Evidence of tradeoffs shaping virulence evolution in an emerging wildlife pathogen

Abstract

In the mid-1990's, the common poultry pathogen Mycoplasma gallisepticum (MG) made a successful species jump to the eastern North American house finch Haemorhous mexicanus (HM). Subsequent strain diversification allows us to directly quantify, in an experimental setting, the transmission dynamics of three sequentially emergent geographic isolates of MG, which differ in the levels of pathogen load they induce. We find significant among-strain variation in rates of transmission as well as recovery. Pathogen strains also differ in their induction of host morbidity, measured as the severity of eye lesions due to infection. Relationships between pathogen traits are also investigated, with transmission and recovery rates being significantly negatively correlated, while transmission and virulence, measured as average eye lesion score over the course of infection, are positively correlated. By quantifying these disease-relevant parameters and their relationships, we provide the first analysis of the tradeoffs that shape the evolution of this important emerging pathogen.

INTRODUCTION

There are at least three component processes that determine whether or not the introduction of an infectious agent into a host population will result in epidemic dynamics: (1) the multiplication rate of the pathogen in a newly infected host; (2) the duration of, and degree to which, a host is infectious; and (3) the rate of transmission of the pathogen between free-living hosts. Host-pathogen systems for which this information can be determined are valuable, but rare; they allow for the study of potentially generalizable infectious disease dynamics, while also providing opportunities for deriving estimates of epidemiologically relevant parameters, and for testing key assumptions of more broadly used epidemiological models (Keeling & Rohani, 2007).

Each different strain of a particular type of pathogen will typically be defined by a suite of parameter values, distinct from each other, that determine the strain-specific host-pathogen interaction. This situation allows for the study of heritable variation in pathogen traits associated with host exploitation, as well as to potentially infer the action of selection and pathogen evolution within a framework that typically concentrates primarily on epidemiological issues (Kryazhimskiy et al., 2007). Multiple strains also provide the grist for evolutionary studies of a pathogen by making it possible to follow temporal or spatial changes in the phenotype of sequentially evolving strains of a pathogen within a host population. This can be a particularly useful property, as numerous fundamental evolutionary questions of epidemiological importance remain under-studied.

For example, a large body of theory has long assumed that pathogen strains can be thought of as variants expressing different genetic trait correlations that impose trade-offs among epidemiological parameters, most commonly transmission and virulence (usually defined as the disease-induced mortality rate (Ebert & Herre, 1996); virulence and recovery are another possibility (Anderson & May, 1982). The appeal of the trade-off formalism is that it provides a plausible and logical explanation for the persistence of seemingly maladaptive traits, like virulence, which reduce pathogen fitness by limiting transmission duration. While this has certainly proven to be a very fertile framework for theoretical studies (Anderson & May, 1982; May & Anderson, 1983; Frank, 1996), empirical examples of such trade-offs are very limited (Alizon et al., 2009). Any system conforming to this trade-off framework would be an important piece of evidence to support and refine the general applicability of this classical assumption. The well-characterized host-pathogen relationship between the house finch Haemorhous mexicanus (HM) and the bacterium Mycoplasma gallisepticum (MG), causal agent of infectious conjunctivitis, provides the requisite ingredients to test this assumption.

Since its emergence in 1994 as a pathogen of house finches and other free-living birds (Dhondt et al., 2008), MG has evolved rapidly (Hawley et al., 2013; Hochachka et al., 2013). Isolates vary greatly in their replication rates in a naive host (Grodio et al., 2012; Hawley et al., 2010; Hawley et al., 2013) leading to higher pathogen loads. More rapidly replicating isolates cause higher levels of morbidity for longer periods, and in a higher proportion of individuals, when held in separate cages and inoculated experimentally with the same dose (Hawley et al., 2010; Hawley et al., 2013). Under these conditions pathogen load (as measured by qPCR; Grodio et al., 2008) and eye lesion severity are significantly positively correlated (Hawley et al., 2010; Grodio et al., 2012; Hawley et al., 2013).

Eye lesion severity is implicated in reducing over-winter survival in wild infected birds (Faustino et al., 2004) and has thus been used as a proxy for virulence in this system (Hawley et al., 2010; Hawley et al., 2013). Similarly, pathogen load has been used as a measure of transmission potential (Hawley et al., 2013). However, as the birds in these previous experiments were housed in isolation from each other, there remains a lack of data on the actual between-host infection process. Experimental transmission studies, in which a small number of birds are initially infected and housed with other uninfected birds, all of which are assayed at regular intervals for disease status, provide the means to follow basic infection dynamics (Saenz et al., 2012).

In this paper we quantify and compare horizontal transmission in groups of house finches in which MG isolates of different virulence were introduced; we also estimate rates of recovery from infection with different pathogen strains. In order to understand the isolate-specific epidemiological dynamics, we derive estimates of these vital parameters under controlled, caged conditions. We also compare disease-relevant parameters for all isolates in order to investigate potential trade-offs as well as to determine whether the MG-HM system might provide the means for testing the classical trade-off model of pathogen evolution (Frank, 1996; Anderson & May, 1982).

MATERIALS AND METHODS

In each of six identical octagonal aviaries inside a closed barn, we introduced 12 wild-caught hatch-year house finches that had been captured in Tompkins County, NY in summer and fall of 2009 and in which we had been unable to detect previous exposure to MG: all birds were negative for MG as determined by the absence of MG-specific antibodies (ELISA, modified from (Hawley et al., 2011), the absence of MG DNA (qPCR following Grodio et al. (2008)) and the absence of external conjunctival signs of disease (eyescore, Sydenstricker et al. (2005)). The sex ratio in all but 2 of those cages was 50:50 (6 males and 6 females). One of the VA94 cages had 7 males and 5 females, and one CA06 cage had 4 males and 8 females. The skew occurred because juvenile birds are difficult to sex prior to the birds molting, when the groups are initially formed. However, there isn't much evidence that sex is important for susceptibility. Previous studies with the house finches show very small, if any, effects of sex on eyescore or pathogen load, and no effects of sex on responses to novel antigens (Altizer et al., 2004; Kollias et al., 2004; Hawley et al. 2006; Hawley et al. 2007).

Each octagon had a ground surface area of 6.87 m² and a volume of 17.87 m³ and contained one six-port tube feeder (re-filled daily with Roudybush Maintenance Diet) hung from the center, two artificial Christmas trees placed in a corner; and several plastic perches attached to the walls. Close to one of the perches we also provided a ceramic heating lamp during winter. On the aviary walls, at about 1.80 m height, we hung five artificial Christmas wreaths that birds used for roosting and nest building. A water bath, heated by a heating lamp in winter, was cleaned and refilled every day, and the cement floor was cleaned twice weekly (for a photograph of the setup, see Appendix in Dhondt et al. 2012). Given that the aviaries were inside a closed barn wild birds could not come into contact with the experimental birds and could not, therefore, be responsible for introducing MG into the system.

On January 27, 2010 all birds were captured and baseline samples were taken to confirm the absence of MG infection (blood, conjunctival swabs). On February 4, 2010 a male and a female in each group were selected at random and inoculated in their conjunctival sacs with 0.05 ml of an inoculum, kept in a paper bag for 10 minutes, and released back into their respective aviary. In each of two aviaries we used one of three MG strains for inoculation: CA06 (2006.052-5 (4p) 1/13/2009; VA94 (1994 7994-1 (7p) 2/12/2009); NC06 (2006.080-5 (4p) 1/9/2009). We will refer to these strains as CA06, VA94, and NC06 respectively. In order to inoculate all index birds with a similar number of MG infectious particles, and because the concentrations of the inocula differed, we needed to dilute VA94 and NC06 inocula to a target concentration of 3.04 × 106/ml as present in the CA06 inoculum. We did this by diluting (volume to volume) VA94 1:7.4, and NC06 1:100.

All birds were recaptured and re-sampled on day 4 post-inoculation (PI), severity of eye lesions scored, and conjunctival swabs taken in both eyes. Because of the large number of individuals to be sampled and the cold temperatures in the aviary during the winter months we recorded eye score and took conjunctival samples on days 11, 18, 25, 32, 39, 46, and 58 in one of the replicates of each treatment, and on days 12, 19, 26, 33, 41, 47, and 59 in the second aviary of each treatment. All individuals were sampled together on days 70, 85, 99, 113, and 127 since the warmer weather permitted longer hours in the aviary. To simplify the reporting of the results we use the days PI from the first group as the time PI for both replicates. The experiment was terminated on 8 June 2010 (day 127 PI).

Infection by MG was measured as the presence of pathogen DNA, detected using conjunctival swabs from both eyes pooled per individual and tested using MG-specific qPCR (Grodio et al., 2008). In order to estimate the date of first exposure to MG we used qPCR results and used the first date a bird was positive as the date of transmission. When inconsistencies arose (i.e. disease fade out followed by disease recurrence), we inferred false positive infections and false negative infections in order to construct a consistent infection history.

The presence of conjunctivitis was scored on a scale of 0 (no lesions) to 3 (severe lesions) (following Sydenstricker et al. (2005)), and individual morbidity was described by summing the scores from both eyes. The severity of an infection (i.e. its virulence) was quantified by computing the average virulence (AV), calculated by first summing the eye scores for left and right eyes for each finch in an aviary and then taking the average during the period of infection. Since we were interested in horizontal transmission between the inoculated individuals and the naïve birds, we expressed MG load in each group as the sum of the MG load in the two index birds.

Quantification of transmission and recovery rates

Based on collected infection data, maximum likelihood estimates (MLE) were obtained for transmission and recovery rates for each cage of the different strains. We used a stochastic S-I-S (Susceptible-Infectious-Susceptible) model (Andraud et al., 2011), wherein transmission was assumed to be density-dependent mass action (see Appendix 1), and the rates of disease recovery were treated as constants; recovery times were thus assumed to have negative exponential distributions. The potential for recovery from the infection raises the possibility that epidemiological parameters might change upon reinfection, therefore our initial parameter estimates were obtained using only data from the first 26 days PI, during which time no re-infection occurred as determined by inspection of the recorded infection history (for comparison, parameters were also calculated using data for the full 127 days). Thus, the simple stochastic S-IS model was sufficient for parameter estimation.

Statistical analyses

Analysis of variance was used to compare variables among strains and regression analysis was used to investigate the relationship between epidemiological parameters. A Kaplan-Meier survival analysis was used to estimate the probability of birds becoming infected with MG and of developing eye lesions over the course of the experiment. Epidemiological parameters and confidence intervals were estimated using the bbmle package in R (Bolker, 2008).

RESULTS

Variation among isolates in pathogen load and infection probability

All 12 index birds were infected successfully, although not all index birds developed eye lesions. The degree to which they responded to inoculation varied between individuals inoculated with the same isolate; nevertheless, the MG loads for index birds differed significantly between isolates by day 26 post infection (F_{2, 5} = 14.16, P = 0.005; Table 1).

Table 1.

Estimates of key epidemiological parameters for three geographic isolates of Mycoplasma gallisepticum. Columns 6 and 7 give the proportions of 10 naive individuals per aviary exposed to different Mycoplasma gallisepticum isolates that tested positive for infection by horizontal transmission according to whether they express MG DNA (6) and/or eye lesions (7).

Strain	Aviary	Aviary sex ratio	Relative virulence	Pathogen load	Average virulence (AV)	Presence of MG DNA using qPCRc	Presence of eye lesions	β (95% CI)a,c	γ (95% CI)b,c
CA 06	1	6M:6F	Low	3.159	0.00	60%	0%	0.005 (0.002-0.012)	0.061 (0.012-0.119)
	2	8M:4F		1.544	1.00	10%	0%	0.003 (0.00-0.008)	0.111 (0.027-0.300)
VA 94	3	7M:5F	Intermediate	6.311	2.00	70%	30%	0.007 (0.003-0.014)	0.034 (0.010-0.078)
	4	6M:6F		5.618	1.48	40%	10%	0.003 (0.001-0.007)	0.030 (0.00-0.073)
NC 06	5	6M:6F	High	8.238	4.12	100%	100%	0.034 (0.016-0.062)	0.000 ( - )
	6	6M:6F		6.268	3.54	100%	90%	0.014 (0.007-0.025)	0.006 (0.00-0.018)

^aTransmission rate is measured in (bird^-1 day^-1)

^bRecovery rate is measured in (day^-1)

^cParameter estimates based on cage data up to 26 days post-infection (PI)

The Kaplan-Meier survival analysis revealed that naïve birds became infected more rapidly in the groups with NC06, than in groups both with VA94 and CA06 (χ²₂ = 23.28, P< 0.001; Fig. 1a). The proportion of individuals that developed eye lesions also varied between isolates (χ²₂ = 42.65, P< 0.001; Fig. 1b). In particular, in the aviaries in which the CA06 isolate had been introduced, none of the naïve birds developed eye lesions, although in several MG was detected (Table 1). Finally, among isolates, there was a significant positive correlation between pathogen load and eye lesion severity measured at day 26 (r² = 0.65, P < 0.001; see also Table 1).

Figure 1.

Probability of infection as a function of time since introduction of Mycoplasma gallisepticum in the group. (a) using MG DNA presence as indicator of infection; (b) using presence of eye lesions only. In the groups exposed to NC06 all naïve birds became infected and all developed eye lesions; in groups exposed to CA06, about 30% of the individuals became infected, but none developed eye lesions. In groups exposed to VA94 an intermediate proportion of individuals became infected and developed eye lesions.

Estimation of epidemiological parameters and comparisons among strains

Maximum likelihood estimates were computed for the transmission and recovery rates in all 6 cages, at both day 26 post-infection (to eliminate the possibility of reinfection following recovery so as to fit an S-I-S model; see Appendix 1) and at the end of the experiment (day 127, for comparison; see Table 1). Parameter estimates varied among cages but, overall, there was a significant negative relationship between transmission and recovery (Fig. 2a), such that highly transmissible strains were associated with low rates of recovery (r² =0.85, P < 0.001). Results were similar using data collected across the entire experiment (r² =0.89, P < 0.001). Recovery and virulence also negatively covaried (r² =0.84 and 0.54, P <0.001 at days 26 and 127, respectively; Fig.2b).

Figure 2.

Plot of (a) estimated recovery rate versus estimated density-dependent transmission rate; (b) estimated recovery rate versus average virulence; and (c) estimated density-dependent transmission rate versus average virulence of 3 different Mycoplasma gallisepticum strains. Open objects indicate parameters measured at day 26 post-infection (PI); closed objects indicate transmission rate measured at day 127 PI. Circles, triangles and squares represent strains CA06, VA94, and NC06, respectively. Solid black line is non-linear regression line measurements up to day 26, while dashed line is non-linear regression line for measurements up to day 127.

Among strains, transmission rates varied markedly, such that NC06 had greater transmissibility than CA06. However the transmissibility of VA94 and CA06 was virtually identical (Table 1). The estimated recovery rates for NC06 (the high virulence strain) were also much lower than those estimates from the intermediate and low virulence strains. Combining data on average virulence (AV) and transmission rates, we find a significant positive relationship between virulence and transmission (r² =0.89 and 0.74, P <0.001 at days 26 and 127, respectively), such that the high virulence strain exhibits higher transmission rates (Fig. 2c).

DISCUSSION

Detailed investigations of the epidemiological interaction between hosts and their parasites requires knowledge of the transmission parameters governing the spread of infection, as well as the factors that determine the infectious period: virulence rates induced by the pathogen, and rates of recovery from infection by the host. Such information not only provides insight into particular disease systems, but can also aid in the development of a deeper understanding of epidemiological issues at large. Estimates of these same parameters across isolates of the same pathogen additionally allow for the identification of evolutionary factors that have shaped, and are likely to continue to shape, the host-pathogen interaction of interest (Doumayrou et al., 2013).

Here, we used experimental transmission trials to make precise measurements of MG transmission, and HM recovery, as well as MG-induced eye lesions, a measure of pathogen virulence. We then used a stochastic, susceptible-infected-susceptible (S-I-S) model and maximum likelihood techniques to estimate transmission and recovery rates, the parameters relevant to the dynamics of the epizootic pathogen MG in free-living populations of their finch host (HM). We also utilized extant standing variation in replication rates of MG to investigate trade-offs between these parameters. We were thus able to execute a quantitative study of both the epidemiological and evolutionary interactions that define the MG-HM system.

MG is adept at corrupting various aspects of its original poultry hosts’ immune system, from inducing inflammatory responses at the site of infiltration to promote lesions (Ley, 2008), to suppressing components of host immunity, including host recovery rates (Javed et al., 2005). In contrast, previous theoretical investigation of the relationship between transmission and recovery has assumed a tradeoff structure: a higher rate of the host exploitation by the pathogen allows for the production of more transmissible particles, which enhances transmission success; but greater pathogen production also stimulates a more rapid, or more vigorous, up-regulation of a hosts’ immune defenses, truncating transmission duration (Alizon, 2008). The scenario outlined in this theoretical treatment assumes that recovery is entirely determined by the hosts’ phenotype. However, in poultry this fails to be the case, with MG being equipped to suppress recovery rates by some, as yet, undetermined means.

Our analysis indicates that this also holds for strains adapting to HM, with high transmission rates, and thus high virulence, being associated with low recovery rates, and hence longer infectious periods (1/γ)(see Table 1). All else being equal, this relationship favors more virulent pathogens, as MG accrues the additional benefit of increasing its transmission duration via reducing the rate at which HM clears the infection for a given increase in virulence. A similar situation might have arisen in the the European rabbit – myxoma virus systems of Australia and Great Britain (Anderson & May, 1982; Fenner & Ratcliffe, 1965). Analyzing data first presented by Fenner & Ratcliffe (1965), Anderson & May (1982) determined the relationship between recovery and virulence and found a qualitative dependence similar to that found in the HM-MG system (Fig. 2b). While their analysis does not explicitly include an analytical treatment of transmission, they do note that “(t)here is good qualitative evidence, however, that high virulence...is typically associated with lots of open lesions, and hence that mosquitoes (the vectors in Australia) or fleas (the vectors in Britain) can more easily bite infected wounds and acquire the virus. Low virulence...is correspondingly associated with poor transmission.” Other authors have attempted to describe a transmission-virulence tradeoff in this system, suggesting that transmission is, for part of its range, an increasing function of virulence, and thereafter decreasing, reaching a maximum at some intermediate value (Mead-Briggs & Vaughan 1975).

“Virulence” in the present situation is operationally defined in terms of the severity of conjunctiva inflammation. This correlates positively with house finch morbidity which, in turn, correlates with mortality under natural conditions (Grodio et al., 2012; Hawley et al., 2010; Hotchkiss et al., 2005; Roberts et al., 2001b; Faustino et al., 2004). The pattern of eye lesions with respect to pathogen load allows for a more in-depth analysis of virulence in this system. There is considerable difference between the MG strains in the pattern and degree to which they induce eye lesions (Fig. 1b). In all infections with the low virulence strain (CA06), eye lesions are absent, even in the experimentally infected birds. In finches infected with the intermediate virulence strain (VA94), only the experimentally infected index birds experience bilateral infections, while naturally infected birds are subjected to mild, unilateral infections. Only in the virulent strain (NC06) are infections bilateral and severe. If any component of virulence is enacted via corrupted vision, these results would indicate that the virulent strain would experience very strong negative selection, adding support to this system as a representative of the trade-off model of pathogen evolution.

An implicit assumption of this last point is that transmission and virulence positively covary across strains. While our results, using eye lesion severity as a proxy for virulence (Table 1; Fig. 2c), are consistent with this assumption, it must be noted that considerable uncertainty surrounds the determination of MG virulence in experimental settings. While virulence quite generally refers to the negative fitness effects induced by an infectious agent on its host (O'Keefe & Antonovics, 2002), this effect is usually defined as the increase in the host mortality rate induced by infection. In an early natural infection experiment during 1995-96, finches housed under ‘benign conditions’ still experienced high levels of morbidity and mortality, suggesting high virulence (Luttrell et al., 1998). This conclusion remains tentative, however, as very high stocking density and insufficient food supply in this experiment greatly obscure the supposed causal relationship of infection to observed mortality (Sydenstricker et al., 2006), although an interaction between infection and these other factors cannot be ruled out. When a similar experimental infection study was repeated in 1998, it failed to elicit the same response, with most birds experiencing a short course of infection and many eventually recovering (Roberts et al., 2001a). Similarly, in another artificial-infection experiment in 2001, in which individuals were housed alone, most exposed birds developed conjunctivitis, although in only 20% of finches was infection severe (5% mortality), with recovery occurring in nearly 80% (Kollias et al., 2004).

Despite the caveats regarding the housing and husbandry practices in (Luttrell et al., 1998), the above findings are broadly consistent with the emergent isolate of MG evolving reduced virulence following its initial introduction. This initial high degree of severity of MG was also found in a 1999 experiment by (Farmer et al., 2002), where 10 of the 11 birds displaying outward signs of MG died at some point during the course of the experiment. Moreover, the finches used in this experiment were from a western population, without any history of previous exposure to epizootic isolates of MG. It has also been suggested that MG infection has been a strong source of selection on eastern finch populations, resulting in enhanced resistance relative to western populations (Bonneaud et al., 2011). Others have disputed this possibility, arguing that the age-structure and slow population growth rates of house finch populations imply that too little time had elapsed between the aforementioned experiments to detect a signal in resistance evolution (Sydenstricker et al., 2006).

Regardless, the work of Luttrell et al. (1998) and Farmer et al. (2002) seems to indicate that shortly after the emergence of epizootic MG, a large component of infection-induced mortality was environmentally context-independent, being expressed even under protected conditions. Over the following few years, this component of mortality seems to have been largely removed by selection. As seen in the present study, the benign conditions provided by population cages generally preclude the death of experimental birds due to any cause, including infection, even by the most virulent (i.e. the most rapidly replicating) isolate, NC06.

Herein lies one of the major difficulties with constructing accurate R₀ estimates by extrapolating from cage experiments to natural settings: in natural environments, the infectious periods for each of the isolates will be determined not only by the rates of recovery from MG, but also from the mortality rates resulting, either directly or indirectly, from the infection. Virulence is a property not only of the host-parasite interaction, but of the environment as well, and will typically exhibit a norm of reaction across environments (Williams & Day, 2001). In the present case, MG infection, by causing impaired sight and immunosuppression, enhances infected birds susceptibility to starvation, predation, and secondary infections (Roberts et al., 2001b; Hotchkiss et al., 2005; Faustino et al., 2004).

All such factors elevate mortality in the presence of infection, and thus correspond to the more common concept of pathogen virulence. The tendency will be for these factors to reduce R₀ values calculated under benign conditions by diminishing the infectious period. This will be particularly true for the most rapidly replicating isolates, who should suffer the largest reductions relative to their values in the absence of natural sources of mortality. Figure 2 is thus broadly consistent with the existence of a transmission-virulence trade-off when such considerations are taken into account: high transmission strains experience high virulence (NC06), while low transmission strains experience low virulence (CA06). However, determining the precise shape of this trade-off is not possible with only three different strains; moreover, we have no way of inferring how the virulence proxy chosen is transformed into a mortality rate in the natural environments in which the HM-MG system is embedded (Hawley et al., 2013). This possibility could be explored further by including more isolates in the analysis, as well as by assaying infection-induced mortality in an environment relevant to the host and pathogen's natural history (Williams & Day, 2001). Nevertheless, the characterization of epidemiologically relevant parameters in the MG-HM system for multiple isolates has shed light not only on the dynamics of disease spread of each type, but also suggesfts that classical trade-offs between transmission, virulence, and recovery are shaping the evolution of this pathogen.

Appendix 1. Maximum likelihood estimation of transmission

Given that we consider closed population cages (very little mortality was detected, none of which could be unequivocally attributed to infection), we can roughly model the deterministic infection dynamics according to:

$$\begin{array}{cc}\hfill \frac{dS}{dt}=& -\beta SI+\gamma I\hfill \\ \hfill \frac{dI}{dt}=& \beta SI-\gamma I.\hfill \end{array}$$ (1)

with the force of infection being given by βI, where β is the transmission parameter, defined as the mean number of new infections caused by a typical infectious individual per unit of time, γ the constant recovery rate from infection, I the number of infected hosts and S the total number of susceptible hosts at time t, and parameters being strain-specific. Individuals were classified as susceptible or infectious according to the presence/absence of MG DNA found in conjunctival discharge. Changes in infectious status, from susceptible to infectious, or infectious to susceptible, were assumed to occur at the end of each time interval.

In the stochastic S-I-S model, the probability for a susceptible animal to escape infection over the course of the time interval dj is given by e−βIjdj+1, so that and the probability of becoming infected during the same time interval is therefore p = 1 – e−βIjdj+1. Therefore, the expected number of new infections generated by this process, Cj+1 , is approximately binomial, i.e. Cj+1 ~ Bin(Sj , pj), where S j is the number of susceptible hosts during the jth time period. The negative log likelihood function is thus $L={\sum }_j\beta S_j{d}_{j+1}-C_{j+1}\log (e^{\beta I_j{d}_{j+1}}-1)$, where j ranges j over the number of time periods for which measurements of S and I are made. Minimization of this quantity provides a point estimate of the transmission parameter β (Bolker 2008).

By a similar process, MLEs of recovery rates were obtained by minimizing the quantity $L={\sum }_j\gamma I_j{d}_{j+1}-R_{j+1}(\log (1-e^{-\gamma d_{j+1}})+\gamma d_j)$, where R_j+1 is the number of birds that recovered from infection during the j + 1 ^th time interval, again over the first 26 days.