Effects of Intrinsic and Extrinsic Host Mortality on Disease Spread

Abstract

The virulent effects of a pathogen on host fecundity and mortality (both intrinsic and extrinsic mortality due to predation) often increase with the age of infection. Age of infection often is also correlated with parasite fitness, in terms of the number of both infective propagules produced and the between-host transmission rate. We introduce a four-population partial differential equations (PDE) model to investigate the invasibility and prevalence of an obligately killing fungal parasite in a zooplankton host as they are embedded in an ecological network of predators and resources. Our results provide key insights into the role of ecological interactions that vary with the age of infection. First, selective predation, which is known both theoretically and empirically to reduce disease prevalence, does not always limit disease spread. This condition dependency relies on the timing and intensity of selective predation and how that interacts with the direct effects of the parasite on host mortality. Second, low host resources and intense predation can prevent disease spread, but once conditions allow the invasion of the parasite, the qualitative dynamics of the system do not depend on the intensity of the selective predation. Third, a comparison of the PDE model with a model based on ordinary differential equations (ODE model) reveals a parametrization for the ODE version that yields an endemic steady state and basic reproductive ratio that are identical to those in the PDE model. Our results highlight the complexity of resource–host–parasite–predator interactions and suggest the need for additional data–theory coupling exploring how community ecology influences the spread of infectious diseases.

Appendices

Appendix A

$$\begin{aligned} Q_e= & {} \mu \frac{f_S(A_e)}{A_e} S_e Z_e \int _{0}^{a_0} e^{-\int _0^{a} (d+v(\tau ) +p_S \theta (\tau )) \mathrm{d} \tau } \mathrm{d}a\\= & {} \mu \frac{f_S(A_e)}{A_e} S_e Z_e J_0 \end{aligned}$$

is the total infected host population and $J_0$ is the expected life span of an infected host. Then, $E_e$ is described by the relationships

$$\begin{aligned} S_e= & {} \frac{\lambda +r \left( 1-\frac{A_e}{K}\right) }{\sigma e_S f_S(A_e) (\mathrm{d} \phi +\chi )} \frac{1}{\mu J_0 \frac{f_S(A_e)}{A_e}} \nonumber \\ Q_e= & {} \frac{r \left( 1-\frac{A_e}{K}\right) A_e}{f_S(A_e)} -S_e = \frac{e_S f_S(A_e) -(d+p_S)}{\frac{1}{J_0} - \psi e_S f_S(A_e)} S_e,\nonumber \\ Z_e= & {} \frac{\sigma e_S f_S(A_e) (\mathrm{d} \phi +\chi )}{\lambda +r \left( 1-\frac{A_e}{K}\right) } Q_e. \end{aligned}$$

(9)

Here,

$$\begin{aligned} J_1 = \int _{0}^{a_0} W(a) \exp \left( -\int _{0}^{a} (d+v(\tau ) +p_S \theta (\tau )) \mathrm{d} \tau \right) \mathrm{d}a \end{aligned}$$

(10)

is the expected value of within-host pathogen load in the life span of an infected host,

$$\begin{aligned} J_2 = \int _{0}^{a_0} v(a) W(a) \exp \left( -\int _{0}^{a} (d+v(\tau ) +p_S \theta (\tau )) \mathrm{d} \tau \right) \mathrm{d}a \end{aligned}$$

(11)

and

$$\begin{aligned} J_3 = \int _{0}^{a_0} \rho (a) \exp \left( -\int _{0}^{a} (d+v(\tau )+p_S \theta (\tau )) \mathrm{d} \tau \right) \mathrm{d}a \end{aligned}$$

(12)

is the expected value of the fecundity reduction parameter over the life span of the infected host,

$$\begin{aligned} \phi = \frac{J_1}{J_0}, ~~~\chi = \frac{J_2}{J_0}, ~~~\text{ and }~~~\psi = \frac{J_3}{J_0}. \end{aligned}$$

(13)

Solving (9) for $A_e$ yields a cubic polynomial which in turn can be used to find the equilibrium values for the remaining population densities.

Appendix B

If $\varLambda$ is real in (6), then it follows by a simple differentiation that $\frac{\mathrm{d} R}{\mathrm{d} \varLambda }<0$. Hence, if $R_0=R(0)>1$, it follows that the equation $R(\varLambda ) =1$ has real positive solutions $\varLambda$. Therefore, $R_0>1$ implies instability. Similarly, if $R_0<1$, then the same argument shows that the equation $R(\varLambda ) =1$ has no real positive solutions. It remains to be shown that it does not have complex solutions with positive real part either. Let $\varLambda = x + i y$ be complex. Then, if $x \ge 0$ and if one sets $\varLambda _0 = \lambda + \frac{f_S(A_{df})}{A_{df}} S_{df}$ and

$$\begin{aligned} g(a)= \sigma e_S f_S(A_{df}) W(a)(d+v(a)) e^{-\int _{0}^{a} (d+v(\tau )+p_S \theta (\tau )) \mathrm{d} \tau } \mu \frac{f_S(A_{df})}{A_{df}} S_{df}, \end{aligned}$$

it follows $1=|R(\varLambda )| = \left| \frac{\int _{0}^{\infty } g(a) e^{-xa} e^{-iya} \mathrm{d}a}{x+\varLambda _0+ i y} \right| \le \frac{\int _{0}^{\infty } g(a) e^{-xa} \mathrm{d}a}{x+\varLambda _0} =R(x) \le R(0)=R_0$. Hence, if $R_0<1$, then the disease-free equilibrium is locally asymptotically stable, and if $R_0>1$, it is unstable. For a more rigorous justification of this result, one needs to study the semigroup properties of the generator (Martcheva and Thieme 2003).