On the definition and utilization of heritable variation among hosts in reproduction ratio r0 for infectious diseases

ABSTRACT Infectious diseases have a major role in evolution by natural selection and pose a worldwide concern in livestock. Understanding quantitative genetics of infectious diseases,

therefore, is essential both for understanding the consequences of natural selection and for designing artificial selection schemes in agriculture. The basic reproduction ratio, _R_0, is the

key parameter determining risk and severity of infectious diseases. Genetic improvement for control of infectious diseases in host populations should therefore aim at reducing _R_0. This

requires definitions of breeding value and heritable variation for _R_0, and understanding of mechanisms determining response to selection. This is challenging, as _R_0 is an emergent trait

arising from interactions among individuals in the population. Here we show how to define breeding value and heritable variation for _R_0 for genetically heterogeneous host populations.

Furthermore, we identify mechanisms determining utilization of heritable variation for _R_0. Using indirect genetic effects, next-generation matrices and a SIR (Susceptible, Infected and

Recovered) model, we show that an individual’s breeding value for _R_0 is a function of its own allele frequencies for susceptibility and infectivity and of population average susceptibility

and infectivity. When interacting individuals are unrelated, selection for individual disease status captures heritable variation in susceptibility only, yielding limited response in _R_0.

With related individuals, however, there is a secondary selection process, which also captures heritable variation in infectivity and additional variation in susceptibility, yielding

substantially greater response. This shows that genetic variation in susceptibility represents an indirect genetic effect. As a consequence, response in _R_0 increased substantially when

interacting individuals were genetically related. SIMILAR CONTENT BEING VIEWED BY OTHERS GENOME-WIDE INSIGHTS ON GASTROINTESTINAL NEMATODE RESISTANCE IN AUTOCHTHONOUS TUNISIAN SHEEP Article

Open access 29 April 2021 DETECTION OF SELECTION SIGNATURES FOR RESPONSE TO ALEUTIAN MINK DISEASE VIRUS INFECTION IN AMERICAN MINK Article Open access 03 February 2021 SPATIALLY STRUCTURED

ECO-EVOLUTIONARY DYNAMICS IN A HOST-PATHOGEN INTERACTION RENDER ISOLATED POPULATIONS VULNERABLE TO DISEASE Article Open access 13 October 2022 INTRODUCTION Infectious diseases are widespread

in humans, animals and plants. In natural populations, infectious diseases have a major role in the process of evolution by natural selection (Haldane, 1949; O'Brien and Evermann,

1988). In domestic populations, particularly in livestock, infectious diseases are imposing a worldwide concern owing to their impact on the welfare and productivity of livestock, and in the

case of zoonosis, also because of the threat for human health. To contain the threat imposed by infectious diseases, different control strategies such as vaccination, antibiotic treatments

and management practices have been implemented widely. However, the evolution of resistance to antibiotics by bacteria, evolution of resistance to vaccines by viruses and undesirable

environmental impacts of antibiotic treatment put these strategies under question (Gibson and Bishop, 2005). Thus, there is a need to investigate additional control strategies, so as to

extend the repertoire of possible interventions. A greater repertoire is favourable (1) because it allows for a change in approach when certain control measures fail and (2) because the use

of combinations of control measures make emergence of resistance against control more difficult. Several studies have demonstrated the existence of genetic variation for different disease

traits for a wide variety of infectious diseases. Examples are clinical mastitis and _Mycobacterium bovis_ infections in dairy cattle (Heringstad et al., 2005). Such studies usually focus on

estimating the genetic variance in individual disease status. As this approach connects an individual’s own disease status to its own pedigree, it only captures heritable variation in

susceptibility (or resistance) to disease (Lipschutz-Powell et al., 2012). However, host genetic variation may be present also in other traits that affect the dynamics of infectious diseases

in populations. Thus, to use a general term for such other traits, infectivity will also have an impact on the transmission of infectious diseases. There clearly exists (phenotypic)

variation in infectivity as it can be seen from the occurrence of superspreaders (Lloyd-Smith et al., 2005). Thus, it is most likely that the classical quantitative genetic analysis based on

individual disease status captures only part of the possible heritable variation in the host underlying infectious disease dynamics (Lipschutz-Powell et al., 2012). The ultimate goal of

selective breeding for disease traits is to reduce the risk of an epidemic and/or to reduce the level of the endemic equilibrium. In epidemiology, the key parameter determining the risk and

size of an epidemic and/or the level of the endemic equilibrium is the basic reproduction ratio, _R_0. _R_0 is the average number of secondary cases produced by a typical infectious

individual during its entire infectious life time, in an otherwise naive population (Diekmann et al., 1990)_. R_0 has a threshold value of 1, which determines whether a major disease

outbreak can occur or whether the endemic equilibrium exists. When _R_0<1, the epidemic will die out. On the other hand, when _R_0>1 major outbreaks or an endemic equilibrium

(persistence) can occur. Hence, breeding strategies to reduce the risk and prevalence of an infectious disease should aim at reducing _R_0, preferably to below a value of 1. Breeding to

reduce _R_0 raises a conceptual difference between quantitative genetics and epidemiology: _R_0 is an epidemiological parameter referring to an entire population, whereas quantitative

genetics rests on the concept of breeding value, which refers to a single individual. It is clear that in a genetically heterogeneous population, _R_0 is a function of individual genotypes

in the population, which in turn are a function of allele frequencies. Moreover, a change in allele frequencies will change _R_0, indicating _R_0 can respond to selection. Genetic

improvement aiming to reduce _R_0 should ideally be based on the effects of an individual’s genes on _R_0, which would require defining individual breeding values for _R_0. Moreover,

defining a breeding value for _R_0 would also allow defining heritable variation in _R_0, that is, the variation in individual breeding values for _R_0, which would give an indication of the

prospects for genetic improvement with respect to _R_0. For domestic populations, the subsequent question would be how to design breeding programs, so as to utilize optimally heritable

variation in _R_0 and achieve the greatest possible rate of reduction in _R_0. The equivalent issue for natural populations would be what ecological conditions are favourable for efficient

reduction of _R_0 by natural selection. For emergent traits that depend on multiple individuals, research in the field of indirect genetic effects (IGEs) suggests that group selection and

relatedness among interacting individuals (‘kin selection’) can be used to increase response to selection (Griffing, 1976; Bijma, 2011). This suggests that relatedness and group selection

may be important mechanisms affecting the utilization of heritable variation in _R_0, either by natural or artificial selection. Here we show how to define breeding value and heritable

variation for _R_0 for a genetically heterogeneous host population, where individuals differ for susceptibility and infectivity. For that purpose, we have adapted the theory of IGEs commonly

applied to socially affected traits, using the epidemiological concept of next-generation matrices (NGMs) (Diekmann et al., 1990, 2010). Furthermore, we examine the mechanisms determining

the utilization of heritable variation in _R_0, focusing on the effects of kin selection on response in _R_0, and in susceptibility and infectivity. MATERIALS AND METHODS DYNAMIC MODEL OF

INFECTION In a completely naive population where a microparasitic infection is introduced, the disease dynamics can be modelled with a basic compartmental stochastic SIR (Susceptible,

Infected and Recovered) model. In this model, individuals move through the states in the order S→I→R (Anderson et al., 1992). Therefore, the possible events that an individual may encounter

are infection and recovery. With stochasticity, these events occur randomly at a certain rate (probability per unit of time) specified by the model parameters. In the SIR model, these

parameters are the transmission rate parameter (_β_) for S→I, and the recovery rate parameter (_α_) for I→R. The transmission rate parameter _β_ is the probability per unit of time that a

typical infected individual infects another individual in a totally susceptible population (Diekmann et al., 1990; Anderson et al., 1992). When constant population density is assumed, the

rate at which the susceptible population becomes infected is _βSI_/_N_, where _S_ denotes the number of susceptible individuals, _I_ the number of infectious individuals and _N_ the total

number of individuals in the population (Kermack and McKendrick, 1991). The recovery rate parameter _α_ is the probability per unit of time for an infective to recover from an infection. In

other words, for constant _α_, the infectious period is exponentially distributed with a mean duration of _α_−1 time units. The transmission rate parameter, _β_, depends on the infectivity

of infectious individuals and on the susceptibility of uninfected recipient individuals. Thus, in a homogeneous population where all individuals have the same level of infectivity and

susceptibility, there is a single _β_ that applies to the whole population, which can be defined as a function of these parameters, where _γ_ is susceptibility, _ϕ_ is infectivity and _c_ is

average number of contacts an infectious individual makes per unit of time (see Table 1 for a notation key). DYNAMIC MODEL OF INFECTION WITH GENETIC HETEROGENEITY In a genetically

heterogeneous population, however, the transmission rate parameter _β_ may vary among pairs of individuals. This pairwise transmission rate will depend on the infectivity genotype of the

infectious individual and on the susceptibility genotype of the recipient susceptible individual. The assumption that transmission depends on the infectivity of only the infectious

individual and on the susceptibility of only the recipient individual is known as separable mixing (Diekmann et al., 1990). Thus, we may define the pairwise transmission rate parameter

_β__ij_ from an infectious individual _j_ to a susceptible individual _i_ as where γ_i_ denotes susceptibility of susceptible individual _i_ and _ϕ__j_ denotes infectivity of infectious

individual _j_. In Equation (2), _c_ represents the average contact rate; any variation in contact rate among susceptible and infectious individuals is included in _γ__i_ and _ϕ__i_ because

of the assumption of separable mixing. In the following, we model genetic heterogeneity in a diploid population using two biallelic loci, one locus for susceptibility effect (_γ_) and the

other locus for infectivity effect (_ϕ_). The susceptibility locus has alleles _G_ and _g_, with susceptibility values _γ__G_ and _γ__g_, respectively, and the infectivity locus has alleles

_F_ and _f_, with infectivity values _ϕ__F_ and _ϕ__f_, respectively. Furthermore, both loci are assumed to have additive allelic effects without dominance. Thus, genotypic values are given

by _γ__GG_=_γ__G_+_γ__G_=2_γ__G_, _γ__gg_=_γ__g_+_γ__g_=2_γ__g_ and _γ__Gg_=_γ__gG_=_γ__G_+_γ__g_ for susceptibility, and _ϕ__FF_=_ϕ__F_+_ϕ__F_=2_ϕ__F_, _ϕ__ff_=_ϕ__f_+_ϕ__f_=2_ϕ__f_ and

_ϕ__Ff_=_ϕ__fF_=_ϕ__F_+_ϕ__f_ for infectivity. As we assumed additive gene action, average susceptibility in the population is given by and average infectivity is given by where _p__f_ is

the frequency of the _f_ allele, _p__g_ the frequency of the _g_ allele and the ‘2’ arises because each individual carries two alleles. Note that and are average susceptibility and average

infectivity over individuals, and not average of allele effects. In a population as defined here, there are nine genotypes of individuals because of the combinations of their genotype for

susceptibility and infectivity. For this heterogeneous population, we can now construct the NGM. The NGM describes the number of infectious individual of each type in the next generation of

the epidemic, produced by infectious individuals of each type in the current generation. Then, we can calculate _R_0 as the dominant eigenvalue of the NGM. Under the assumption of separable

mixing, the dominant eigenvalue equals the trace of a matrix, and thus _R_0 can be obtained as the trace of the NGM (Diekmann et al., 2010). Appendix 1 shows the NGM for the population with

linkage equilibrium and in Hardy–Weinberg Equilibrium (HWE) described by Equations (2), (3), (4). _R_0 is given by the trace of the NGM: where _α_ is the recovery rate, which is assumed to

be the same for all individuals in the population. The NGM was also constructed for the more general case of a population that deviates from HWE and linkage equilibrium. For that case, _R_0

is given by (Appendix 2) where _F_IS is the inbreeding coefficient and measures deviation of the population from HWE. It is a function of observed heterozygosity (_H_o) and expected

heterozygosity (_H_e) in the population, The _D_ measures the deviation of the population from linkage equilibrium and expresses the excess of coupling phase haplotypes (Falconer and Mackay,

1996), The second term in brackets in Equation (6) is the covariance between susceptibility and infectivity of individuals in the population. When either (i) _D_=0 or (ii) _F_IS=−1, that

is, full disassortative ordering of alleles over diploid organisms (_H_o=2_H_e=1, which requires _p_=1/2) or (iii) there is no variance in either of the two traits ( or ), then there is no

covariance between the two traits and _R_0 is given by Equation (5). INDIVIDUAL BREEDING VALUE FOR _R_0 Equation (5) gives _R_0, which is an emergent trait of the population, that is, a

trait that arises when the different individuals (susceptible and infectious) interact (Dawkins, 2006). The objective here, however, is to define individual breeding values for _R_0. We use

results from the field of IGEs to define breeding value for _R_0. An IGE is heritable effect of an individual on the trait value of another individual (Griffing, 1967, 1976, 1981; Moore et

al., 1997; Wolf et al., 1998; Muir, 2005). Hence, infectivity is an IGE, as an individual’s infectivity affects the disease status of its contacts. Moore et al. (1997) and Bijma et al.

(2007) show how breeding value and genetic variance can be defined for such traits. Bijma (2011) shows how the approach can be generalized to any trait, including traits that are an emerging

property of a population, such as _R_0. They propose a (total) breeding value that follows from the genetic mean of the population, rather than from individual trait values. In classical

quantitative genetics, breeding value is the sum of the average effects of an individual’s alleles on its trait value, where the average effects equal the partial regression coefficients of

individual trait values on individual allele count (Fisher, 1919; Lynch and Walsh, 1998). For traits affected by IGEs, the total breeding value is the sum of the average effects of an

individual’s alleles on the mean trait value of the population (Bijma, 2011). For an emergent trait, however, there is only a single trait value for the entire population, and the average

effects of alleles on that trait follow from the partial derivatives of the trait value with respect to allele frequency, rather than from partial regression of individual trait values on

allele count. This is analogous to the derivation of economic values in livestock genetic improvement. Applying this approach to _R_0 (Equation (5)) with linkage equilibrium and HWE, average

effect of the _g_ allele equals and the average effect of the _f_ allele on _R_0 equals Consequently, the individual breeding value for _R_0 is given by where _p__g_,_i_ and _p__f_,_i_

refer to the allele frequencies in individual _i_, thus taking values of 0, 1/2 or 1. The equation for for the population that deviates from HWE and with linkage disequilibrium (LD) is

presented in Appendix 2. In the following, we will refer to as the breeding value for _R_0 of individual _i_. Note that, in contrast to the pairwise transmission rate parameter _β__ij_, an

individual’s breeding value for _R_0 is entirely a function of its own genes. This is because an individual transmits its own genes to its offspring, which may differ from the genes

affecting its own disease phenotype. The relationship between the breeding values of the individuals in a population of _n_ individuals and _R_0 of that population is The first term in

Equation (8) is the intercept that determines the magnitude of _R_0, but it does not depend on the allele frequencies and is not needed in the breeding value. The last term is there because

of the nonlinear relationship between _R_0 (Equation (5)) and susceptibility and infectivity. From Equation (8), it can be seen that changes in breeding value for _R_0 will lead to

corresponding changes (in magnitude and direction) in _R_0 itself. Only when also the frequencies in whole populations (_p__g_, _p__f_) are changing, the change in _R_0 will be more than the

change in breeding values due to this last term. In that case, selection that reduces both susceptibility and infectivity will lead to a greater reduction in _R_0 than predicted by the

breeding values. Response to selection in _R_0 will equal the change in average individual breeding value for _R_0, Hence, a (small) change in average individual breeding value for _R_0 due

to selection will generate the same change in _R_0. Thus, just as with an ordinary breeding value (Fisher, 1919; Lynch and Walsh, 1998), for a small change in allele frequency, the change in

mean breeding value for _R_0 equals response to selection in _R_0. HERITABLE VARIATION IN _R_0 Response to selection in any trait, including emergent traits such as _R_0, can be expressed

as the product of intensity of selection _ι_, accuracy of selection _ρ_T and total genetic standard deviation for that trait (Bijma, 2011), In the above equation, response to selection _R_

is change in mean trait value from one generation to the next. The selection intensity _ι_ is the selection differential expressed in standard deviation units. Accuracy of selection _ρ_T is

the correlation between the total breeding value and the selection criterion in the candidates for selection, and is the standard deviation in total breeding value for the trait in the

candidates for selection. Selection intensity and accuracy of selection are scale-free parameters and do not include any information about the heritable variance in the trait. Standard

deviation in total breeding value, on the other hand, reflects the potential of the population to response to selection. Note that heritable variation in the context of Equation (10)

strictly refers to the potential of a population to respond to selection, and may differ from the classical additive genetic variance in a trait. _R_0, for example, has no classical additive

genetic variance, as there exist no individual phenotypes for _R_0. Thus, in the following, heritable variation in _R_0 will refer to the potential for genetic change in _R_0, and not to

the additive genetic component of phenotypic variation in _R_0 among individuals. This conceptual difference is discussed in detail in Bijma (2011). From the above, it follows that heritable

variation in _R_0 equals the variance in breeding value for _R_0 among individuals in the population. We drop the prefix ‘total’ from breeding value and heritable variation, as _R_0 has no

classical breeding value. Taking the variance of Equation (7c), assuming linkage equilibrium, shows that heritable variation in _R_0 equals where is the variance among individuals in

breeding value for _R_0. Hence, Equation (11) shows how heritable variation in _R_0 depends on the susceptibility and infectivity effects of alleles and on the allele frequencies in the

population. The expression in Equation (11) may be recognized as the sum of the additive genetic variances at two independent loci. Additive genetic variance at a single locus is

traditionally written as 2_p_(1−_p_)_α_2, where _α_ denotes the average effect of an allele substitution (Falconer and Mackay, 1996). In Equation (11), the average effect at the

susceptibility locus equals , and average effect at the infectivity locus equals (see also Equations (7a–c)). UTILIZATION OF HERITABLE VARIATION IN _R_0 Efficient reduction of _R_0 by means

of selective breeding requires selection schemes that optimally utilize the heritable variation in _R_0. Because an individual’s infectivity represents an IGE, that is, a heritable effect of

the individual on the disease status of other individuals within the same epidemiological unit, optimal breeding schemes for traits affected by IGEs may provide a clue for the design of

optimal schemes for reducing _R_0. For traits affected by IGEs, group selection and relatedness among interacting individuals (‘kin selection’) increase response to selection (Griffing,

1967, 1976; Bijma and Wade, 2008). Moreover, Bijma (2011) shows that relatedness among interacting individuals in general tends to increase response to selection for traits that have an IGE.

We, therefore, considered a group-structured population, where group mates can be genetically related. The objective of this section is not to precisely quantify or predict response to

selection, but to identify and illustrate important factors affecting it. To investigate mechanisms affecting response in _R_0, a simulation study was performed on a population with discrete

generations. The genetic model was the same as described above. The population was subdivided into 100 groups of 100 individuals each. In each group, an epidemic was started by a single

randomly infected individual. After the end of an epidemic, selection was based on individual disease status (0/1), where only those that escaped the infection were selected from each group

to be parent of the next generation. For the next generation, selected parents were mated randomly and offspring genotypes were randomly sampled based on the parental genotypes. The size and

the number of groups were kept constant throughout the generations. Each group in the population was set up in such a way that group mates showed a certain degree of genetic similarity,

which we refer to as ‘relatedness’, _r_, here. The term ‘relatedness’ has different meanings in different scientific disciplines. In animal breeding, for example, relatedness is implicitly

understood as ‘pedigree relatedness’. In sociobiology, such as in studies on the evolution of altruism, on the other hand, relatedness is interpreted as a more general measure of genetic

similarity, irrespective of the cause of that similarity, for example, as a genetic regression coefficient (Hamilton, 1970; see also Frank, 1998). Here we define relatedness as the

correlation between the allele count of group mates, irrespective of the cause of that correlation. This definition agrees with the use of relatedness in animal breeding applications, such

as selection index theory and genomic relationship matrices, where the current population is treated as the base population (Falconer and Mackay, 1996). Relatedness at the susceptibility

locus, _r__γ_, and at the infectivity locus, _r__ϕ_, were allowed to differ. To achieve a certain relatedness among group mates, a fraction _f_ of fully related individuals was added to each

group, supplemented by a fraction 1−_f_ of randomly selected individuals. We did not consider negative values for relatedness, because the lower bound for relatedness is practically zero

when group size equals 100 individuals (_r_min=−1/99). Appendix 3 shows that the required fraction equals the square root of relatedness. Thus, a fraction of individuals that were fully

related to each other at the susceptibility locus, and a fraction of individuals that were fully related to each other at the infectivity locus were added to each group. As each individual

carries both loci, these additions cannot be done independently; details of the strategy to jointly make those additions are given in Appendix 4. The simulation was further extended to allow

for a certain degree of LD between both loci. However, for a given LD in the population, there exists an upper and lower bound for _r__γ_ given _r__ϕ_ and vice versa. For example, when both

loci are in strong positive LD and relatedness is zero at the susceptibility locus, then it is not possible to have very high relatedness at the infectivity locus. Appendix 5 provides

expressions for those bounds. Four different scenarios were simulated (Table 2). First, a scenario with heritable variation at both the susceptibility and the infectivity locus and groups

created randomly with respect to relatedness _r_ among group mates. No LD and a recombination rate _θ_ of 0.5 between both loci were further assumed. Second, varying degrees of relatedness

were used, which were the same at both loci. Third, to investigate a potential effect of relatedness on response in susceptibility, heritable variation was simulated at the susceptibility

locus only, for varying degrees of relatedness among group mates. Finally, to investigate the potential effect of relatedness on response in _R_0 in the case where there is strong negative

LD between both loci and no recombination, a scenario with a relatedness of either 0 or 0.1 at both loci was simulated. SIMULATION RESULTS In the first scenario, which had unrelated group

mates, a response to selection was observed only at the susceptibility locus, where the _G_ allele became fixed after an average of 100 generations. At the infectivity locus, in contrast,

only a random fluctuation of allele frequency was observed (Figure 1). Thus, with groups composed at random with respect to relatedness, no response was observed at the infectivity locus. As

a result, in the final generation, the response in _R_0 was limited. In the second scenario, which had related group mates, response to selection was observed at both loci, and the

population became fixed for the G-allele at susceptibility locus and for F-the allele at the infectivity locus (Figures 2 and 3). In this case, selection resulted in a greater reduction of

_R_0 than in the first scenario (Figure 4 vs Figure 1). As relatedness among group mates increased, response was much faster in all three traits. As it was also faster on the susceptibility

locus, this suggested that also the susceptibility locus showed an IGE. To verify this IGE in susceptibility in the third scenario, we chose to have variation in the susceptibility only.

Also in this case, the response at the susceptibility locus increased substantially when relatedness among group mates increased (Figure 5). For selection on individual phenotype, it is

known that relatedness increases response in the IGEs, but not in the direct genetic effects (Griffing, 1976; Bijma and Wade, 2008). Thus, this result suggests that (1) susceptibility not

only has a direct genetic effect on the disease status of the individual itself but also has an IGE on the disease status of its groups mates, and that (2) this indirect genetic variance is

utilized by kin selection (see Discussion), even in the absence of genetic variance in infectivity. In the fourth scenario, which had strong negative LD and no recombination, the direction

of response in _R_0 depended on the relatedness among group mates. Without relatedness, selection fixed the _G_ allele irrespective of the linked allele at the infectivity locus. As a

consequence, selection increased the frequency of _f_ allele, yielding an increase rather than a decrease of _R_0. When relatedness _r__γ_=_r__ϕ_=0.1 was used, however, selection caused

fixation of the GF haplotype, resulting in a decrease in _R_0 (Figure 6). This result shows that kin selection can prevent a maladaptive response to selection. DISCUSSION The aim of this

study was to define the breeding value and heritable variation for _R_0. This was done for a diploid host population with genetic variation for susceptibility and infectivity. Breeding

values of individuals were derived by finding the _R_0, linearizing this value in the allele frequencies and substituting the individual’s allele frequencies. The heritable variation that

measures the potential for response in _R_0 can then be found by taking the variance of the breeding values in the population. We applied this approach to a simple SIR model with genetic

variation in susceptibility and infectivity, and assuming separable mixing. The second focus of this paper was to investigate the mechanisms that affect response in _R_0. As genetic

relatedness between interacting individuals is expected to increase response in the general case (Bijma, 2011), we hypothesized that this result would extend to _R_0 and considered a

group-structured population with related group members. Our results show that, with unrelated group members and no LD between both loci, selection based on individual disease status yields

response in susceptibility only. In the absence of relatedness, response in infectivity depends entirely on the correlation with susceptibility, which was zero in the absence of LD.

Relatedness among group members increased response in _R_0 in two ways. First, with related group members, selection for individual disease status captures the heritable variation in

infectivity. This occurs because an individual that carries the favourable allele for infectivity has group mates with a below-average infectivity, which increases its probability of

escaping the epidemic, and thus being selected. Second, relatedness among group mates increases response in susceptibility. This occurs because an individual that carries the favourable

allele for susceptibility on an average has fewer infected group mates, which increases its probability of escaping the epidemic and being selected. These results show that not only

infectivity but also susceptibility exhibits an IGE; at the same level of infectivity, individuals with lower susceptibility have a reduced chance of infecting others simply because they

have a lower chance of being infected themselves. The net result of both mechanisms is a strong increase in response to selection in _R_0 when relatedness increases. To quantify the impact

of relatedness on the accuracy of selection for _R_0, we calculated the correlation between the selection criteria (healthy/infected) and the breeding value for _R_0. Using the parameter

values presented in Scenario 2, Table 2, accuracy of selection increased from 0.05 to 0.24 when relatedness increased from 0 to 1. Thus, our study further supports the claim of Bijma (2011)

that relatedness is an important factor in utilization of heritable variation in traits affected by IGEs. Our results suggest that relatedness among interacting individuals can be used in

livestock breeding programs aiming to reduce disease incidence. In current breeding strategies in livestock, data on individual disease status is connected to the pedigree of individuals to

estimate breeding values. When interacting individuals are unrelated, those breeding values capture only the direct genetic effect, that is, the direct genetic part of susceptibility.

Breeding values can be improved by also considering IGEs, for example, by fitting direct–indirect genetic effects models to data on disease status (Lipschutz-Powell et al., 2012). However,

estimating direct and indirect breeding values for disease status is methodologically challenging because the linear mixed models traditionally used in quantitative genetics do not fit the

nonlinear dynamics of infectious diseases (Lipschutz-Powell et al., 2012). The use of related group members may offer a low-tech solution, for capturing more of the heritable variation in

_R_0 without the need to explicitly model IGEs. In this work, we have assumed that the selection objective is to reduce _R_0. While this is probably the obvious choice for epidemiologists,

it may be unexpected for breeders who are not very familiar with _R_0. For breeders, reducing disease incidence might be the more common objective. For example, in the context of our

two-locus model, breeders might specify an objective _H__i_=_v_γ_p__g_,_i_+_v__ϕ__p__f_,_i_, where _v__γ_ and _v__ϕ_ are the so-called economic values for susceptibility and infectivity,

respectively, which would be the partial derivatives of disease incidence with respect to the population allele frequencies _p__g_ and _p__f_. However, both objectives are very similar, both

for epidemic and endemic diseases. For epidemic diseases, the ultimately affected fraction of the population, known as the final size 1−_s_(∞), is determined by _R_0, as is shown by the

final size equation: ln _s_(∞)=_R_0(_s_(∞)−1) (Kermack and McKendrick, 1991). For endemic diseases, the equilibrium-affected fraction is given by: 1−_s_(∞)=1−1/_R_0. Hence, the relationship

between disease incidence and allele frequency occurs entirely _via R_0, both for epidemic and endemic diseases. Thus, when the objective is to decrease incidence, the economic values for

any disease trait, say _x_, that is, the partial derivatives of incidence with respect to that trait, can be written as In this expression, the _∂i_/_∂R_0 is a constant that is the same for

all individuals in the population, and is independent of the disease trait considered (e.g. susceptibility or infectivity). Thus, the ranking of individuals will be the same, irrespective of

whether they are ranked on breeding value for incidence or on breeding value for _R_0. Beware that breeding for incidence is not the same as breeding for susceptibility. When comparing

breeding for susceptibility to breeding for _R_0 or incidence, the latter is to be preferred because it also covers the heritable variation originating from infectivity (e.g. Figure 4 vs

Figure 1). With respect to the evolution of parasite virulence, also the key role of kin selection has been recognized (Levin and Pimentel, 1981; Frank, 1996; Galvani, 2003). Much less

attention has been given to the potential for kin selection acting on the host population. Using Monte Carlo simulation, Fix (1984) showed that the presence of kin groups in a small-scale

human population considerably accelerated the increase in frequency of a resistance allele. Schliekelman (2007) seems to be the first who used rigorous mathematical modelling to investigate

the impact of kin selection on the frequency of mutant alleles conferring resistance to the host. Moreover, despite the evidence of heterogeneity in infectivity (Woolhouse et al., 1997;

Lloyd-Smith et al., 2005; Doeschl-Wilson et al., 2011), little attention has been given to the effect of kin selection on the frequency of alleles affecting infectivity in the host

population. Our simulations show that, at least in theory, kin selection can greatly accelerate the evolution of _R_0, because it utilizes the indirect genetic variance in both

susceptibility and infectivity in the host population. For any actual case, the potential impact of kin selection will of course depend critically on the magnitude of this indirect genetic

variance. Particularly, the component due to genetic variation in infectivity is unknown at present, but first steps towards estimating this component have recently been made

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This study was financially supported by EU Marie Curie NematodeSystemHealth (ITN-2012-264639). The contribution of PB was supported by the foundation for applied sciences (STW) of the Dutch

science council (NWO). AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Animal Breeding and Genomics Centre, Wageningen Institute of Animal Sciences (WIAS), Wageningen University, Wageningen,

The Netherlands M T Anche & P Bijma * Quantitative Veterinary Epidemiology Group, Wageningen Institute of Animal Sciences (WIAS), Wageningen University, Wageningen, The Netherlands M T

Anche & M C M de Jong Authors * M T Anche View author publications You can also search for this author inPubMed Google Scholar * M C M de Jong View author publications You can also

search for this author inPubMed Google Scholar * P Bijma View author publications You can also search for this author inPubMed Google Scholar CORRESPONDING AUTHOR Correspondence to M T

Anche. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no conflict of interest. APPENDICES APPENDIX 1 This appendix shows the construction of the NGM (Diekmann et al., 2010) and

_R_0 for a diploid population where there is no LD between the locus affecting susceptibility and the locus affecting infectivity. In such population, we have nine types of individuals for

the combination of their genotype for susceptibility (_gg_, _gG_, _GG_) and infectivity (_ff_, _fF_, _FF_). Thus, the NGM has nine rows and nine columns. The column of the matrix represents

the contributions to the next generation by infectious individuals of the genotype written above the column (‘cause’). The rows indicate the genotypes of the susceptible individuals that

become infected (‘consequence’). In the following, we present the NGM on three rows: the first row gives columns 1–3, the second columns 4 –6 and the final row columns 7–9. The NGM uses the

transmission rate parameters between genotypes, which are given by _R_0 is the dominant eigenvalue of the NGM. As we have the so-called separable mixing, where elements of the NGM are

products of the rows and columns, the NGM has a single eigenvalue only, which therefore equals the trace of the NGM. Thus, _R_0 is the sum of the diagonal elements of the NGM (given in bold

above), in which and . APPENDIX 2 The NGM was also constructed for a population that deviates from LD and HWE. Because of LD, the genotype _gGfF_ has to be partitioned into the two possible

haplotypes for this genotype, _gfGF_ and _gFGf_. Hence, when accounting for LD, the NGM includes 10 distinct genotypes, rather than the 9 considered in Appendix 1 (Table A2-1). To avoid over

presentation of results, we only give the trace of the NGM, which equals _R_0 because of the separable mixing assumption, Here _β__vwxy_ represents the transmission rate parameter within a

genotype, that is, from genotype _vwxy_ to genotype _vwxy_, For example, _β__gFGF_=_γ__gG__ϕ__FF__c_. The haplotype frequencies are where _D_ is the usual measure of LD (see main text). The

genotype frequencies are After few steps of algebraic manipulation, Equation (A2-1) will reduce to Individual breeding values for _R_0 were obtained by linearizing _R_0 in the allele

frequencies, using partial first derivatives, and subsequently substituting individual allele frequencies (i.e. 0, 1/2 or 1) APPENDIX 3 As mentioned in the main text, relatedness at the

susceptibility locus, _r__γ_, and at the infectivity locus, _r__ϕ_, were allowed to be different. With a single biallelic locus, pairwise relatedness between individuals takes only three

discrete values. However, our interest is in a continuum of the average relatedness among the individuals that together make up a group. To achieve a certain average relatedness among group

mates, a fraction _f_ of fully related individuals was added to each group, supplemented by a fraction 1−_f_ of randomly selected individuals. In this appendix, we show that the required

fraction equals the square root of relatedness at each locus, that is a fraction of random individuals will be replaced by individuals that were fully related to each other at the

susceptibility locus, and for the infectivity locus this is a fraction . We defined relatedness as the correlation between the genotypes of two group mates, say _x_ and _y_, As the same

theory applies to both loci, we will show the derivation for the susceptibility locus only. Because the addition strategy should not change allele frequency in the population nor affect the

HWE, the population needs to have three types of groups. The first type has _gg_ individuals added to the group. The second type has _gG_ individuals added and the third type has _GG_

individuals added. The number of groups of the first type equals _no. groups_ × _p_2, the number of groups of the second type equals _no. groups_ × 2_p_(1−_p_), and the number of groups of

the third type equals _no. groups_ × (1−_p_)2, where _p_ is the frequency of the _g_ allele. The frequency of _g_ in the three types of groups is then To derive the correlation, we first

derive the covariance between genotypic values of group members, where, for example, _E_(_xy_|1) denotes the expectation of the product of the genotypic values of two group members in a

group of type 1. To simplify the derivation, without loss of generality, _g_ was given an effect of 1 and _G_ an effect of 0. As we are interested in additive genetic relationship, resulting

genotypic values are 2 for _gg_, 1 for _gG_ and 0 for _GG_. Thus, _x_ and _y_ denote genotypic values, taking values of either 0, 1 or 2. The possible genotypes of two individuals and the

corresponding values for _E_(_xy_|_group type_) are presented in the table below. Since the genotypic value for _GG_ equals zero, any pair of individuals involving at least one _GG_

individual has _E_(_xy_)=0, and is therefore left out of the table. If we insert Equations (A4-2) for _p_1 and Equation (A4-3) and (A4-4) for _p_2 and _p_3, respectively, and sum up all the

elements in each of the three column for _E_(_xy_), we find And as Then, Next, we need to calculate _E_(_x_) and _E_(_y_): Then, Then, covariance will be Next, the variances are given by

Then, Equations (A3-1) becomes Simplifying this expression yields Thus, to achieve a certain relatedness, a fraction of fully related individuals should be added to each group. APPENDIX 4

This appendix contains an example demonstrating the strategy to make additions in each group, so as to achieve a certain relatedness for susceptibility and infectivity among group mates. We

considered 100 groups, each with 100 individuals. Let us assume that LD (_D_)=0.15, and that the allele frequency at susceptibility locus is 0.5 and allele frequency at infectivity locus

equals 0.6. Thus, _P__g_=0.5 and _P__f_=0.6. The _r__γ_=0.75 and _r__ϕ_=0.6. It is assumed that the population is in Hardy–Weinberg equilibrium. The haplotype frequencies will be As

_r_=_f_2, then the fraction _f__γ_ of individuals that are fully related at their susceptibility locus will be . And the fraction _f__ϕ_ of individuals that are fully related at their

infectivity locus will be . Because the required fraction is lowest for the infectivity locus, we start with the infectivity locus. Thus, in each of the 100 groups we added individuals that

are fully related at their susceptibility and infectivity locus. The first 100 × _f__gf_2 groups will contain 77 individuals with _gfgf_ genotype, 100 × 2_f__gf__ f__gF_ groups will contain

77 individuals with _gfgF_ genotype, 100 × 2_f__gf__ f__Gf_ groups will contain 77 individuals with _gfGf_ genotype, 100 × 2_f__gf__ f__GF_ groups will contain 77 individuals with _gfGF_

genotype, groups contain 77 individuals with _gFgF_ genotype, 100 × 2_f__gF_ _f__Gf_ groups will contain 77 individuals with _gFGf_ genotype, 100 × 2_f__gF_ _f__GF_ groups contain 77

individuals _gFGF_ genotype, groups will contain 77 individuals with _GfGf_ genotype, 100 × 2_f__Gf_ _f__GF_ groups will contain 77 individuals with _GfGF_ genotype and finally, groups will

contain 77 individuals with _GFGF_. With respect to the infectivity locus, there are _p__f_2 × 100=36 groups that contain a fraction of individuals that are of _ff_, 2_p__f_(1−_p__f_) ×

100=48 groups that contain a fraction of individuals that are of _fF_ genotype and (1−_p__f_)2 × 100=16 groups that contain a fraction of individuals that are of _FF_ genotype at their

infectivity locus. Thus, the desired additions for the infectivity locus are achieved. With respect to the susceptibility locus, we have _p__g_2 × 100=25 groups that contain 77 individuals

that are of _gg_, 2_p__g_(1−_p__g_) × 100=50 groups that contain 77 individuals that are of _gG_ genotype and (1−_p__g_)2 × 100=25 groups that contain 77 individuals that are of _FF_

genotype at their infectivity locus. For the susceptibility locus, however, the required number of individuals to be added equals . As we have already added 77 individuals that are fully

related at their susceptibility locus, what is left to add to the group is 87−77=10 individuals Thus, the next addition will be 10 individuals that are fully related at their susceptibility

locus, but taken at random with respect to their infectivity locus (so that relatedness as the infectivity locus is not affected). Therefore, for those groups that already have a fraction of

individuals with _gg_ genotype, we will add 10 more individuals that are off _gg_ genotype. Analogously, to groups that already have a fraction of individuals with a certain genotype, 10

more individuals with that genotype are added. As the groups size is assumed to be 100, the rest of the group, which are 100−87=13 individuals, will be assigned randomly. APPENDIX 5 In this

appendix we presented the lower (min) and upper (max) bound for _r__γ_ given _r__ϕ_ and vice versa for a given LD, _D_. These bounds follow from the fraction of available individuals for the

second addition step (see Appendix 4), which depends on the allele frequencies, _D_, and relatedness at the locus in the first addition step. When _D_>0, When _D_<0, When _D_=Max

(_D_)=±0.25, RIGHTS AND PERMISSIONS Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Anche, M., de Jong, M. & Bijma, P. On the definition and utilization of heritable

variation among hosts in reproduction ratio _R_0 for infectious diseases. _Heredity_ 113, 364–374 (2014). https://doi.org/10.1038/hdy.2014.38 Download citation * Received: 05 December 2013 *

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