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- Abstract
- 1. Introduction
- 2. Roles of the reproductive number in a mass action epidemic
- 3. A model for homogeneous individuals in a community of households of varying sizes
- 4. Five reproductive numbers
- 5. The relative magnitude of the reproductive numbers
- 6. Proofs of inequalities in Theorem 1
- 7. Numerical simulations
- 8. Discussion and implications
- References

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Math Biosci. Author manuscript; available in PMC 2010 September 1.

Published in final edited form as:

Published online 2009 June 25. doi: 10.1016/j.mbs.2009.06.002

PMCID: PMC2731010

NIHMSID: NIHMS127743

E. Goldstein,^{a,}^{1} K. Paur,^{b} C. Fraser,^{c} E. Kenah,^{d} J. Wallinga,^{e,}^{f} and M. Lipsitch^{a}

The publisher's final edited version of this article is available at Math Biosci

See other articles in PMC that cite the published article.

Many of the studies on emerging epidemics (such as SARS and pandemic flu) use mass action models to estimate reproductive numbers and the needed control measures. In reality, transmission patterns are more complex due to the presence of various social networks. One level of complexity can be accommodated by considering a community of households. Our study of transmission dynamics in a community of households emphasizes five types of reproductive numbers for the epidemic spread: household-to-household reproductive number, leaky vaccine-associated reproductive numbers, perfect vaccine reproductive number, growth rate reproductive number, and the individual reproductive number. Each of those carries different information about the transmission dynamics and the required control measures, and often some of those can be estimated from the data while others cannot. Simulations have shown that under certain scenarios there is an ordering for those reproductive numbers. We have proven a number of ordering inequalities under general assumptions about the individual infectiousness profiles. Those inequalities allow, for instance, to estimate the needed vaccine coverage and other control measures without knowing the various transmission parameters in the models. Along the way, we’ve also shown that in choosing between increasing vaccine efficacy and increasing coverage levels by the same factor, preference should go to efficacy.

Over the last few years, there has been much interest in analyzing data from SARS, past influenza pandemics, and other diseases to estimate the reproductive numbers of these infections. A practical goal in making such estimates is that reproductive numbers can be used to gauge the effort that would be required to control such infections, either by classical public health measures such as isolation and quarantine, by biological interventions such as antivirals or vaccines, or by social interventions to reduce contact. A fundamental insight from simple models of infectious disease transmission is that the critical proportion *p _{c}* of transmission events that must be blocked by such measures to halt the growth of an epidemic is given by the equation 1

$${p}_{c}=1-\frac{1}{{R}_{0}}$$

(1.0.1)

Here *R*_{0} is the basic reproductive number, or the mean number of infections caused by a typical infectious case. The validity of equation 1.0.1 depends on the mass action assumptions for epidemic transmission; equation 1.0.1 is often applied to an early stage of an epidemic in a large community. We note that equation 1.0.1 is also valid under certain other assumptions besides mass action, such as no small loops and no depletion of susceptibles.

Another known consequence of the mass action assumptions, often used to compute *R*_{0}, is that the reproductive number can be related to the epidemic curve, and to the growth rate of an emerging epidemic; this will be discussed further in Section 2. All this makes the mass action model a convenient (and widely used) set-up for estimating the various epidemiological quantities, and for the subsequent specification of the required control measures. In reality, these mass action assumptions are violated due to the existence of complex social networks. Hence, it is possible that estimates of the effort needed to halt the growth of an epidemic are significantly biased. It would be useful to know the directions and magnitude of these biases, and to provide quantitative bounds on the required measures even though the estimates of transmission are made under models that involve simplifications of reality.

In this paper we focus on one important departure from mass action mixing, the existence of small, closely connected groups of people in which transmission is localized and possibly quite intense – that is, individuals within the group tend to mix preferentially with others in the group, and to subject other members of their group to a sustained risk of transmission. The classic example of such a group is a household, though the notions presented here may be generalized to other, similar settings; the basic idea is that there are two levels of mixing (Ball et al. (1997)), one local and one global. In this context, the universal role played by the reproductive number in the mass action model, both in characterizing the epidemic’s dynamics and the required control measures, is no longer valid. In fact, given the two levels of mixing, there are several reproductive numbers describing the spread of an epidemic and the various control measures that can be implemented to stop it. This paper deals with those reproductive numbers and the relations between them. It is organized as follows:

In Section 2 we review the roles played by the reproductive number under the simplifying assumptions of mass action. In Section 3 we present a model for the stochastic transmission dynamics in a community of households, following the one of Ball et al. (1997); Britton and Becker (2000) - see also House and Keeling (2008); Ferguson and Dodd (2007) for a treatment of related issues in a deterministic model. In Section 4 we define five reproductive numbers that arise in this context. Three of these reproductive numbers are quantities that may be measured in an emerging epidemic, while two others encapsulate the extent of control measures directed at individuals that would be required to stop the growth of such an epidemic. While all five numbers are equal in the setting of a mass-action epidemic, their magnitudes diverge in a population with a household structure. In Section 5 we note that all five are epidemic thresholds (the requirement for epidemic spread is that each of them exceeds one), but when they exceed one, their magnitudes diverge. Numerically, we show that there is in most cases a consistent ordering among them, so that the three measurable reproductive numbers provide bounds on the two relevant for control. Analytically, we demonstrate that most of these ordering relations are generally true (Theorem 1). Along the way, we also prove that in the household model, increasing the vaccine efficacy is better than scaling the vaccination coverage level by the same factor (Proposition A.2.3), with the two measures having the same effect in a mass action model. The rest of the paper deals with proving Theorem 1 (Section 6 and Appendix A), analyzing the limiting cases for the remainder of the ordering relations (Appendix B), and presenting the numerical model for computing those reproductive numbers under various assumptions on individual infectiousness profiles (Section 7). We conclude by discussing the relevance of these comparisons for public health, and in particular for estimating important quantities during emerging epidemics (Section 8).

In a mass action model, the critical proportion of infectious contacts that must be blocked to halt the growth of an epidemic is given by equation (1.0.1). If a vaccine is to be used as the sole control measure, then effective immunization of a fraction *p _{c}* of the population is required; likewise, preventing

Under mass-action mixing, or more generally, under the assumption of no small loops and no depletion of susceptibles, there are several known techniques for estimating the reproductive number. The available information often includes the epidemic curve (number of new cases on each day) and a known distribution *w*( ) for the infectious contact interval. The latter can be understood in several ways: under no depletion of susceptibles, it represents the distribution of times from appearance of one case in the epidemic curve to appearance of secondary cases caused by it, Kenah et al. (2008); alternatively, *w*(*t*) equals the proportion of the average individual’s infectivity which falls on day *t* since infection. Given that information, one may estimate the effective daily reproductive number *R*(*t*), which equals the mean number of infections caused by an individual, who got infected at time (day) *t* - see Wallinga and Teunis (2004). Essentially, this technique produces a local estimate of how fast the epidemic is growing on each day by attributing cases in the present to their likely infectors in the past, and then estimating how many cases are attributed to each infector on a given day. When detailed epidemic curve data are not available, or when one is interested in the average value of *R*(*t*) over some interval (*t*_{1}, *t*_{2}), a special case of this approach is to estimate the exponential growth rate *r* of the epidemic over that interval as

$$r=\frac{lnN({t}_{2})-lnN({t}_{1})}{{t}_{2}-{t}_{1}}$$

(2.0.2)

where *N*(*t*) is the number of persons infected on day *t*, and then to estimate the value of *R* during that interval using the expression

$$R=\frac{1}{{M}_{w}(-r)}$$

(2.0.3)

where the moment-generating function

$${M}_{w}(z)=\underset{a=0}{\overset{\infty}{\int}}w(a){e}^{za}da$$

(2.0.4)

These two approaches are closely related (Wallinga and Lipsitch (2007)), converging to the same answer if *R*(*t*) remains constant during the interval for which data are analyzed.

The definition of *R* has been rigorously extended to the setting in which the population is stratified into a number of groups (e.g., age groups or sexes or sexual activity groups) with different mixing patterns: in this case, members of certain groups are more likely than others to become infected, and these individuals may in turn be more or less likely to infect others. The appropriate definition of a “typical” infective in this setting is a weighted average over the groups, in which the weights are derived from the leading right eigenvector of the next generation matrix (*R _{ij}*), where

In summary, for a mass action model, the same reproductive number may be used for four different purposes: as the mean number of secondary cases caused by a typical infectious person, as a well-defined function of the exponential growth rate of the epidemic and the infectious contact interval distribution, as the proportion of contacts that must be blocked to halt the growth of the epidemic (all-or-nothing transmission reduction), and as the proportional reduction in the probability of each transmission that is required to halt the growth of an epidemic (leaky intervention).

In the next section, we define a model for transmission in a community of households, and then in Section 4 we discuss these four reproductive numbers and one additional one in the context of a community of households.

The model used here builds on the seminal paper of Ball et al. (1997); see also Britton and Becker (2000).

We consider a population of households, in which the infection is spreading. The relative frequency of individuals in the population who live in a household of size *h* is *π _{h}*, with Σ

$${\beta}_{G}(t)=E(I(t))$$

(3.0.5)

The number

$$C{I}_{G}={\int}_{0}^{\infty}{\beta}_{G}(t)dt=E({\int}_{0}^{\infty}I(t)dt)$$

(3.0.6)

is the expected cumulative infectiousness of an infected individual. Having introduced the individual infectiousness profile, we can now describe the infectiousness hazard that an infected individual poses, both outside of his/her household and within the household.

Outside of the household, for a person infected *t* time units ago, his/her infectiousness hazard to the community is *μ _{G}I*(

$$\mathrm{\Delta}t\xb7{\mu}_{G}\xb7I(t)$$

(3.0.7)

Thus the total number of people that one infected person is expected to infect outside of his/her household is

$${R}_{G}=E({\int}_{0}^{\infty}{\mu}_{G}I(t)dt)={\mu}_{G}{\int}_{0}^{\infty}{\beta}_{G}(t)dt={\mu}_{G}C{I}_{G}$$

(3.0.8)

Within a household of size *h*, given a person *A* infected *t* time units ago and having an individual infectiousness profile *I*(*t*), and an uninfected person *B*, the probability that *A* will infect *B* during time Δ*t* is

$${\mu}_{h}\xb7\mathrm{\Delta}t\xb7I(t)$$

(3.0.9)

Here *μ _{h}* > 0 is a number, which depends on the household size. Note that the out-of-household hazard, given by equation 3.0.7, and the within-household hazard, given by equation 3.0.9 are proportional. This represents the assumption that the overall shape of the infectiousness curve is determined by biological (e.g. pathogen shedding, fluid production) and behavioral (e.g. amount of time spent sleeping, social contact) characteristics of the infection, and that household contacts simply get a different, but proportional, exposure to the infectious individual than outside contacts. We also note that while in- and out-of household infectiousness hazards have a similar form, the infectiousness process taking place within an infected household is quite different from the mass action (branching) process for out-of-household infections, due to a fixed number of susceptibles in the household. Some insight into that process will be gained via a Sellke-type construction in Appendix A. In the meantime, let us introduce some notation to be used later.

Consider a household of size *h* infected *t* time units ago - this means that the first, or index case got infected *t* time units ago. Let *I _{h}*(

$${\beta}_{h}(t)=E({I}_{h}(t))$$

(3.0.10)

be the expected intensity of infectiousness of a household of size *h* infected *t* time units ago. Let

$$C{I}_{h}={\int}_{0}^{\infty}{\beta}_{h}(t)dt$$

(3.0.11)

be the expected cumulative infectiousness of an infected household of size *h*. From equation 3.0.7 we see that the outside hazard posed by a household of size *h* infected *t* time units ago is *μ _{G}I_{h}*(

$${R}_{h}=E({\int}_{0}^{\infty}{\mu}_{G}{I}_{h}(t)dt)={\mu}_{G}{\int}_{0}^{\infty}{\beta}_{h}(t)dt={\mu}_{G}C{I}_{h}$$

(3.0.12)

We conclude this section with the following well-known fact (see Andersson and Britton (2000), p. 15), whose proof we include for reader’s convenience:

$$C{I}_{h}={f}_{h}\xb7C{I}_{G}$$

(3.0.13)

where f_{h} is the expected number of people who will be eventually infected in a household of size h. Thus we also have

$${R}_{h}={f}_{h}\xb7{R}_{G}$$

(3.0.14)

Let the individuals in the household be 1,…,*h*. Define the random variable *A _{i}* to be 1 if person

$${f}_{h}=E(\mathrm{\sum}{A}_{i})=\mathrm{\sum}P({A}_{i}=1)$$

Let *Tl _{i}* be the cumulative (total) infectiousness of the

$$E(T{l}_{i}{A}_{i}=1)=C{I}_{G}$$

Thus

$$\begin{array}{l}C{I}_{h}=E(\mathrm{\sum}{Tl}_{i})=E(\mathrm{\sum}{A}_{i}\xb7{Tl}_{i})=\mathrm{\sum}P({A}_{i}=1)\xb7E(T{l}_{i}{A}_{i}=1)=& =C{I}_{G}\xb7\mathrm{\sum}P({A}_{i}=1)=C{I}_{G}\xb7{f}_{h}\end{array}$$

The fact that *R _{h}* =

In this section, we define five possible reproductive numbers that are relevant to transmission of infection in a community of households. All except the second (to our knowledge) have been described previously in various contexts. We also assume that we are dealing with an early stage of an epidemic in a large community - see Ball et al. (1997) for the limiting behavior in that transmission model as the community size goes to infinity. In particular, there is no depletion of susceptible households, and no household is infected more than once from outside.

One definition of the reproductive number that has been widely used in the literature on transmission in communities of households is the “household reproductive number” (we call it *R _{H}*; it is also occasionally called

We know from equation 3.0.14 that the expected number of households infected by an infected household of size *h* is *R _{h}* =

$${R}_{H}=\sum _{h}{\pi}_{h}{R}_{h}=\sum _{h}{\pi}_{h}{f}_{h}{R}_{G}f\phantom{\rule{0.16667em}{0ex}}{R}_{G}$$

(4.1.1)

where as before, *R _{G}* is the mean number of out-of-household infections by a single infected individual;

This reproductive number is usefully thought of as corresponding to the extent of reduction in between-household transmission required to stop the epidemic from spreading (Becker and Dietz (1995); Fraser (2007)).

This reproductive number ignores the distinction between within- and between- household infections and counts the expected number of secondary cases caused by an average infected individual from an average infected household, including those outside and inside the household. There is a caveat in this definition, stemming from double averaging - see Becker and Dietz (1995) for an alternative way of counting. Our approach is as follows:

First, for each household size *h*, pick *N* infected households of size *h*, where *N* is large. Let *A _{N}* be the total number of people eventually infected in those households, and let

We have

$${A}_{N}N\xb7{f}_{h},\phantom{\rule{0.16667em}{0ex}}{B}_{N}{A}_{N}-N+{A}_{N}\xb7{R}_{G}$$

We define

$${R}_{hI}=li{m}_{N\to \infty}E(\frac{{B}_{N}}{{A}_{N}})={R}_{G}+\frac{{f}_{h}-1}{{f}_{h}}$$

(4.2.1)

to be the expected number of people infected by one infected person in a household of size *h*. Now we need to average this out over various household sizes. We will choose a stratified average. The proportion of people living in households of size *h* among all people is *π _{h}*; this is also the proportion of infected households of size

$${R}_{HI}=\sum {\pi}_{h}{R}_{hI}=\sum {\pi}_{h}(\frac{{f}_{h}-1}{{f}_{h}}+{R}_{G})g+{R}_{G},$$

(4.2.2)

where $g\sum {\pi}_{h}{\scriptstyle \frac{{f}_{h}-1}{{f}_{h}}}$.

There are merits and demerits to this definition of *R _{HI}*, discussed further in Appendix A.4.2. We note that

*R _{H}* defines the extent to which transmission must be reduced between households to halt growth of an epidemic; for instance if we effectively vaccinate

$${R}_{V}=1/(1-{p}_{C})$$

(4.3.1)

In contrast to an “all-or-nothing” vaccine, a “leaky” vaccine with efficacy *E* is defined as one that reduces the instantaneous probability (rate) of infection given infectious contact (both for in-household and out-of-household contacts) by a proportion *E* (multiplies it by 1 − *E*, see Ball and Becker (2006)). Moreover it has no effect on transmission by the vaccinated individuals who did get infected. If *E* = 1, the vaccine is called perfect. A leaky vaccine would have effects on both within- and between-household transmission. For out-of-household transmission, the expected number of people infected by an index case (*R _{G}*) scales down by a factor of 1 −

We can define a **minimum (critical) efficacy** *E _{C}* for a leaky vaccine, where

$${R}_{VL}=1/(1-{E}_{C})$$

Note that this need not refer only to vaccines. One could also consider antiinfective treatments, hygiene measures, or masks that reduce the rate of infection of susceptibles by a 1 − *E*, and consider the critical value *E _{C}* for such an intervention to be the one which, if administered to the whole population, reduces the reproductive number to 1.

One can define the notion of a reproductive number *R _{E}* for any vaccine efficacy

$${R}_{E}=\frac{1}{1-E\xb7{p}_{E}}$$

Note that we recover the perfect vaccine-associated reproductive number *R _{V}* by setting

Recall the definition of *β _{G}*(

$${w}_{G}(t)=\frac{{\beta}_{G}(t)}{{\int}_{0}^{\infty}{\beta}_{G}(s)ds}=\frac{{\beta}_{G}(t)}{C{I}_{G}}$$

(4.5.1)

We note that *w _{G}*( ) can often be estimated from the data, using the distribution of times from appearance of one case in the epidemic curve to appearances of secondary out-of-household cases caused by it (Kenah et al. (2008), Lipsitch et al. (2003), Wallinga and Teunis (2004), Mills et al. (2004), Ferguson et al. (2005)).

As a consequence of the Euler-Lotka equation (Wallinga and Lipsitch (2007)), infection spreading in a mass-action population with a reproductive number *R* and an infectious contact interval distribution *w _{G}*( ) will grow exponentially at a rate

$$\frac{1}{R}={\int}_{0}^{\infty}{e}^{-rt}{w}_{G}(t)dt$$

(4.5.2)

or alternatively,

$$R=\frac{1}{{M}_{{w}_{G}}(-r)}$$

(4.5.3)

where *M _{wG}* is the moment-generating function for the density

$${w}_{H}(t)=\frac{\mathrm{\sum}{\pi}_{h}{\beta}_{h}(t)}{\mathrm{\sum}{\pi}_{h}{\int}_{0}^{\infty}{\beta}_{h}(s)ds}=\frac{\mathrm{\sum}{\pi}_{h}{\beta}_{h}(t)}{\mathrm{\sum}{\pi}_{h}C{I}_{h}}$$

(4.5.4)

where as before, *β _{h}*(

$$\frac{1}{{R}_{H}}={\int}_{0}^{\infty}{w}_{H}(t){e}^{-rt}dt$$

(4.5.5)

or alternatively,

$${R}_{H}=\frac{1}{{M}_{{w}_{H}}(-r)}$$

(4.5.6)

where *M _{wH}* is the moment-generating function for the density

$${R}_{r}=\frac{1}{{M}_{{w}_{G}}(-r)}$$

(4.5.7)

Here *w _{G}*( ) is the individual infectious contact interval distribution from equation 4.5.1 and

It is useful to note that the reproductive numbers *R _{H}*,

To begin comparing all the reproductive numbers defined so far, we note that they all serve as an epidemic threshold:

$${R}_{H}=1{R}_{r}=1{R}_{V}=1{R}_{VL}=1$$

(5.0.1)

and for a single household size, R_{H} = 1 R_{HI} = 1.

*R _{H}* = 1

The situation is quite different for a growing epidemic. In this paper we’ll prove

In a growing epidemic

$${R}_{H}\ge {R}_{VL}\ge {R}_{V}\ge {R}_{HI}\phantom{\rule{0.16667em}{0ex}}\mathit{and}\phantom{\rule{0.16667em}{0ex}}\mathit{also}\phantom{\rule{0.16667em}{0ex}}{R}_{H}\ge {R}_{r}$$

(5.0.2)

We note that Theorem 1 doesn’t characterize well the relation between *R _{r}* and the rest of the reproductive numbers. In fact, as we will show in the next section, in most numerical simulations, we also have

$${R}_{VL}\ge {R}_{r}\ge {R}_{V},$$

suggesting a strict ordering between the reproductive numbers, with the dynamical ones (*R _{H}*,

The proof of Theorem 1 is given in the next section, with several more technical results deferred to Appendix A. Some of the consequences of Theorem 1 are discussed in Section 8.

We have *R _{H}* =

$${R}_{H}({E}_{C})={R}_{G}({E}_{C})\xb7f({E}_{C})$$

where *R _{G}*(

$${R}_{G}({E}_{C})=(1-{E}_{C}){R}_{G}$$

(outside hazard scales by a factor of (1 − *E*)); and *f*(*E _{C}*) ≤

$$1={R}_{H}({E}_{C})={R}_{G}({E}_{C})\xb7f({E}_{C})\le (1-{E}_{C}){R}_{G}f=(1-{E}_{C}){R}_{H}$$

So
$1-{E}_{C}\ge {\scriptstyle \frac{1}{{R}_{H}}}$, which is equivalent to *R _{VL}* ≤

Note that *R _{H}* and

$${W}_{G}(t)={\int}_{0}^{t}{w}_{G}(s)ds,\phantom{\rule{0.38889em}{0ex}}{W}_{H}(t)={\int}_{0}^{t}{w}_{H}(s)ds$$

(6.2.1)

We have *W _{G}*(0) = 0 =

$$\frac{1}{{R}_{H}}={\int}_{0}^{\infty}r{e}^{-rt}{W}_{H}(t)dt,\phantom{\rule{0.38889em}{0ex}}\frac{1}{{R}_{r}}={\int}_{0}^{\infty}r{e}^{-rt}{W}_{G}(t)dt$$

(6.2.2)

We see from the equation above that the *R _{r}* ≤

The household infectious contact interval distribution *w*_{H}( ) is always longer than the individual infectious contact interval distribution *w*_{G}( ): *W*_{H}(*t*) ≤ *W _{G}*(

The infectious contact intervals from household to household will be a “rightward smear” of the infectious contact intervals from any given individual to others outside the household.

Let us introduce the following definitions, both for a household of size *h* and for an individual:

$${B}_{h}(t)={\int}_{0}^{t}{\beta}_{h}(s)ds,\phantom{\rule{0.38889em}{0ex}}{B}_{G}(t)={\int}_{0}^{t}{\beta}_{G}(s)dt$$

Thus *CI _{G}* =

$${B}_{h}(t)\le {f}_{h}{B}_{G}(t)$$

(6.2.3)

To understand why the latest equation is true, define the function *y _{h}*(

$${f}_{h}-1={\int}_{0}^{\infty}{y}_{h}(s)ds$$

- the above represents the expected number of people ever infected in the household (besides the index case). Also for each *a* ≥ 0,

$${\beta}_{h}(a)={\beta}_{G}(a)+{\int}_{0}^{a}{y}_{h}(a-s){\beta}_{G}(s)ds$$

(6.2.4)

The first factor on the right hand side above is the contribution from the index to the household’s expected infectiousness at time *a*, and the integral gives the contribution of everybody else. Integrating equation 6.2.4 from zero to *t* we get

$$\begin{array}{l}{B}_{h}(t)={\int}_{0}^{t}{\beta}_{G}(a)da+\underset{a=0}{\overset{t}{\int}}\underset{{a}^{\prime}=0}{\overset{a}{\int}}{y}_{h}(a-{a}^{\prime}){\beta}_{G}({a}^{\prime})d{a}^{\prime}da=\\ ={\int}_{0}^{t}{\beta}_{G}(a)da+\underset{{a}^{\prime}=0}{\overset{t}{\int}}{\beta}_{G}({a}^{\prime})\underset{a={a}^{\prime}}{\overset{t}{\int}}{y}_{h}(a-{a}^{\prime})\mathit{dad}{a}^{\prime}\le \\ \le {\int}_{0}^{t}{\beta}_{G}(a)da+\underset{{a}^{\prime}=0}{\overset{t}{\int}}{\beta}_{G}({a}^{\prime})\underset{a={a}^{\prime}}{\overset{\infty}{\int}}{y}_{h}(a-{a}^{\prime})\mathit{dad}{a}^{\prime}\\ ={\int}_{0}^{t}{\beta}_{G}(a)da+({f}_{h}-1){\int}_{0}^{t}{\beta}_{G}({a}^{\prime})d{a}^{\prime}={f}_{h}{\int}_{0}^{t}{\beta}_{G}(a)da={f}_{h}{B}_{G}(t)\end{array}$$

This inequality was previously demonstrated by Ball et al. (2004) - see also Ball and Becker (2006), where the model is extended to allow for two degrees of infection - mild and severe. In the Appendix A we provide a proof of a slightly generalized version of this claim. It is generalized in two ways. First, it is valid for any population structure, not only for a population of households. Second, we generalize the results of Ball et al. (2004), which in effect compared only perfect vs. leaky vaccines, to consider a continuum of vaccines with leaky efficacy between perfect and a minimal (critical) leaky efficacy *E _{C}* - see Proposition A.2.3.

Recall the notion of a vaccine of efficacy *E* from Section 4.4. Let us define the notation *R*[*E; p*] to denote the value of a reproductive number realized when a fraction *p* of the population receives a vaccine with efficacy *E*. For any *E _{C}* ≤

$${R}_{E}=\frac{1}{1-{p}_{E}E}$$

In the appendix we prove that *R _{E}* is a non-increasing function of

The basic finding *R _{V L}* ≥

The proof that it is always better to vaccinate a fraction *p*_{1} with a vaccine of efficacy *E*_{1} than to vaccinate a proportion *p*_{2} with a vaccine of efficacy *E*_{2}, if *p*_{1}*E*_{1} = *p*_{2}*E*_{2} but *p*_{1} < *p*_{2} is fully general, not relying on household structure in particular. It stems from a simple observation that for any realization of individual infectiousness profiles *I*(*t*), and for any individual *i*, the probability of *i* getting infected by a given time is lower under the (*E*_{1}, *p*_{1}) scenario than under the (*E*_{2}, *p*_{2}) scenario - see Proposition A.2.3 for full details. The argument is then formalized by coupling the two scenarios so that in the first case there are always less infected people than in the second case.

The non-increasing nature of *R _{E}* as a function of

This is the proof in Appendix A that is longest, most mathematically involved, and least illuminating intuitively. Here we’ll sketch an argument for a community of homogeneous households of the same size. The basic approach is to note that *R _{HI}* =

To prove the above, pick any *q** ≤ *x* ≤ 1, and let *f*(*x*) be the expected number of infected people in an infected household in a population, where each individual is vaccinated with a perfect vaccine with probability 1 − *x*. Let
$g(x)={\scriptstyle \frac{f(x)-1}{f(x)}}$ as before. We need *g*(*q**) ≥ *q***g*(1). Well show more generally that for any number *C* > 1, *g*(*Cx*) ≤ *Cg*(*x*). Some intuition for why this inequality holds may be gained from considering limiting cases.

As the first limiting case, suppose that each household is very large with *N* persons and the probability of each infecting the other is *μ*/*N* for some *μ* < 1. Thus the dynamics can be well approximated by a branching process, since the depletion of susceptibles within a household has negligible impact on the transmission process in that household. So each person will infect on the average *μ* more people in the next generation, etc., so the expected total number of infected people is
${\scriptstyle \frac{1}{1-\mu}}$. If a proportion 1 − *x* is vaccinated with a perfect vaccine then the expected number of infected people is
$f(x)={\scriptstyle \frac{1}{1-x\mu}}$, *g*(*x*) = *μx* and *g*(*Cx*) = *Cg*(*x*).

The spirit of the general proof in the appendix is to demonstrate that this branching process limit is the “worst-case scenario.” To show this, we’ll consider an infected household where each person is vaccinated with a perfect vaccine with probability 1 − *x*, and define *H _{k}*(

$$x{f}^{\prime}(x)\le f(f-1)$$

This in turn implies that *g*(*Cx*) ≤ *Cg*(*x*) for *C* > 1.

In the preceding sections, we have shown mathematically that in a growing epidemic, *R _{H}* ≥

- We show the magnitude of these reproductive numbers for a wide range of assumptions about the natural history of the disease and the household size distribution in order to assess how much they differ under particular assumptions.
- We show that in most plausible parameter regions, we have a strict ordering$${R}_{VL}\ge {R}_{r}\ge {R}_{V},$$despite the fact that we can show mathematically that conditions exist in which
*R*<_{r}*R*(Appendix B), and that we have found no mathematical argument to demonstrate that_{V}*R*≥_{V L}*R*._{r} - We show that a simple approximation to
*R*works well for reasonable parameters for several diseases._{V}

We took two approaches to numerical studies. In section 7.1 we evaluate the reproductive numbers in a Markov SEIR model in which the parameters may be varied in a simple fashion to explore a wide range of parameter space. In section 7.2 we use more realistic distributions of parameters, similar to those for measles and influenza (Fraser (2007)), to assess how these reproductive numbers may perform for real diseases.

To explore parameter space and compare the magnitudes of the five reproductive numbers, we numerically implemented a model for within- and between-household transmission of an infection with a simple susceptible-exposed-infectious-recovered (SEIR) natural history. Upon infection, an individual remains exposed but not infectious for a latent period whose length is exponentially distributed with mean 1/*u*, and then becomes infectious for a period that is also exponentially distributed with mean 1 (without loss of generality). Infectiousness *I*(*t*) (see section 3) during the infectious period is constant (and can be assumed equal to 1), and is zero before and after. Thus *CI _{G}* = 1 (see equation 3.0.6). External infections follow standard mass-action dynamics, but we limit our consideration to the early phase of the epidemic such that new (uninfected) households are plentiful, and all secondary infections outside the household occur in households that have never before been exposed (no depletion of susceptible households).

Households are assumed to be of a single fixed size *h*, and the infectiousness profile of a household is calculated using a master equation for within-household infections. In this setting, using “standard” (deterministic) SEIR infection dynamics within a household is inappropriate because of the small numbers of individuals, hence we keep track explicitly of the probability *p*(*σ*, *ε*, *ι*) that the number of S, E, I and R individuals in the household at a particular time is (*σ*, *ε*, *ι*, *h*−*σ* −*ε*−*ι*). Infectious individuals exert constant infectiousness *μ _{h}* on other, uninfected individuals within the household (see equation 3.0.9, with

The appropriate master equation for **within-household** dynamics is:

$$\begin{array}{l}\frac{dp(\sigma ,\epsilon ,\iota )}{dt}=-{\mu}_{h}\iota \sigma p(\sigma ,\epsilon ,\iota )+{\mu}_{h}\iota (\sigma +1)p(\sigma +1,\epsilon -1,\iota )\\ -\iota p(\sigma ,\epsilon ,\iota )+(\iota +1)p(\sigma ,\epsilon ,\iota +1)\\ -\epsilon up(\sigma ,\epsilon ,\iota )+u(\epsilon +1)p(\sigma ,\epsilon +1,\iota -1)\end{array}$$

(7.1.1)

Here we define *p*(*σ*, *ε*, *ι*) = 0 if *σ*+*ε*+*ι* > *h* or if any of *σ*, *ε*, *ι* is negative. To simulate the dynamics within a community of households, we allow these dynamics to run in each household while also allowing transmission between households. If we now define *z*(*σ*, *ε*, *ι*) as the expected number of households with *σ* susceptibles, *ε* latently infecteds, *ι* infectious, and *h* − *σ,* − *ε,* − *ι* recovered, we can write these dynamics identically to the within-household dynamics, adding a term for between-household infections

$$\begin{array}{l}\frac{dz(\sigma ,\epsilon ,\iota )}{dt}=-{\mu}_{h}\iota \sigma z(\sigma ,\epsilon ,\iota )+{\mu}_{h}\iota (\sigma +1)z(\sigma +1,\epsilon -1,\iota )\\ -\iota z(\sigma ,\epsilon ,\iota )+(\iota +1)z(\sigma ,\epsilon ,\iota +1)\\ -\epsilon uz(\sigma ,\epsilon ,\iota )+u(\epsilon +1)z(\sigma ,\epsilon +1,\iota -1)\\ +\delta (\sigma -h+1)\delta (\epsilon -1)\delta (\iota )\sum _{{\sigma}^{\prime}+{\epsilon}^{\prime}+{\iota}^{\prime}\le h}{\iota}^{\prime}{R}_{G}z({\sigma}^{\prime},{\epsilon}^{\prime},{\iota}^{\prime})\end{array}$$

(7.1.2)

where *δ* (*x*) = 1 if *x* = 0 and 0 otherwise. The last term represents transmission to other households, which feeds new households into the epidemic with *h* −1 susceptibles, 1 latent, and no infectious individuals. For this last term, note that the transmission coefficient for newly infected households is *R _{G}* =

For a household of size *h* there are
${\scriptstyle \frac{h(h+1)(h+2)}{6}}$ possible configurations for the family, of which two (all susceptible and all recovered) are not tracked in our dynamics, the former being assumed constant and the latter being irrelevant. We can thus exactly track the early phase of the epidemic with
${\scriptstyle \frac{h(h+1)(h+2)}{6}}-2$ linear differential equations defined by equation 7.1.2. The largest eigenvalue of this system constitutes the exponential growth rate of the epidemic and is used, along with the serial interval distribution in the calculations of the main text. This system was implemented in Mathematica 6.0, and the eigenvalues taken as the eigenvalues of the Jacobian matrix of the system of linear equations. Code is available on request.

Figure 1 shows plots of the five reproductive numbers for varying household sizes (different rows), varying values for the out-of-household reproductive number *R _{G}*, and varying rates of within-household transmission

Magnitudes of the 5 reproductive numbers for various household sizes (rows) and values of *R*_{G}, the number of outside-household secondary cases per infectious individual (columns). The horizontal axis shows *μ*_{h}, the within-household transmission **...**

Several features are notable in Figure 1. First, the five reproductive numbers follow a strict ordering

$${R}_{H}\ge {R}_{VL}\ge {R}_{r}\ge {R}_{V}\ge {R}_{HI}$$

for nearly all parameter values chosen, with *R _{r}* nearly overlapping

The ratio *R*_{r}/*R*_{V} as a function of the latent period (with all other parameters fixed) in a household of size 4. As the latent period becomes (unrealistically) large, *R*_{r} becomes slighty smaller than *R*_{V}, as predicted in Appendix B.3.

Second, it is clear that *R _{H}* and

Third, *R _{r}*, which has been measured in various forms for infections such as SARS (Lipsitch et al. (2003); Wallinga and Teunis (2004)), influenza (Ferguson et al. (2005); Mills et al. (2004)) and Ebola (Chowell et al. (2004)), gives in general a conservative indication of how much transmission must be blocked to halt the growth of the epidemic (as indicated by the

Fourth, whenever there is significant within-household transmission, *R _{V}* and

Fifth, there is significant variation in *R _{r}* depending on the household size distribution and the degree of within-household transmission. It has been suggested that if different models of a disease are calibrated to give the same initial growth rate of an epidemic, they should then produce similar estimates for the effect of interventions. Our model suggests this may not be correct, since different models of the same disease might make different assumptions about parameters such as household size distributions and the relative contribution of within-household transmission (Halloran et al. (2008)), producing different values of

In a second set of numerical simulations, we studied the five reproductive numbers in the context of a disease natural history calibrated more closely to two specific diseases, measles (which has a very high reproductive number and a highly peaked infectiousness profile) and influenza (which has a modest reproductive number and a wider infectiousness profile). These simulations followed the protocol described by Fraser (2007). Individual cumulative intensities of infectiousness were Gamma-distributed, with the shape parameter set to.22 for measles and 1 for flu. Stochastic simulations were used (rather than the master equation approach described in the previous section) to estimate the mean infectiousness profile for each household size, and from these profiles and an assumed value of *R _{G}* (out of household reproductive number, set to equal 1.5 for flu and 15 for measles), the values of the other reproductive numbers are calculated for a household size distribution based on the 2000 US census. The within-household transmission parameter

Magnitudes of the 5 reproductive numbers for influenza (left) and measles (right), as a function of the within-household transmission parameter *μ*_{h}. Red - *R*_{H}, orange - *R*_{V L}, green - *R*_{r}, brown - *R*_{V A}, purple - *R*_{V}, blue - *R*_{HI}. In both cases the individual **...**

Here, in addition to the five reproductive numbers, we also calculated an approximate reproductive number *R _{V A}*, which approximates

$${R}_{V}\approx {R}_{VA}={(1-p)}^{-1}=\frac{2(f-1)}{-1+\sqrt{4(f-1)/{R}_{G}}}$$

The following can be observed from Figure 3:

*R*is a very close approximation to_{r}*R*for all parameters considered, and when it is wrong, as in the simulations in the previous section, it is conservative (_{V}*R*predicts that more people need to be vaccinated than the actual number). Moreover,_{r}*R*is also an excellent approximation, indicating within-household herd immunity effects are minor compared to the direct effects of vaccination in the household and the effects in the population as a whole._{V A}*R*is significantly larger than_{V L}*R*whenever within-household transmission is substantial._{V}- For flu,
*R*significantly underestimates the other reproductive numbers, and in both cases_{HI}*R*significantly overestimates them._{H}

In this paper we have defined five reproductive numbers that may be of epidemiologic interest when an infection is spreading in a population of households. The reproductive numbers *R _{H}*,

The upper and lower bound values, the household reproductive number *R _{H}* and the individual reproductive number

As Figure 1 shows, these two numbers may in general give extremely divergent values, especially when household sizes are large and within-household transmission is important. An intermediate value *R _{r}* may be obtained by estimating the exponential growth rate of the epidemic, then calculating a reproductive number from the Euler-Lotka equation 4.5.7, using the individual infectious contact distribution.

The main purpose of this paper was to compare these three (dynamical) quantities that can be estimated from data against two other quantities that are important for epidemic control: namely, the proportion *p _{C}* of the population that must be successfully immunized to stop epidemic growth with a perfect vaccine, or the minimum efficacy

Conveniently, the reproductive numbers that can be estimated from data provide bounds on these reproductive numbers associated with control. *R _{V L}* lies between

These findings suggest that while no simple measure of epidemic progress is exactly indicative of the effort needed to control epidemic spread in a community of households, it is nonetheless possible to obtain some bounds on, and reliable estimates of, the effort required. For instance, even if no good contact data exists for most cases, *R _{r}* may be obtained by observing the exponential growth rate of the epidemic, then calculating it from the Euler-Lotka equation 4.5.7, using the individual infectious contact distribution. Several studies in the past (Lipsitch et al. (2003), Wallinga and Teunis (2004), Mills et al. (2004), Ferguson et al. (2005) etc.) took that approach, using the growth rate of an epidemic along with an infectious contact interval distribution, in various ways, to estimate the reproductive number. In at least the SARS cases, the infectious contact interval distribution was derived from known contacts of brief duration, and thus may be approximately equal to the infectious contact distribution.

Another consequence of our findings is that we can give a lower bound for the quantity of a vaccine needed to halt the epidemic. The argument runs as follows: first we can observe the epidemic’s impact on infected households, estimating the numbers *f _{h}*. Using the inequalities we’ve proven we get that for any vaccine of efficacy

$${R}_{E}\ge {R}_{V}\ge {R}_{HI}={R}_{G}+g=\frac{{R}_{H}}{f}+g\ge \frac{{R}_{r}}{f}+g$$

(8.0.1)

where as before, *f* = Σ*π _{i}f_{i}* and
$g=\mathrm{\sum}{\pi}_{i}{\scriptstyle \frac{{f}_{i}-1}{{f}_{i}}}$.

Several aspects of the story remain incomplete. First, we have no proof yet of the inequality *R _{V L}* ≥

This work is supported by the US National Institutes of Health cooperative agreement 5U01GM076497 “Models of Infectious Disease Agent Study”; Ruth L. Kirchstein National Research Service Award 5T32AI007535 “Epidemiology of Infectious Diseases and Biodefense”; and a Royal Society Research Fellowship.

In this section we’ll present a framework to understand the spread of infection on some network, on which each person receives a leaky (or perfect) vaccine with a given probability *p*, and those vaccinations are independent of one another. The main idea behind this contruction appeared essentially in Sellke (1983). We’ll use two variants of the Sellke contruction, needed for our proofs: one with individual thresholds, which is more common (see Andersson and Britton (2000), p. 14), and one with pairwise thresholds, to trace who infected whom.

Let the nodes (persons) in the network be labelled as *P*_{1}, …, *P _{N}*. Once infected, individual

For a **calendar** time *t* and a person *P _{i}* infected at time

$${A}_{i}(t)={\int}_{0}^{t-{t}_{i}}{I}_{i}(s)ds$$

(A.1.1)

The integral (which is zero if *t _{i}* >

Given two individuals *P _{i}* and

$${\mu}_{ij}{I}_{i}(t-{t}_{i})\mathrm{\Delta}t={\mu}_{ij}\mathrm{\Delta}{A}_{i}$$

(A.1.2)

Note that “receiving an infectious contact” is not the same thing as getting infected as *P _{j}* could already be infected by time

For a person *P _{j}* and a calendar time

$${D}_{j}(t)={\mathrm{\sum}}_{i\ne j}{\mu}_{ij}{A}_{i}(t)$$

(A.1.3)

Recall the notion of a leaky vaccine from Section 4.4. Given a number 0 ≤ *E* ≤ 1, we say that a vaccine has efficacy *E* if for a person *P _{j}* receiving this vaccine, the transmission coefficient for infectious contacts between any person

We assume that each person receives such a vaccine with probability *p* and those events are independent. We now describe two variants of a constuction for the spread of infection:

This construction is well known, see Andersson and Britton (2000), p. 12, Ball et al. (2004). For a person *P _{j}*, let

If *P _{j}* is unvaccinated, equations A.1.2 and A.1.3 say that

$$P(D<{Q}_{j}\le D+\mathrm{\Delta}D{Q}_{j}D)=\mathrm{\Delta}D$$

Thus (with probability *q*), *Q _{j}* is an exponential variable

The construction of the spread of infection works as follows: suppose we have a network as above with individuals *P*_{i1}, …, *P _{il}* initially infected. Consider the following collection of independent random variables:

*I*_{1}(*s*), …,*I*(_{N}*s*) are trajectories of the stochastic processes*M*_{1}, …,*M*(those are individual infectiousness profiles)._{N}*Q*=_{j}*Exp*(1, 1 −*E, q*).

Given any realization for those variables, we reconstruct the dynamics of the spread of infection. We go along the time line until the first susceptible *P _{s}*

For any individual *P _{i}*, define a random variable

For persons *P _{i}* and

$${D}_{ij}(t)={\chi}_{q}({X}_{i}){\mu}_{ij}{A}_{i}(t)+(1-{\chi}_{q}({X}_{i}))(1-E){\mu}_{ij}{A}_{i}(t)$$

(A.1.4)

Let *Q _{ij}* be the dosage of infection that

It is also useful to state measure-theoretically that the whole stochastic process is represented by the probability space

$$S=\mathrm{\Pi}{M}_{i}\times \mathrm{\Pi}{ij}_{\times}$$

(A.1.5)

*M _{i}* is the space of trajectories of the stochastic process of

Any point in *S* represents a particular realization of the dynamics of infection on our network. Thus, for instance the total number of infected people under a particular dynamic is an integer-valued function of *S*, and the integral of that function over *S* gives the expected number of people, who are eventually infected etc.

Suppose that in an emerging epidemic in a community of households, we **randomly** distribute a vaccine of efficacy *E* (0 ≤ *E* ≤ 1) - see Section 4.4 for more details. Recall (see Remark 2 in Section 4.4) that the reproductive number for a vaccine of efficacy *E* is 1

$${R}_{E}=\frac{1}{1-E\xb7{p}_{E}}$$

Note that the definition makes sense only for *E* ≥ *E _{C}*. The main result of this section is the following

The function R_{E} is a **non-increasing** function of E as E ranges between E_{C} and 1. In particular, R_{V L} ≥ R_{V}.

This lemma and its proof are a slight generalization of a proposition in Ball et al. (2004), where it is essentially proved that for any vaccine efficacy *E*, *R _{E}* ≥

The idea is as follows: suppose we have two vaccines of efficacy *E*_{1} and *E*_{2} with *E*_{1} > *E*_{2}, and we distribute them at random for proportions *p*_{1} and *p*_{2} of the population, with *p*_{1}*E*_{1} = *p*_{2}*E*_{2}. Then the spead of infection is less severe in the first case than in the second case. This is true for any finite network, as we’ll show below. A simple argument will then establish Lemma A.2.1 for a community of households. First we prove the following elementary

*Consider two numbers* 1 > *q*_{1} > *q*_{2} ≥ 0, *and two numbers* 0 ≤ *c*_{1} < *c*_{2} < 1*. If* (1−*q*_{1})(1−*c*_{1}) = (1−*q*_{2})(1−*c*_{2})*, then the random variable X* = *Exp*(1, *c*_{1}, *q*_{1}) *is stochastically larger than the random variable Y* = *Exp*(1, *c*_{2}, *q*_{2}).

Let *p _{i}* = 1 −

$$P(X\ge a)\ge P(Y\ge a)$$

Now *P*(*X* ≥ *a*) = *q*_{1}*e*^{−}* ^{a}* +

The next proposition is well-known (see Andersson and Britton (2000), p. 20)

(Coupling) Suppose a random variable X is stochastically larger than a random variable Y. Then there is a probability space J and realizations : J (∞) and Ŷ: J (∞) such that has the same distribution as X, Ŷ has the same distribution as Y and (z) ≥ Ŷ (z) for every point z J.

From this we derive the following proposition, which generalizes the result in Ball et al. (2004):

Consider a finite network with the initially infected P_{i1},…,P_{ik}. Consider the two scenarios: In the first case, each of the susceptibles receives a leaky vaccine of efficacy E_{1} with probability p_{1}; in the second case, each receives a leaky vaccine of efficacy E_{2} with probability p_{2}. Suppose E_{1} > E_{2} and E_{1}p_{1} = E_{2}p_{2}. Then one can couple the two processes for the spread of infection so that for any realization and for each point in time, the set of people, who are infected by then in the first scenario is contained in the corresponding set for the second scenario.

For each person *P _{i}*, we assign a pair (
${Q}_{i}^{1},{Q}_{i}^{2}$) of individual thresholds for the two scenarios (thus
${Q}_{i}^{1}=\mathit{Exp}(1,1-{E}_{1},{q}_{1})$ and
${Q}_{i}^{2}=\mathit{Exp}(1,1-{E}_{2},{q}_{2})$), coupled according to propositions A.2.1 and A.2.2, so that
${Q}_{i}^{1}\ge {Q}_{i}^{2}$. We construct the spread of infection in time as described in section A.1.1. We claim that for a particular realization of the individuals infectiousness profiles and thresholds and for each time

We can now prove Lemma A.2.1, which is simplest to see for a community of homogeneous households. Consider a community of homogeneous households of size 1,…,*n* and suppose that *π _{i}* is the proportion of people, living in households of size

$${R}_{H}={R}_{G}\xb7\mathrm{\sum}{\pi}_{i}{f}_{i}$$

Here *R _{G}* is the expected number for out-of-household infections by one infected person, and

$${R}_{H}[{E}_{1},{p}_{1}]=(1-{p}_{1}{E}_{1}){R}_{G}\xb7\mathrm{\sum}{\pi}_{i}{f}_{i}[{E}_{1},{p}_{1}]$$

Here *f _{i}*[

$${f}_{i}[{E}_{1},{p}_{1}]\le {f}_{i}[{E}_{2},{p}_{2}]$$

Now pick a portion *p*_{E1} of the first vaccine to have the reproductive number *R _{H}*[

In this section we’ll establish several auxiliary results towards proving that *R _{V}* ≥

Let *S* be the space of all realizations of infectiousness profiles, pairwise thresholds and individual vaccination statuses as in equation A.1.5. Pick a particular realization *z* in *S*. We’ll now define the notion of generations of infections for *z*. The 0* ^{th}* generation

Consider a network Net^{q} as above and let C > 1 be a real number such that Cq ≤ 1. Consider also the network Net^{Cq}. Then for each k ≥ 1,
$E({G}_{k}^{Cq})\le {C}^{k}E({G}_{k}^{q})$

For each person *P _{j}*, who is infected in generation

$${\kappa}_{j}={P}_{j,0},\dots ,{P}_{j,k}$$

where *P _{j,}*

Let S be the probability space as in equation A.1.5. For a given path κ of length k, let L^{q}(κ) S to be the set of all points z S such that κ is a transmission path for z in Net^{q}.

For any *z* *S*, the number of persons in the *k ^{th}* generation

$$E({G}_{k}^{q})={\mathrm{\sum}}_{\kappa}P({L}^{q}(\kappa ))$$

Here the sum is taken over all possible paths of length *k* starting from one of the initially infected persons. Thus Lemma A.3.1 will follow from

For any path κ of length k, P(L^{Cq}(κ)) ≤ C^{k}P(L^{q}(κ))

Let *S* be the space as in equation A.1.5. The same space works for both networks *Net*^{q} and *Net*^{Cq}, only the spreads under a particular scenario *z* *S* are different due to different definitions for the pairwise dosages in equation A.1.4. We have the subsets *L*^{q}(*κ*), *L*^{Cq}(*κ*) *S*. We’ll establish the proposition by constructing a map *W: S* → *S* with the following properties:

- W(L
(κ)) L^{Cq}(κ).^{q} - For any subset
*B*of*S*,*P*(*W*(*B*)) =*C*^{−}(^{k}P*B*).

To construct *W*, recall that *S* = Π*M _{i}* × π

If *P _{i}* is a vertex in the path

If not, *W* is an identity map on [0, 1]* _{i}*.

Clearly property b) holds. Also for any *z* *S*, *W*(*z*) has the same infectiousness profiles and pairwise thresholds as *z*, while χ* _{Cq}*(

For the rest of this section we specialize to the case when our network is homogeneous, namely all *μ _{ij}* are equal and all the individual infectiousness profiles are trajectories of the same stochastic process

In a homogeneous network, Σ_{l}_{≥ 0}*H _{k}*

For each individual *P _{i}*, let

$${\mathrm{\sum}}_{l\ge 0}{H}_{k+l}(q)={\mathrm{\sum}}_{i}P({U}_{i})E(\mathit{infected}\phantom{\rule{0.16667em}{0ex}}\mathit{via}\phantom{\rule{0.16667em}{0ex}}{P}_{i}{U}_{i})$$

Here a person is “infected via *P _{i}*” means that

$$E(\mathit{infected}\phantom{\rule{0.16667em}{0ex}}\mathit{via}\phantom{\rule{0.16667em}{0ex}}{P}_{i}{U}_{i})\le f(q)$$

This is because under any spread of infection in time till *P _{i}* gets infected in the

Suppose we have an epidemic in a community of households, with all households having a single size *n*. Let *f* be the expected number of infected people in an infected household. Thus
${R}_{HI}={R}_{hI}={R}_{G}+{\scriptstyle \frac{f-1}{f}}$. We have

*R _{V}* ≥

Let *p* = 1 − *q* be the critical proportion of people, which needs to be vaccinated to bring the reproductive numbers to 1. By definition, *R _{V}* = 1/

The theorem will hold via the following

Let C > 1. Then

$$1-\frac{1}{f(C\xb7t)}\le C(1-\frac{1}{f(t)})$$

Consider
$g(x)=1-{\scriptstyle \frac{1}{f(t\xb7{e}^{x})}}$ for *x* ≥ 0. We’ll show that *g*′(*x*) ≤ *g*(*x*). It will follow from Gronwall’s inequality that *g*(*x*) ≤ *e _{x}g*(0), which is the statement of the lemma. Now
${g}^{\prime}(x)={\scriptstyle \frac{t\xb7{e}^{x}{f}^{\prime}({e}^{x}\xb7t)}{{f}^{2}({e}^{x}\xb7t)}}$. We’ll now show that for any number

$$y{f}^{\prime}(y)\le f(y)(f(y)-1)$$

which is equivalent to *g*′(*x*) ≤ *g*(*x*).

The above inequality becomes an equality for a branching process, in which case
$f(y)={\scriptstyle \frac{1}{1-y}}$ for *y* < 1 (here *y* is the expected number of people that one infected individual will infect) and *f*(*y*) is infinite if *y* > 1.

Let *H _{k}*(

$$y{f}^{\prime}(y)={\mathrm{\sum}}_{y}{H}_{k}^{\prime}(y)\le {\mathrm{\sum}}_{k\ge 1}k{H}_{k}(y)$$

Now we re-write the latest sum as

$${\mathrm{\sum}}_{k\ge 1}k{H}_{k}(y)={\mathrm{\sum}}_{k\ge 1}{\mathrm{\sum}}_{l\ge 0}{H}_{k+l}(y)$$

By Proposition A.3.2, Σ_{k}_{≥ 1}Σ_{l}_{≥ 0}*H _{k}*

Suppose we have households of size 1,…,*n*, and the proportion of people living in households of size *i* is *π _{i}* with Σ

$${R}_{HI}=\mathrm{\sum}{\pi}_{i}({R}_{G}+\frac{{f}_{i}-1}{{f}_{i}})={R}_{G}+\mathrm{\sum}{\pi}_{i}\frac{{f}_{i}-1}{{f}_{i}}$$

At the same time, if we forget about the stratification by the household size, we note that the proportion *π _{i}* of people in households of size

$${R}_{HI}^{\prime}={R}_{G}+\frac{\mathrm{\sum}{\pi}_{i}{f}_{i}-1}{\mathrm{\sum}{\pi}_{i}{f}_{i}}$$

The latter reproductive number serves as an epidemic threshold. However one can construct examples of household structures with ${R}_{HI}^{\prime}>{R}_{VL}\ge {R}_{V}$. On the other hand we always have

For any household size distribution, R_{V} ≥R_{HI}.

First we need to establish the following

For any non-negative numbers π_{1}, …, π_{n} with Σπ_{i} = 1, and for any positive numbers f_{i},

$$\mathrm{\sum}{\pi}_{i}\frac{{f}_{i}-1}{{f}_{i}}\le \frac{\mathrm{\sum}{\pi}_{i}{f}_{i}-1}{\mathrm{\sum}{\pi}_{i}{f}_{i}}$$

Define a random variable *X*, which takes values *f _{i}* with probabilities

The above proposition tells that
${R}_{HI}\le {R}_{HI}^{\prime}$. Now we can argue as in the case of single size households. First we vaccinate the proportion *p* = 1 −*q* of the population to have
${R}_{H}(q)=1={R}_{HI}^{\prime}(q)$. By definition, *R _{V}* =

Here we demonstrate the inequality at two extremes. First consider the extreme where within-household transmission is very low, the limit as *μ _{h}* →0 (see equation 3.0.9). The household model converges to the mass action model, and all reproductive numbers converge; thus at this extreme equality holds.

Now, consider the extreme where within-household transmission is very high, *μ _{h}* →∞. Then everyone in the household becomes infected, and a leaky vaccine will change that fact only negligibly. Thus, the only effect of a leaky vaccine is to reduce between-household transmission. Hence at this extreme,

Another comparison result between *R _{V L}* and

For households of size h = 2 with exponentially distributed infectiousness periods and no latent periods, R_{V L} = R_{r}

We note that in this case one can explicitly compute both reproductive numbers to see that they are equal - we omit the derivations.

Again, we note that equality holds at the extreme of no within-household transmission, *μ _{h}* →0 - in this case dynamics converge to mass action model.

Now we consider the limiting case where *μ _{h}* → ∞. Here, all persons within the household will become infected (

$${w}_{G}(t)=w(t)$$

- see equations 4.5.1 and 4.5.4. Thus in this limit, *R _{r}* →

In this appendix we prove the following

If we add an “infinite” latent period (a limiting case of adding very long latent periods) to individual infectiousness profiles, then *R*_{V} ≥*R _{r}*, with equality if all households have sizes up to 3.

Suppose we add a large latent period to individual infectiousness profiles (without changing anything else). We can scale down time and scale up intensities of infectiousness by the same factor, so that the reproductive numbers don’t change, the latent period is 1, and the infectiousness period is very short. In this limit the individual infectious contact interval distribution *w _{G}*(

$${M}_{{w}_{G}}(-r)={e}^{-r}$$

To understand the household infectious contact interval distribution, we observe that in the limit, the *k ^{th}* generation of infections in the household are the people infected at time

$${R}_{G}{\mathrm{\sum}}_{h}{\pi}_{h}({\mathrm{\sum}}_{k=0}^{h-1}{G}_{k}(h){e}^{-r(k+1)})=1$$

(B.3.1)

with *R _{r}* =

$${R}_{V}=\frac{1}{q}$$

Let us denote by
${G}_{k}^{q}(h)$ the expected number of people infected in *k ^{th}* generation is an infected household of size

$$1=q\xb7{R}_{G}\xb7{\mathrm{\sum}}_{h}{\pi}_{h}{f}_{h}(q)={R}_{G}{\mathrm{\sum}}_{h}{\pi}_{h}({\mathrm{\sum}}_{k=0}^{h-1}q{G}_{k}^{q}(h))$$

By Lemma A.3.1, ${G}_{k}^{q}(h)\ge {q}^{k}{G}_{k}^{1}(h)={q}^{k}{G}_{k}(h)$. Using this, we rewrite the latest equation as

$$1\ge {R}_{G}{\mathrm{\sum}}_{h}{\pi}_{h}({\mathrm{\sum}}_{k=0}^{h-1}{q}^{k+1}{G}_{k}(h))$$

(B.3.2)

Comparing equations B.3.1 and B.3.2, we see that *q* ≤*e*^{−}* ^{r}*. Thus

$${R}_{V}=\frac{1}{q}\ge {e}^{r}={R}_{r}$$

Finally we note that if all households are of size up to 3, it is easy to see that
${G}_{k}^{q}(h)={q}^{k}{G}_{k}^{1}(h)$. Thus the inequality in equation B.3.2 becomes an equality. Comparing equations B.3.1 and B.3.2, we conclude that *e*^{−}* ^{r}* =

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