inst/Outline for SI-Macro Paper.md

In noamross/age-infects: Package for manuscript and code of age-infection interaction paper

author: Noam Ross date: 'October 8, 2014' title: 'Comparative dynamics of SI and multiple-infection models: Outline with draft methods and results.' ...

Introduction

1. Response to disease outbreak often requires prediction of medium- and long-term behavior from early-phase dynamics of disease. However, at early phases, appropriate models for disease dynamics may not be known.

2. Several forms of emerging wildlife and forest diseases (SOD, beech bark disease, chytridiomycosis, white nose syndrome), have been found to be driven by spore loads. [@Briggs2010, @KateSporeLoad, @DaphniaPapersbyDuffy]

   (Here I use the term multi-infection to refer to such diseases)

3. The multi-infection, or macroparasite framework [@Anderson1978] may be a more appropriate model for such diseases than SIR models.

   1. Plant diseases potentially have macro-parasite-like mechanisms [@Dobson1994], but they are rarely modeled this way, even though some patterns justify it (See Martinez1996 for examples in mistletoe, @Waggoner1981 and @McRoberts2003 for more general examination of infection count distribution).

4. Stage and age structure is often an important factor in the dynamics of these diseases (chytrid, SOD).

   There is a considerable literature on the dynamics of disease dynamics of in age-structured populations (e.g. Castillo-Chavez et al. 1989; Busenberg and Hadeler 1990; Diekmann and Heesterbeek 2000; Hethcote 2000; Dietz and Heesterbeek 2002; Thieme 2003), with a somewhat smaller literature on stage-structured populations [@Klepac2010].

The role of age structure in multi-infection (or macroparasite) models.

@Krasnov2006 showed, for instance, that parasite counts increase with age in rodents. @Pacala1998 created a method to detect the mortality effect of macroparasites based of the distribution of parasites among different age groups. @Duerr2003 showed how a age-infection relationships could be modified by a variety of age- time- and density-dependent processes, but also showed that interpretation of such age-infection patterns was ambiguous if more than one such process was operating.

@Briggs2010 created a stage-structured individual-based model of Batrachochytrium mycosa (Bd) infections in frogs, finding that low-susceptibility tadpoles could act as disease reservoirs and promote disease persistence.

@Krkosek2013, using the @Anderson1978 simplication demonstrated that macroparasites can induce an Allee effect.

In the case of an emerging disease, how important are the differences in dynamics between multi-infection and

1. Questions: How do the dynamics of SI and multi-infection models differ?

   1. How do models with similar early-phase behavior compare in long-term behavior, and vice versa?

2. How do patterns of disease in stage-structured populations differ between SI and multi-infection models

Methods

Model Structure

I compared dynamics in 3 ODE-based disease models: A simple $SI$ model, an multi-infection model based on @Anderson1978, and an intermediate $SIV$ (susceptible-infected-very infected) model.

Each model has a two-stage population structure (population $N$ = juveniles $J$ + adults $A$). New individuals enter the uninfected, juvenile stage via density-dependent recruitment ($fN(1-N/K)$, where $f$ is fecunidity and $K$ carrying capacity). Individuals move from juvenile to adult classes at the transition rate $g$.

Disease transmission is density-dependent; susceptible individuals $(J_S, A_S)$ become infected $(J_I, A_I)$ at a rate equal to the density of other infected individuals times the transmissivity of the disease $(\lambda)$. All individuals die at the a base rate $(d)$, and diseased individuals have additional mortality $(\alpha)$

The complete $SI$ model is

$$\begin{aligned} \frac{dJ_S}{dt} &= fN(1 - N/K) - J_S(d + g + \lambda J_I + \lambda A_I) &\frac{dA_S}{dt} &= g J_S - A_S(d + \lambda N) \ \frac{dJ_I}{dt} &= \lambda J_S (J_I + A_I) - J_I(d + g + \alpha) &\frac{dA_I}{dt} &= g J_I + \lambda A_S (J_I + A_I) - A_I(d + \alpha) \ N &= J_S + A_S + J_I + A_I \end{aligned}$$

Note that this is a null model of age structure; neither demographic nor epidemiological parameters vary with age. When juvenile and adult classes are summed, the growth term $g$ drops out, and $dN/dt$ is independent of $g$.

The other two models are extensions of the $SI$ model with additional disease classes representing degrees of infection. In the multi-infection model, there are an infinite number of disease classes designated $i = 0, 1, 2, \dots, \infty$. For purposes of simulation, the number of classes is truncated, with a maximum value of $k$. Transmissivity $(\lambda)$ and mortality $(\alpha)$ and are additive in these models. Trees advance to the next disease class at rate $\Lambda$, the overall force of infection, which is the sum of each tree's contribution, $i\lambda$. Trees in each stage die at rate $d + i\alpha$. The complete multi-infection model.

$$\begin{aligned} \frac{dJ_0}{dt} &= fN(1 - N/K) - J_0(d + g + \Lambda) &\frac{dA_0}{dt} &= g J_0 - A_0(d + \Lambda) \ \frac{dJ_i}{dt} &= \Lambda dJ_{i-1} - J_i(d + g + i\alpha + \Lambda) &\frac{dA_i}{dt} &= g J_i + \Lambda A_{i-1} - A_i(d + i\alpha + \Lambda) \ \frac{dJ_k}{dt} &= \Lambda dJ_{k-1} - J_k(d + g + k\alpha) &\frac{dA_k}{dt} &= g J_k + \Lambda A_{k-1} - A_k(d + k\alpha) \ N &= \sum_{i=0}^k J_0 + A_0 &\Lambda &= \lambda \sum_{i=1}^k i(J_i + A_i) \end{aligned}$$

The $SIV$ model is merely a truncated version of the multi-infection model, with $k = 2$. For this model I refer $N_0$ as $S$, $N_1$ as $I$ and $N_2$ as $V$, and use $S$, $I$, and $V$, as subscripts for $J$, and $A$ as well.

In this paper, parameters (e.g., $\lambda$ and $\alpha$) are subscripted with $\text{param}{SI}$, $\text{param}{SIV}$, or $\text{param}_\text{multi}$ when referring to their values in each of the three models. I also use the term "infected" to refer to individuals of either the $I$ class in the $SI$ model, or having at least one infection in the $SIV$ or multi-infection models.

Multi-infection models typically assume a distribution of infections in order to reduce the system of equations [@Anderson1978]. Negative-binomial distributions of infections allow tractable analysis of such models and match empirical studies of infection distribution in the wild [@REF]. However, the reduced model only approximates the full model asymptotically [@Adler1992], and key assumptions of the reduced model break down in the presence of age structure (See Appendix.) Instead, I avoided making such assumptiosn by simulating the the infinite system of equations truncating at $k$.

Comparative parameterization

I compared the models' behaviors under "equivalent" parameterizations. As the models have different structures, their parameters in the models have different interpretations. Specifically, $\lambda$ and $\alpha$ operate on a per-individual basis in the $SI$ model, while they operate on a per-infection basis on the $SIV$ and multi-infection models. Thus, they are not identical parameterizations.

In order to determine equivalent parameterizations between models, I set parameters for the $SI$ model to those in Table 1. I then fit the $SIV$ and multi-infection models so that they would exhibit identical behavior to the $SI$ model under different criteria. The behavior of SIV and multi-infection models were adjusted by multiplying both the infectivity $(\lambda_{SIV}, \lambda_{multi})$ and disease-induced mortality $(\alpha_{SIV}, \alpha_{multi})$ parameters by a constant $c$. Where there there were dual behavior criteria (behaviors (4) and (5) below), the $\lambda$ and $\alpha$ values were allowed to vary independently.

Initial conditions in simulations were set at the disease-free equilibrium of the system, modified with 1% of both juveniles and adults having a single infection.

Parameter Symbol Base Case Value

fecundity $f$ 1 carrying capacity $K$ 1 transition rate $g$ 0.1 mortality $d$ 0.01 disease-induced mortality $\alpha$ 0.2 transmissivity $\lambda$ 3 max number of infections (SIV/multi-infection) $k$ 3 / 150

Table 1: Base parameters for disease models

Equilibrium mortality rate. The first behavioral criterion was identical equilibrium mortality rate across models. $c$ was varied to match the overall disease-induced mortality rate (and thus the total mortality rate) between models. That is, at steady state, $$\alpha_{SI} = \alpha_{SIV} \frac{I + 2V}{I+V} = \alpha_\text{multi} \frac{1}{N} \sum_i i N_i$$

Initial growth and acceleration rates of infected individuals. $c$ was adjusted to match behavior under initial conditions. This is,

$$\begin{aligned} \frac{dI}{dt} &= \frac{d(I+V)}{dt} = \frac{dN_{i > 0}}{dt} \text{, and} \ \frac{d^2 I}{dt^2} &= \frac{d^2(I+V)}{dt^2} = \frac{d^2 N_{i > 0}}{dt^2} \end{aligned}$$ at initial conditions of $S \approx N$, $I_{SI} = I_{SIV} = N_{1\, multi} \approx 0$ and $I_{SIV} = N_{i \geq 2\, multi} = 0$.

Note that the first condition, of the initial growth rate of infected individuals, is identical at all cases under these initial conditions. Thus, we only fit to the second derivative. Derivatives were determined numerically using the numDeriv packages [@REF].

Time to 10% infection. This criterion was selected to match behavior among models during the early transient period of disease. $c$ was adjusted so that the $SIV$ and multi-infection models would reach 10% infection in the same time period as the $SI$ model. That is,

$$t\big|{\frac{I}{S+I} = 0.1} = t\big|{\frac{I+V}{S+I+V} = 0.1} = t\big|{\frac{N{i \geq 1}}{N} = 0.1}$$

All simulations simulations were performed in R (@REF), using the deSolve package (@REF) for simulation and the ggplot (@REF) package for plotting. Code to reproduce these results is archived online [@Ross2014].

Results

Aggregate dynamics

Models with similar equilibrium behavior differ in initial transient behavior. Figure 1 shows the dynamic behavior of the $SI$, $SIV$, and multi-infection models calibrated to equivalent mortality at equilibrium. At equilibrium. Under the base case parameterization, all models reach an internal equilibrium with a population level suppressed from the disease-free equilibrium at which they started. As all other rates are equal, the equilibrium populations are identical between the models, as well.

Under this parameterization, the ratio of $\alpha$ values between the models is the inverse of the mean number of infections at equilibrium in the multi-infection and $SIV$ models.

In the $SIV$ and multi-infection models, the apparent mortality rate of infected individuals increases over time. Early in the epidemic, individuals have small numbers of infections, thus the mortality rate across individuals with any level of infection is low. As the epidemic progresses, the mean number of infections per infected individual increases, raising the mortality rate of the infected class until equilibrium is reached.

The change in mortality rates is driven by changes in the distribution of infections over time, shown in Figures 2 and 3. As the disease progresses through the population, the proportion of individuals in teh $I$ and $V$ classes increases for both juveniles and adults. Similarly, in the multi-infection model, the mean number of infections in each individual increases over time, increasing the mortality rate.

While equilibrium behaviors are identical and models start at the same initial conditions, transient behavior differs. The time to equilibrium is greater in the multi-infection model than the $SIV$ model, and greater in both than the $SI$ model. It takes longer in the $SIV$ model, and longest in the multi-infection model, for the disease to emerge.

Models with similar initial behavior reach different equilibrium conditions. Figure 4 shows the dynamics of the three models in the case where the initial first and second derivatives are equivalent. As in the matching-equilibrium parameterizations, mortality for infected individuals increases over time until equilibrium is reached. Unlike that parameterization, per-infection parameters in the $SIV$/multi-infection models (mortality and infectivity), are higher than than per-individual parameters in the $SI$ model. Total mortality rates for the $SIV$ and multi-infection models start at higher levels than the $SI$ model and further diverge over time.

The $SIV$ and multi-infection models have nearly identical behavior. At equilibrium, their populations are suppressed to lower levels than in the $SI$ model, and a smaller number of the individuals are infected. This is because the difference in mortality rates of infected individuals between the $SI$ and the other models is greater, increasing turnover of infected individuals. In this case, the $SIV$ and multi-infection models reach equilibrium before the $SI$ model.

Fig 5 shows the results when models were parameterized to have equivalent time until 10% of the total population was infected. Here, the initial mortality rates of the $SIV$ and multi-infection models are similar to those in the $SI$ model; per-infection parameters in $SIV$/multi-infection models are similar to per-individual parameters in the $SI$ model.

Patterns in the time-to-10%-infection parameterizations are qualitatively similar to the matched-second-derivative parameterization. Mortality rates for individuals in $SIV$ and multi-infection models increase before reaching equilibrium, resulting in lower population sizes and lower populations of infected individuals at equilibrium. Dynamics for the $SIV$ and multi-infection models are again very similar, though not as similar as in the matched-derivative case. Also, in this case, the number of infected individuals reaches a peak before going down to reach equilibrium levels.

Age effects

Multi-infection models generate age-dependent effects not found in SI models.

All three models, under all three parameterization exhibit some common patterns in the dynamics of population stages. From the disease-free equilibrium dominated by adults, disease outbreak decreases the population of adult stages and and increases both the relative and absolute population of the juvenile stages. The infected population of both stages increases, with the adult infected stage reaching a peak before equilibrium and the juvenile infected stage reaching a smaller equilibrium with no peak.

In the equivalent equilibrium mortality parameterization, The $SIV$ and multi-infection stage structures are slower to reach an equilibrium than the $SI$ stage structure, with the multi-infection case being slowest. This is similar to the aggregate dynamics for this parameterization. In the case of equivalent initial derivatives, as will as the case of equivalent time to 10% infection, the change in age structure from the disease-free equilibrium is greater in the $SIV$ and multi-infection models than the $SI$ model. at equilibrium, there are more juveniles and fewer adults in the $SIV$/multi-infection cases.

In the $SIV$ and multi-infection models, the mortality rate of infected juveniles and adults increases as the disease progresses, and their mortality rate diverges, with adults having greater mortality rates than juveniles at equilibrium. This occurs in all parameterizations. The reason for this can be found in figures 2 and 3, which show the distribution of infections for both adult and juvenile populations over the course of the epidemic in $SIV$ and multi-infection models. Adults and juveniles begin with equal mean numbers of infections, but as the epidemic continues, adult trees accumulate more infections than juveniles by both new infections on adult trees and already-infected juveniles recruiting into the adult population.

In a multi-infection model with age structure, individuals accumulate infections over time, resulting in more infections, and thus greater mortality and infectivity, among older individuals than younger individuals. Even in the absence of age-driven variation in how individuals respond to disease (that is, in a "null model"), different behavior is observed between age groups. In an SI model, these differences do not arise.

Discussion

1. Epidemics that appear to be well represented by $SI$ models during their outbreak phase may no longer be well represented in later stages.

   A multi-infection model that behaves like an $SI$ model in early stages will have greater mortality rates at equilibrium, suppressing population more. A smaller fraction of the population will be infected at equilibrium, because it

2. If a disease is driven by multi-infection mechanisms, age effects may (in part) be artifacts of infection accumulation in older individuals, rather than biological differences.

   @McCallum1995 suggested that comparison of infection counts between different aged individuals may be useful in inferring disease mortality rates. Similarly, these results suggest that, in the absence of infection counts, differences between age groups in effects of disease, such as mortality, may be indicative of load-driven disease mechanisms.

3. Reduced version of multi-infection model (SIV) is sufficient for many applications.

   Qualitative differences between SI and multi-infection models are largely present in the differences between SI and SIV models. As the number of infection classes increases, model behavior approaches the behavior of the infinite-class multiple-infection model.

4. Future work

   1. Can the models be distinguished in noisy data?
   2. Detection thresholds higher than $i=1$
   3. Compare variation in mortality rates across models with explicit variation/stochasticity.
   4. Ultimate measure of the loss due to incorrect model is in relative utility of management presuming $SI$ or macroparasite models.

Appendix:

1. Show Klieber-Ludwig divergence of multi-infection model from negative binomial distribution during transient phase

2. Show analysis of divergence of simplified version of multi-infection model [@Anderson1978] from full model when age structure.

noamross/age-infects documentation built on May 23, 2019, 9:30 p.m.