cycle. Continuing across LPC, the two cycles collide (forming a limit point cycle, where LC _u = LC₂, at LPC) then disappear, leaving only the stable disease equilibrium ( EQ₂) as the lone attractor in region ①. As a result, the basin of attraction for the stable interior equilibrium, EQ₂, is vanishingly small near the subcritical Hopf bifurcation, H ₊ (i.e., most trajectories result in cycling), while the basin of attraction for EQ₂dominates near LPC. BPC separates regions ③ (cycling with disease) and ④ (without disease) and marks the threshold at which disease can no longer persist in the cycling host population.

Appendix B: Model derivation

The general model (Eq. 1) is a rescaled version of the following model, which is based on spore-based fungal parasitism of Daphnia sp. N is producer density (per liter), X is susceptible consumer density (per liter), Y is infected consumer density (per liter), Z is infectious spore density (spores per liter), and time τ is in days. Table 1 contains parameter descriptions, ranges, and values.

$$ \frac{dN}{d\tau} = \tilde r N (1 - N/K) - \tilde \alpha (N) N (X+\rho Y)$$

(B1a)

$$\frac{dX}{d\tau} = \chi \tilde \alpha (N) N (X + \tilde f \rho Y) - (d_x+\tilde m)X - \tilde \beta(N) X Z $$

(B1b)

$$\frac{dY}{d\tau} = \tilde \beta(N) X Z - (d_{x} + \tilde \nu + \tilde m) Y$$

(B1c)

$$\frac{dZ}{d\tau} = \tilde \sigma(N) (d_x+\tilde \nu) Y - \tilde \mu Z - \eta \tilde C(N) Z (X+\rho Y) $$

(B1d)

The rescaled model (Eq. 1) is given by $t = \frac {\tau }{\frac {1}{d_{x}}}$, $r=\frac {\tilde r}{d_{x}}$, $\alpha =\frac {\chi \tilde \alpha }{d_{x}}$, $k=\tilde k / K$, $f=\tilde f \rho$, $m=\frac {\tilde m}{d_{x}}$, $C=\eta \tilde C \chi K\frac {1}{d_{x}}$, $C_c=\eta \tilde C_{c} \frac {\chi }{d_{x}}$, $\beta =\tilde \beta \chi K \frac {1}{d_{x}}$, $\nu =\frac {\tilde \nu }{d_{x}}$, $\sigma ={\tilde \sigma }$, and $\mu =\frac {\tilde \mu }{d_{x}}$. This leaves the rescaled variables as n = N / K, $x=\frac {X}{\chi K}$, $y=\frac {Y}{\chi K}$, and $z=\frac {Z}{\chi K}$. We further simplify this model by assuming that consumption does not deplete spores in the environment, i.e., that C ≡ 0. See Table 2 for parameter descriptions.

Table 1 Parameter values and descriptions for the unscaled model

Table 2 Parameters of the scaled model (Eq. 1)

Reduced model

Model (1) can be simplified by assuming that spore turnover is fast ( σ, μ very large). In that case, spore density z(t) tracks the equilibrium of Eq. 1d obtained by holding the density of infected hosts y(t) constant, $z(t) = \left ( \frac {\sigma (n(t))(1+\nu )}{\mu } \right )y(t)$. Substituting this expression for z(t) into Eqs. 1b and 1c gives a reduced model with direct disease transmission,

$$\frac{dn}{dt} = r n (1-n) - \alpha(n) (x+ y) n$$

(B2a)

$$\frac{dx}{dt} = \alpha(n) (x+f y) n - (1+m) x - B(n) x y $$

(B2b)

$$\frac{dy}{dt} = B(n) x y - (1 + m\theta + \nu) y,$$

(B2c)

where B(n) is the direct transmission rate, i.e., the rate of new infections per infectious individual (Eq. 5).

With this simplification, the force of infection now depends on the current density of infectives, by assuming that spores are so short-lived that the current spore density is proportional to the current density of infectives. The reduced and general models have the same equilibria for n, x, and y, and all of the bifurcations mentioned in the main text occur in both the general and reduced models.

Appendix C: Parameter values

Parameter values and ranges were determined based upon their biological interpretations, using published values of those quantities when available. Other values come from previously published models of Daphnia parasitism or algae consumption. Within biologically plausible parameter ranges, certain parameter values were further specified in order to yield dynamics consistent with field and laboratory observations or produce specific dynamic behaviors.

Algal growth and consumptions rates are based on the wide range of natural variability in green algae and their interactions with Daphnia. Intrinsic growth rate $\tilde {r}$ is based on 1–4 doublings day ^{− 1} for green algae (e.g., Sorokin and Krauss 1958) and maximum algal densities are based on naturally occurring levels during typical algal blooms.

Daphnia feeding (or filtering) rates were based on previously published population growth rates and observed feeding rates (Hall et al. 2007; Duffy et al. 2005; Porter et al. 1982, for example), though such rates in reality likely depend on other factors including temperature and food quality and thus are only loosely defined here. Under the units of milligrams dry weight per liter (as in Hall et al. 2007), we have maximum consumption rates a ≈ 1– 2 orders of magnitude smaller than N < K, with k roughly of the same order of magnitude as K, though slightly smaller. Based on Duffy et al. (2005) where the maximum birth rate was 0. 4 day ^{− 1} (with generation times of 1– 3 weeks), we can assume that the maximum birth rate χ a is equal to b _max = 0. 4 day ^{− 1} from Duffy et al. (2005), which implies (by a ≈ 10²above) that χ = 0. 4 × 10^{− 2}.

Disease parameters given in Table 1 yield an $\mathcal {R}_{0,\text {dis}}$ only slightly larger than 1 (though simulation suggests that this does not guarantee disease persistence in the presence of host–resource cycles). Thus, the spore-based infection rate was allowed to be somewhat flexible in order to attain prevalence levels consistent with those observed in naturally occurring M. bicuspidata epizootics.

The maximum per consumer filtering rate (liters per day) for Daphnia magna is taken from Fig. 1 of Porter et al. (1982), which suggests nearly 4 mL h ⁻1. Converting to the proper units and rounding yields approximately 0. 1 L day$^{-1} = \frac {\tilde C_{c}}{\tilde k}$. With the above, we compute $\tilde C_{c}$ as 0. 1k.

To estimate plausible values of ϕ, we rely on the data in Fig. 1f of Hall et al. (2009a). The best fit line to those data gives σ(1) / σ(0) = (1 + ϕ n) ≈ 0. 03 / 0. 02, a quantity independent of σ ₀. To use this to estimate ϕ, we must first know what weight corresponds to the algal carrying capacity (i.e., what n corresponds to 1 mg C/L?). Assuming that 1 mg C/L corresponds to the carrying capacity n = 1—a conservative (low) estimate—these data imply ϕ ≈ 0. 5. If 1 mg C/L corresponds to some density below carrying capacity, then ϕ = 0. 5 / n > 0. 5. Thus, if carrying capacity corresponds to W (milligrams C per liter), then ϕ = W · 0. 5. To avoid extrapolating beyond the available data, larger values of W may require a saturating or other functional form of σ(n). In the text, we assume the conservative estimate of ϕ = 0. 5.

Appendix D: Consumer–resource models with slow disease dynamics

Disease can reduce consumer fecundity, increase consumer mortality, or both. How does this affect the consumer–resource dynamics? How do those changes affect disease dynamics? How is bistability maintained between cycling and steady-state dynamics in the presence of disease? To answer these questions, we consider a limiting case of model B2 with constant B(n) = B.

Assuming x + y > 0, we transform Eq. B2 to be in terms of resource density n, total consumers p = x + y, and fraction of infected consumers i = y / (x + y). This is done by substituting f = 1 − g ∈ (0, 1), x = i p and y = (1 − i)p into d p / d t = d x / d t + d y / d t and d i / d t = (d y / d t) / p − i(d p / d t) / p. Defining R = B / (1 + ν), this yields

$$\frac{dn}{dt} = \left ( \frac{r}{\alpha(n)} \left (1-n \right ) - p \right ) \alpha(n) n$$

(D1a)

$$\frac{dp}{dt} = \left ( \alpha(n) n - \frac{1 + \nu i}{1 - g i} \right ) (1 - g i) p$$

(D1b)

$$\frac{di}{dt} = \left ( R p (1-i)- 1 - \frac{d\ln(p)}{dt} \frac{1}{1+\nu} \right ) (1+\nu) i.$$

(D1c)

Here, the equations are factored to clarify their nullclines, and without loss of generality, we have assumed no predation on consumers ( m = 0). (Compare Eq. D1 to Hilker and Schmitz (2008) for a similar model with frequency-dependent disease transmission and no bistability).

Assuming that the long-term disease dynamics (under Eq. D1) occur slowly relative to the consumer–resource dynamics, the n-p dynamics approach a quasi-asymptotic state (e.g., either steady-state or cycling dynamics) as i slowly changes. We can understand these quasi-asymptotic consumer–resource dynamics as follows:

Assuming constant i ∈ (0, 1), the dynamics of Eqs. D1a and D1b can be understood from the n and p nullclines:

$$\frac{1}{n}\frac{dn}{dt} = \;0 \quad \Rightarrow \quad p = \frac{r}{\alpha(n)} \left (1-n \right )$$

(D2a)

$$\frac{1}{p}\frac{dp}{dt} = \;0 \quad \Rightarrow \quad G(i) = \alpha(n) n.$$

(D2b)

where $G(i)=\frac {1+\nu i}{1-g i}$ is the fractional increase in the per-consumer mortality rate divided by the decrease in fecundity. Figure 6 shows an example of these nullclines using a type II functional response.

Fig. 6

Nullclines for the Rosenzweig–MacArthur model (type II α (n)) illustrating the consequence of reduced fecundity and/or increased mortality among consumers. Assuming a fixed fraction (i) of the population are diseased, increasing i shifts the p-nullcline to the right, the direction of increasing stability. Changing i has no effect on the n-nullcline. The coexistence equilibrium is unstable where the n-nullcline is increasing (shaded gray, here the populations cycle) and is stable where the n-nullcline is decreasing. This is generally the case for any monotone increasing feeding rate α (n)n under Eqs. D1a and D1b

The p-nullcline (Eq. D2b) for fixed i is a vertical line at the equilibrium value of n = n _eq. The coexistence equilibrium occurs where these two nullclines intersect and is stable when (1) the per-capita feeding rate α(n)n is increasing at n _eq and (2) the n-nullcline (Eq. D2a) is decreasing at n _eq. Cycling dynamics occur if the n-nullcline is increasing at n _eq(where the coexistence equilibrium is unstable).

Prevalence i only affects the p-nullcline (Eq. D2a). Differentiating α(n)n = G(i) with respect to i, it follows that n _∗typically increases as disease prevalence i increases since

$$ \text{sign} \left ( \frac{dn_*}{di} \right ) = \text{sign}\left (\frac{d}{dn}(A(n_*)n_*) \right ) \text{sign} \left (\frac{d}{di} G(i) \right ). $$

(D4)

Consequently, a slow increase in disease prevalence should stabilize the system by increasing n _eq as shown in Fig. 6.

Why bistability?

Having described the consumer–resource dynamics under fixed i, we can clarify how density-dependent disease transmission leads to bistability by simplifying the disease (Eq. D1c). This can be done (albeit crudely) by considering the criterion for disease to invade the disease-free cycle LC₁ (see the next section for details):

$$R\overline{p_{0}} > 1. $$

(D5)

Here, $\overline {p_{0}}$ is the mean consumer density over the disease-free cycle. Note the similarity to the condition ℛ_0d > 1 necessary for invasion of the disease-free steady state EQ₁.

Based upon this observation, we approximate Eq. D1c by taking the “slow disease” limit of Eq. D1 which relies on the mean resource density $\overline {p_{i}}$ over the attracting cycle or equilibrium point determined by Eqs. D1a and D1b with fixed i.

Doing so yields the approximation

$$ \frac{di}{dt} = i \left ( \overline{p_{i}} \cdot (1-i) - 1/R \right ) R. $$

(D5)

Computing $\overline {p_{i}}$ over a range of (fixed) i values, Fig. 7 shows how a disease-induced mortality and reduced fecundity can each increase mean consumer density during cycling regimes, when consumer–resource dynamics are fast relative to changes in i. This “hydra effect” (Abrams 2009) appears to be what allows for bistability when disease transmission is density-dependent. Positive feedback between p and i means that perturbing either a cycling system with little or no disease could sufficiently increase consumer density and push the system above threshold, allowing persistence of disease at steady state.

Fig. 7

Nullclines for the Rosenzweig–MacArthur model (type II α (n)) illustrating how the “hydra effect” (Abrams 2009) in this model allows disease to paradoxically increase the mean resource density during cycling dynamics. The mean and amplitude of the consumer population are shown as the p-nullcline (vertical line) moves right as the fixed disease fraction (i) is increased from i = 0 to i = 1. Also shown is the mean $\overline {p_{i}}$ as a function of i. See Fig. 6 for other details

More precisely, Fig. 8 shows the $\overline {p_{i}}$ curve shown in Fig. 7 which shapes the dynamics of Eq. D5 for differing values of R. Stability is determined by where i is increasing ($\overline {p_{i}} (1-i) > 1/R$) and decreasing ($\overline {p_{i}} (1-i) < 1/R$), and the small peak in Fig. 8 is approximately where the Hopf bifurcation occurs in the consumer–resource model (Eqs. D1a and D1b). As in Fig. 7, equilibria to the left of this peak correspond to cycles under models D1a and D1c and those to the right correspond to the endemic disease steady state EQ₂.

Figure 8 accounts for four of the five qualitatively different cases described for models 1 and B2 in the text. Though not shown, if the “Hopf peak” was to surpass the critical value of $\overline {p_{0}}$ (the triangle in Fig. 8), then any values of 1 / R between the peak’s maximum value and $\overline {p_{0}}$ would yield bistability between disease-free cycles LC₁ and the endemic disease steady state EQ₂ as in region ④ of Fig. 4.

Fig. 8

Example dynamics of Eq. D5 for multiple values of parameter R. Equilibria in the region that is shaded light gray correspond to cycles under Eq. D1. Note $i>0$ requires $R\overline {p_{0}}<1$ (triangle at i = 0). In terms of the dynamics of models D1 and B2, these equilibria correspond to dynamic states where only the disease-free cycle LC₁ is stable ($R_{1}$), only the disease cycle $\text {LC}_{2}$ is stable ($R_{2}$), both $\text {LC}_{2}$ and the endemic disease steady state $\text {EQ}_{2}$ are bistable ($R_{3}$), and only $\text {EQ}_{2}$ is stable ($R_{4}$). These correspond to the dynamics dominating regions ⑤, ②, ③ and ① in Fig. 4, respectively

Criteria for invasion of disease-free limit cycles

Equation D1c is equivalent to

$$ \frac{d(\ln i)}{dt} = (1+\nu)(Rp(1-i)-1) - \frac{d(\ln p)}{dt}. $$

(D6)

A criterion for the local stability of the disease-free limit cycle LC₁(period T) can be obtained by considering the average rate of increase in i over that disease-free limit cycle assuming an arbitrarily small i(0) > 0. Define the instantaneous growth rate on LC₁ as

$$ \lambda_{i}(t) \equiv (1+\nu)(Rp(1-i) - 1) - \frac{d(\ln p)}{dt}. $$

(D7)

The average growth rate over LC₁ is therefore given by

$$ \Lambda_{T} \equiv \frac{1}{T} \int\limits_{0}^{T}{\lambda_{i}(t)dt}. $$

(D8)

thus, LC₁ is locally stable if Θ _T < 0 and disease invades if Θ _T > 0.

Combining the above equations yields

$$\begin{array}{rll} \Lambda_T &=& \frac{1}{T} \int\limits_{0^{T}}(1+\nu)(Rp(1-i) dt - \frac{1}{T} \int\limits_{0^T}\frac{d(\ln p)}{dt} dt\\ &=& (R\overline{s} - 1)(1+\nu) - \frac{1}{T}(\ln p(T)-\ln p(0))\end{array}$$

(D9)

where $\overline {s} \equiv \frac {1}{T} \int \limits _{0}^{T}{p(t)(1-i(t))dt}$. On the limit cycle p(T) = p(0) and $\overline {s}=\overline {s_{0}}$ where s ₀ is the mean susceptible population size over the disease-free cycle. These two facts together with Eq. D9 yield the LC₁stability criterion

$$ R \overline{s_{0}} < 1. $$

(D10)

Note that in cases of bistability, sufficiently high initial levels of disease can push the system beyond the basin of attraction for disease-free cycle LC₁ and result in disease invasion despite $R\overline {s_{0}} < 1$ (Fig. 9).

Fig. 9

Simulation showing the performance of the disease invasion criterion for the disease-free cycle LC₁ under constant–rates model (Eq. 1) with fast spore dynamics. Parameter values were sampled from uniform distributions with β ∈ (5, 11), f ∈ (0, 1), and ν ∈ (0, 3). Disease invasion is shown by the slope of the best-fit line to log(i(t)) over 15 time units, with positive slope indicating disease invasion. The quantity R p¯ is as described in the text. Parameters used: r = 40, α = 9, k = 0.6, m = 0, μ = 1900, and σ = 90