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- Abstract
- 1. Introduction
- 2. Materials and Methods
- 3. Application: 2009 pandemic H1N1 influenza
- 4. Discussion
- References

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Epidemics. Author manuscript; available in PMC 2013 July 7.

Published in final edited form as:

Published online 2011 December 7. doi: 10.1016/j.epidem.2011.11.003

PMCID: PMC3703471

NIHMSID: NIHMS343170

Previous influenza pandemics (1918, 1957, and 1968) have all had multiple
waves. The 2009 pandemic influenza A(H1N1) (pandemic H1N1) started in April 2009
and was followed, in the United States (US) and temperate Northern Hemisphere,
by a second wave during the fall of 2009. The ratio of susceptible and immune
individuals in a population at the end of a wave determines the potential and
magnitude of a subsequent wave. As influenza vaccines are not completely
protective, there was a combined immunity in the population at the beginning of
2010 (due to vaccination and due to previous natural infection), and it was
uncertain if this mixture of herd immunity was enough to prevent a third wave of
pandemic influenza during the winter of 2010. Motivated by this problem, we
developed a mathematical deterministic two-group epidemic model with vaccination
and calibrated it for the 2009 pandemic H1N1. Then, applying methods from
mathematical epidemiology we developed a scheme that allowed us to determine
critical thresholds for vaccine-induced and natural immunity that would prevent
the spread of influenza. Finally, we estimated the level of combined immunity in
the US during winter 2010. Our results suggest that a third wave was unlikely if
the basic reproduction number *R*_{0} were below 1.6,
plausible if the original *R*_{0} was 1.6, and likely if
the original *R*_{0} was 1.8 or higher. Given that the
estimates for the basic reproduction number for pandemic influenza place it in
the range between 1.4 and 1.6 [1,
2, 3, 4, 5, 6, 7], our approach accurately
predicted the absence of a third wave of influenza in the US during the winter
of 2010. *We also used this scheme to accurately predict the second wave
of pandemic influenza in London and the West Midlands, UK during the fall of
2009*.

In the past century, there were three major influenza pandemics (1918, 1957, and 1968) and they all had multiple waves. There is evidence of an early wave in the spring of 1918 in the United States (US) and Europe, followed by a large wave in the fall of 1918 and a third, more mild wave, in the winter of 1919 [8, 9, 10]. In the US and temperate Northern Hemisphere, 2009 influenza pandemic A(H1N1) (pandemic H1N1) started in April 2009, and it was followed by a second wave during the fall of 2009. The ratio of susceptible and immune individuals in a population at the end of a given wave, plays a crucial role in determining the possibility and magnitude of the following wave. While vaccines were not available in the previous pandemics, we now have the ability to produce vaccines quickly and in large quantities [11]. In fact, more than 20 vaccines were developed during late spring to early summer of 2009. In several countries, vaccination started as early as mid-September and continued during the fall of 2009, so that, at the end of the second wave, a fraction of the population had acquired vaccine-induced immunity, while some fraction of the population got infected and hence acquired natural immunity. Since influenza vaccines are not completely protective, vaccine-induced immunity is not expected to be as strong as naturally acquired immunity due to natural infection. Therefore, at the end of the second wave, the population had a combination of natural and vaccine-induced immunity.

For logistical, practical, and economic reasons, estimating the number of
people infected and the number of people vaccinated is not always possible. In the
context of the H1N1 influenza pandemic, these estimates were extremely difficult to
make. It then became important to determine if this mixture of immunity in the
population was going to be enough to prevent a third wave of pandemic influenza.
Motivated by this problem, we developed a simple scheme to determine the possibility
of a third wave of pandemic influenza in the United States in the winter of 2010,
*based on the level of herd immunity at that time*. To do this,
we formulated a mathematical model for pandemic influenza. Using known techniques
for computing the basic reproduction number *R*_{0} and the
effective reproduction number *R _{f}* (defined to be the
reproduction number when a fraction

Influenza has strong seasonality patterns in the temperate Northern and Southern hemispheres, usually peaking during the late fall or early winter. In other regions of the world, the peaks tend to occur during the rainy season, but the seasonality is less pronounced. The basic mechanisms for these changes are not completely understood, but there is some evidence that survival and transmission of the influenza virus are influenced by physical factors, such as humidity and temperature [12, 13, 14, 15]. In addition, influenza spreads best during periods when schools are open. The aim of our work is to predict the occurrence or absence of a new epidemic wave based on the current level of herd immunity. For this reason, we will assume, below, that we are at the beginning of the influenza season, so that, in absence of prior immunity in the population, an epidemic wave is certain.

Our influenza model is based on the standard *SIR* model,
and it is an extension of the model given in [16]. We considered a closed population of
size *N*. Since the time scale for the spread of influenza is
short compared to migration or demographics (births and deaths), none of these
features are included. We divided the population into two sub-populations of
children and adults of size *N*_{1} and
*N*_{2}, so that *N* =
*N*_{1} + *N*_{2}.
Members in each group are either susceptibles *S _{ij}*,
infected asymptomatic

- A fraction
*ρ*of the infected people will develop symptoms. The infected asymptomatic people never develop symptoms but are still able to transmit the infection to others. Infected asymptomatic people have their infectiousness reduced by a factor*m*compared to infected symptomatic people [17], where*m*[0,1]. *c*is the number of contacts per day between people in group_{ik}*i*and people in group*k*, where*i, k*{1, 2}.*p*is the probability of infection given contact; it will be used here as a parameter to vary the transmissibility of the infection.- Children and adults recover at rates γ
_{1}and γ_{2}respectively. - Following the ideas in [18], we consider that vaccination has three major effects in the vaccinee, as follows
- (i) VE
_{S}, the vaccine efficacy for susceptibility is the ability of the vaccine to prevent infection. - (ii) VE
_{I}, the vaccine efficacy for infectiousness (conditioned upon being infected) is the effect of the vaccine on reducing infectiousness. - (iii) VE
_{P}, the vaccine efficacy for pathogenicity (conditioned upon being infected), accounts for the effect of the vaccine on reducing the probability of symptomatic disease given infection.

Based on these assumptions we have the following system of differential equations:

Unvaccinated | Vaccinated |

$$\frac{d{S}_{10}}{dt}=-{\lambda}_{1}{S}_{10}$$ |
$$\frac{d{S}_{11}}{dt}=-{\lambda}_{1}\theta {S}_{11}$$ (1) |

$$\frac{d{S}_{20}}{dt}=-{\lambda}_{2}{S}_{20}$$ |
$$\frac{d{S}_{21}}{dt}=-{\lambda}_{2}\theta {S}_{21}$$ (2) |

$$\frac{d{A}_{10}}{dt}={\lambda}_{1}(1-\rho ){S}_{10}-{\gamma}_{1}{A}_{10}$$ |
$$\frac{d{A}_{11}}{dt}={\lambda}_{1}(1-\rho \psi )\theta {S}_{11}-{\gamma}_{1}{A}_{11}$$ (3) |

$$\frac{d{A}_{20}}{dt}={\lambda}_{2}(1-\rho ){S}_{20}-{\gamma}_{2}{A}_{20}$$ |
$$\frac{d{A}_{21}}{dt}={\lambda}_{2}(1-\rho \psi )\theta {S}_{21}-{\gamma}_{2}{A}_{21}$$ (4) |

$$\frac{d{I}_{10}}{dt}={\lambda}_{1}\rho {S}_{10}-{\lambda}_{1}{I}_{10}$$ |
$$\frac{d{I}_{11}}{dt}={\lambda}_{1}\rho \psi \theta {S}_{11}-{\gamma}_{1}{I}_{11}$$ (5) |

$$\frac{d{I}_{20}}{dt}={\lambda}_{2}\rho {S}_{20}-{\lambda}_{2}{I}_{20}$$ |
$$\frac{d{I}_{21}}{dt}={\lambda}_{2}\rho \psi \theta {S}_{21}-{\gamma}_{2}{I}_{21}$$ (6) |

$$\frac{d{R}_{10}}{dt}={\gamma}_{1}({A}_{10}+{I}_{10})$$ |
$$\frac{d{R}_{11}}{dt}={\gamma}_{1}({A}_{11}+{I}_{21)}$$ (7) |

$$\frac{d{R}_{20}}{dt}={\gamma}_{2}({A}_{20}+{I}_{20})$$ |
$$\frac{d{R}_{21}}{dt}={\gamma}_{2}({A}_{21}+{I}_{21})$$ (8) |

where VE_{S} = 1 − *θ*,
VE_{I} = 1 − *ϕ* and
VE_{P} = 1 − *ψ*. The forces
of infection for children and adults, respectively, are given by

$${\lambda}_{1}=\frac{p{c}_{11}}{{N}_{1}}\left(m{A}_{10}+m\varphi {A}_{11}+{I}_{10}+\varphi {I}_{11}\right)+\frac{p{c}_{12}}{{N}_{2}}(m{A}_{20}+m\varphi {A}_{21}+{I}_{20}+\varphi {I}_{21})$$

(9)

and

$${\lambda}_{2}=\frac{p{c}_{21}}{{N}_{1}}(m{A}_{10}+m\varphi {A}_{11}+{I}_{10}+\varphi {I}_{11})+\frac{p{c}_{22}}{{N}_{2}}(m{A}_{20}+m\varphi {A}_{21}+{I}_{20}+\varphi {I}_{21}).$$

(10)

In this section we follow the ideas of [16] and [19]. Let *f*_{1} be
the fraction of vaccinated children and *f*_{2} be the
fraction of vaccinated adults, where we assume that vaccination occurred before
the (possible) epidemic.

The basic reproduction number *R*_{0} for a given
disease is defined as the expected number of secondary infections resulting from
a single typical infectious person in a completely susceptible population, and
*R _{f}* is defined to be the effective
reproduction number, which is the reproduction number in presence of
vaccination. We use the approach given in [20] and [21, 22] to compute the next generation matrix

If *R _{f}* > 1, the epidemic will grow,
whereas if

We assume natural infection confers complete protection against
reinfection, equivalent to setting *θ* = 0 in the
vaccinated equations (1) and
(2). So, to model the
fraction of children and adults who previously got the infection and who are now
immune, we simply set *θ* = 0 in our
computations. We let *R _{n}* be the effective
reproduction number with natural immunity, that is, the effective reproduction
number when a fraction of the population has natural immunity
(

In this section, we outline the scheme we developed to investigate the possibility of multiple influenza epidemic waves using the model and the thresholds established above. In the next section, we will use this approach to predict the absence of a third wave of pandemic H1N1 influenza in the United States during the winter of 2010.

We begin by noting that if all the parameters of the model are known
except for the values of *f*_{1} and
*f*_{2}, then *R _{f}* becomes
a function of

We define a critical vaccination vector to be a pair
(*f*_{1}, *f*_{2}) such that
*R _{f}*(

In general, finding an analytical solution for the contour lines of
*R _{f}*(

Surfaces representing all the eigenvalues of the next generation matrix for
*R*_{0} = 1.6. All the eigenvalues of the
matrix but two are zero (plotted in yellow). The ones that are non-zero are
plotted above in red and blue. The green plot corresponds to the **...**

First, we assume that the only way to be protected just before the start of a new influenza season is to be vaccinated. We then compute the critical vaccination vectors using the method described above. In this fashion we obtain thresholds for the fraction of vaccinated children and adults needed to prevent a further epidemic wave. While this is not realistic, since people who got influenza in a previous wave would have some degree of protection, it permits us to isolate the effects of vaccination from those of natural immunity at the population level.

Second, we assume that the only way to be protected just before the start of a new influenza season is to be naturally immune, that is, to have acquired the infection in a previous wave. This scenario will only be realistic if a vaccine for the particular influenza strain did not exist, but, as before, it allows us to separate out the effects of natural immunity. We then compute the critical immune vectors, providing us with a threshold for the fraction of naturally immune children and adults needed to prevent a further epidemic.

Finally, we combine this information to establish upper and lower bounds
for the threshold on the fractions of children and adults, either vaccinated or
naturally immune, that would prevent a new wave of influenza. In this way, we
predict the regions in the
*f*_{1}*f*_{2}—plane
that would result in a third wave and the regions in the plane that would
prevent a third wave. We then compute estimates of the number of children and
adults infected and vaccinated, and compare these estimates with our thresholds.
If the estimates lie above the thresholds, there is sufficient combined herd
immunity to prevent a subsequent epidemic wave. If the estimates lie below the
threshold, a new epidemic wave is possible. If the estimate lies between the
lower and upper bound, then the method is inconclusive. This approach will be
clarified in the next section, when we use it to predict the absence of a third
wave of pandemic influenza in the United States during the winter
2009–2010, *and to occurrence of the second wave of pandemic
influenza in England during the fall 2009*.

During the first (Spring 2009) and second (Fall 2009) waves of the pandemic H1N1, a significant fraction of the population got infected and became naturally immune. Meanwhile, several vaccines were developed and during the second wave, a fraction of children and adults got vaccinated. We used the model and approach described above to accurately predict the absence of a third wave of pandemic influenza in the United States during the winter of 2010.

Based on current estimates (for example [1, 2, 3, 4, 5, 6, 7]), we considered *R*_{0} for
pandemic H1N1 to lie in the interval [1.2, 1.8]. We used the
parameter *p* to vary the intensity of the infection by
selecting values of *p* for which the original basic
reproduction number would be in the range [1.2, 1.8]. We
calibrated the model (1)–(8)
for the pandemic H1N1 epidemic in the US according to the Table 2. Using [16] as a guide, we manually computed the
contact rates *c _{ij}* given in the Table 1 so that the final illness attack rates
(defined as the percentage of the population that became ill with influenza)
shown in Table 3 would match the
final illness attack rates observed at the end of the second wave of
pandemic influenza in the United States [23]. These numbers satisfy the following
three conditions: first, the number of contacts within the child group is
higher than the contacts between children and adults and within the adult
group. Secondly, the number of contacts in the diagonal (child-to-child and
adult-to-adult) is higher than the off-diagonal contacts. This is in
agreement with previously published contact studies [24, 25]. Finally, the ratio of the illness attack rate in
children to adults is similar to the one obtained from the CDC estimates
[23]. We further
assumed symmetry in the contacts, i.e.,

*The numbers of previously infected or vaccinated people are
used in our scheme to compute the effective reproductive number. For
this reason, we need to take into account all the people who became
previously infected or were vaccinated against this particular strain of
influenza. This implies that we need to consider the number of infected
(or vaccinated) people that resulted from both the first and the second
waves of pandemic H1N1 in 2009. The estimates for the final attack rates
provided by the CDC are the cumulative numbers for the total number of
people vaccinated and infected up to December 2009. This implies that
these estimates combine the attack rates of the first and the second
wave of pandemic H1N1, giving us the desired estimates*.

We assumed that pandemic H1N1 vaccines have similar efficacies to the ones for seasonal vaccines, and hence we took the vaccine efficacies for susceptibility, infectiousness and pathogenicity as an average between well-matched live attenuated vaccine and a well-matched inactivated vaccine using the estimates given in [26].

For each *R*_{0} in the range given above, we
used symbolic software (Mathematica) to obtain surfaces of the eigenvalues
of the next generation matrix *K* as described in the
previous section. Here, the structure of our model resulted in a next
generation matrix of rank 2. This implies that zero is an eigenvalue of
multiplicity 6 and we were able to compute closed analytic formulas for the
two non-zero eigenvalues. The explicit formulas can be found in the Appendix.

Recall that this analysis is done in three steps. First, we assume
that the only way to be protected at the beginning of the influenza season
is by being vaccinated. For instance, Figure
1 shows all the eigenvalue surfaces in presence of vaccination
together with the plane *P* =
{*z* = 1} and
*R*_{0} = 1.6. We determine the spectral
radius of this matrix *R _{f}*
(

Contour curve for *R*_{f} (*f*_{1},
*f*_{2}) = 1. The points
(*f*_{1}, *f*_{2}) on the
curve are the critical vaccination vectors.

The points above this curve (the green region in Figure 2) correspond to coverages of a vaccinated
fraction *f*_{1} of children and a vaccinated
fraction *f*_{2} of adults that will make the
effective reproduction number be below one, so that no further transmission
of the infection would be possible. For example, if 60% of the
children and 10% of the adults were vaccinated during the second
wave, that would be enough to prevent a new epidemic wave. This would also
be true if 45% of the children and 75% of the adults were
vaccinated.

Then, we repeated this analysis to find the critical immune curves
(here *θ* = 0). Under this scenario, the only
way to be protected at the beginning of the influenza season is by being
naturally immune. An example of the critical immune curves for
*R*_{0} = 1.6, is given in Figure 3. For example, once 45% or more of
children coupled with 10% or more of adults have already been
infected, there would be no chance of transmission; or if 33% or
more of children immune coupled with 75% or more of adults have
previously been infected it would have the same effect.

Contour curve for *R*_{n}(*f*_{1},
*f*_{2}) = 1. The points
(*f*_{1}, *f*_{2}) on the
curve are the critical immune vectors.

Since influenza vaccines are not completely protective, the critical
vaccination curves will always lie above the critical immune curves. The
threshold curve for a mixture of vaccination-induced immunity and natural
immunity should then lie somewhere between the critical vaccination curve
and the critical immune curve. We call this region the **intermediate
region**. Figure 4 shows the
intermediate regions for several basic reproduction numbers.

Suppose that we know that exactly *x*% of the
children got infected during the previous wave and exactly
*y*% of the children got vaccinated, then,
ideally, (*x* + *y*)% of the
children would be protected for the next wave. However, we do not precisely
know the number of children who were infected during the first and second
waves of pandemic H1N1 (spring and fall of 2009) nor the number of children
who were vaccinated during the second wave (fall 2009). Moreover, we cannot
guarantee that children who were vaccinated were not already immune,
especially given the fact that a fraction of the infected children never
develop symptoms. Therefore, we know that the level of protection of
children should lie somewhere between *x*% and
(*x* + *y*)%. A similar
analysis can be done for the adult age group.

We used this argument to estimate the level of combined immunity in
children and adults in the winter 2010. First, we obtained raw estimates for
the total number of people vaccinated and infected by using the information
provided by the CDC [23]. Then, we used the data given by Ross *et
al*. [27]
to calculate the age-specific infection attack rates in children and adults
by computing a weighted average of their estimates, where the weights were
given by the proportions of the population in each age group according to US
census data [28]. We
assumed that vaccination was carried out independently of whether a person
was previously infected or not. We considered the full range of estimates
given by the CDC in [23], hence obtaining a rectangle S in the
*f*_{1}*f*_{2}—plane
of estimates of combined immunity at the end of winter 2009–2010 for
children and adults, shown as a dotted box in Figure 4. The center of this rectangle, corresponding to the
mean of the estimates, gives the combined immune and vaccinated fraction of
children to be about 54% and for adults to be about 37%.

Given an estimate for the level of combined immunity in winter 2010,
we were able to correctly predict the absence of a third wave of pandemic
H1N1. To do this, we looked at the estimated level of combined immunity
together with the intermediate regions. The results are shown in Figure 4. Provided that the rectangle
*S* lies on or above the intermediate regions, the
effective reproduction numbers will be below one and there will be no
further transmission. The rectangle *S* lies above the
intermediate regions for *R*_{0} = 1.2, and
*R*_{0} = 1.4. According to our model,
this implies that a third epidemic wave would not have occurred for these
reproduction numbers. However, for *R*_{0} =
1.6, the rectangle *S* overlaps with the intermediate region,
and our model suggests that a third wave would have been possible. Finally,
if the original *R*_{0} were 1.8 or higher, our
results suggest that substantial spread would have been possible.

There was no third wave of pandemic influenza in the US during the
winter of 2010. Considering that most of the estimates for
*R*_{0} for pandemic H1N1 place it at 1.6 or
below [1, 2, 3, 4, 5, 6, 7], our method correctly predicted the
absence of this new epidemic wave.

For comparison, we present in this section an analogous analysis for the second wave of pandemic H1N1 influenza in London and the West Midlands, UK. Contrary to the US, the UK and in particular, England, experienced a big first epidemic wave of pandemic H1N1 influenza during the summer of 2009 [29]. This provides a unique opportunity for us to investigate whether our method can predict the occurrence of the second wave of pandemic H1N1 influenza in the fall of 2009, in the UK.

The main parameters of the model calibrated for the US epidemic do not change for the epidemic in England (contact rates, vaccine efficacies, etc.), so we only adjusted the proportion of children and adults in the population so that they matched the proportions in the UK. Since vaccine was introduced in the UK during the second wave, we only need to consider natural immunity, acquired through infection during the first wave (spring-summer 2009). We used the data for London and the West Midlands of Miller et al. [30] to estimate the fraction of the children and the adults who became infected during the first wave in this region. Miller et al. [30] partitioned the population into eight age groups (0–4, 5–14, 15–24, 25–44, 45–64, 65–74, 75–79, and ≥ 80 years), so we considered children to be up to 24 years old. The fraction of children previously infected was computed as a weighted average, where the weights corresponded to the proportion of the population in each age group in the UK [31]. We considered the full range of estimates given in [30]. In their paper, they found that there was no difference between the baseline (before the first wave of pandemic H1N1) and September 2009, for people in the older age groups (25 years and older). We then assumed that there were no adults immune after the first wave of H1N1 in London and the West Midlands.

*The results are presented in*
figure 5. The estimate of the natural
immunity in the children population in London and the West Midlands is shown in
red. For R* _{0} = 1.2, the estimate for the natural
immunity lies above the critical immune curve. Our method would then predict
no further transmission. For R_{0} ≥ 1.4, the estimate lies
below the critical immune curve, indicating that a new epidemic wave would
be possible. We acknowledge that the assumption that no adults were infected
during the first wave is unrealistic, but it is important to note that our
method accurately predicts the second epidemic wave for R_{0}
≥ 1.6, for all possible values of natural immunity in the adult
population. If R_{0} = 1.4, then as long as less than
30% of the adults were infected during the first wave, our method
predicts a new epidemic wave (see panel 2*
figure 5). Given that most of the estimates
for R

We performed a one-way sensitivity analysis for each of the contact
rates, for the recovery rates in children and adults and for each of the vaccine
efficacy parameters. Since *c*_{12} =
*c*_{21}, we performed sensitivity analysis for the
contact rates *c*_{11}, *c*_{12},
and *c*_{22} only. We drew values from a uniform
distribution of length 0.2 centered around each value for the given contact
rate. Each contact rate was drawn independently but only the triplets
{*c*_{11}, *c*_{12},
*c*_{22}} that satisfy the following
conditions were chosen:

- – The basic reproduction number obtained by using the contact rates fell into one of the following intervals: 1.15 <
*R*_{0}< 1.25 or 1.35 <*R*_{0}< 1.45 or 1.55 <*R*_{0}< 1.65 or 1.75 <*R*_{0}< 1.85. - –
*c*_{11}>*c*_{12}and*c*_{11}>*c*_{22}(there are more contacts among children than among adults and more than between children and adults).

The results for 500 runs for each reproduction number are shown in Figure 6. The general trends are quite robust
for this sensitivity; if *R*_{0} = 1.2 or
*R*_{0} = 1.4, all of the critical
vaccination curves lie below the rectangle *S*. If
*R*_{0} = 1.6 there is some overlap with
*S*, and for *R*_{0} = 1.8 the
majority of the curves intersect or lie above *S*.

Sensitivity analysis for the contact rates taken from a uniform distribution of
length 0.2, centered at the value of the *c*_{ij} used.
The contact rates were taken independently but only the triplets that gave 1.15
< *R*_{0} < 1.25, or 1.35 <
**...**

For the vaccine efficacies, we analyzed the uncertainty by drawing a
random number from a uniform distribution of length 0.2 centered around the
value used. For each *R*_{0}, we performed 500 runs. The
results are shown in Figure 7. The trends
are very robust with respect to these parameters as well.

Sensitivity analysis for the vaccine efficacies considered. The vaccine
efficacies were taken from a uniform distribution of length 0.2 centered around
the values used. Each vaccine efficacy was drawn independently. The results are
quite robust with respect **...**

*Finally, we performed sensitivity analysis for the recovery
rates for children and adults. For each recovery rate, we drew a random
number from a uniform distribution of length 0.2, centered around the value
used. We used only those pairs of recovery rates for children and adults for
which the basic reproduction number fell into one of the following
intervals: 1.15 < R _{0} < 1.25 or 1.35 <
R_{0} < 1.45 or 1.55 < R_{0} <
1.65 or 1.75 < R_{0} < 1.85. As shown in*
figure 8, the results are robust with
respect to these parameters.

The approach proposed here, using information derived from the next
generation matrix, provides simple thresholds for the vaccine-induced protection and
natural immunity needed to prevent further spread of influenza, once a wave has
passed. This can be particularly useful in a situation where most of the parameters
are difficult to determine accurately. Most of the time we only have ranges of
possible values. For example, determining the number of people infected from
reported influenza illness data is difficult, given that a fraction of infections
are asymptomatic. In addition, serosurveys can be problematic because of cross
reacting antibodies. We have incorporated information about both the
naturally-induced immunity and the vaccination induced immunity, and we have
discussed a possible interpretation of this mixture of immunity and its relationship
to the naturally-induced immunity only and to the vaccine-induced immunity only. The
thresholds proposed here can be calculated exactly and even if a closed form might
not always be available, symbolic software can help us in interpreting and using
this information. We parametrized our influenza model for the pandemic H1N1 and used
estimates given by the *US Centers for Disease Control and
Prevention* (CDC) [23] and serosurvey data [27] to estimate the level of combined immunity due to
vaccination and previous infection in the United States at the end of the second
wave of pandemic influenza H1N1 (Fall 2010). Our computations suggested that for
this epidemic, a third wave in the United States was unlikely if the original
*R*_{0} was 1.4 or lower, plausible for the low estimates
of mixed immunity if the original *R*_{0} was 1.6, and likely
if the original *R*_{0} was 1.8 or higher. Our results
accurately predicted the absence of a third wave of pandemic influenza in the United
States during the winter of 2010. *It is worth noting that these results are
conservative. This is because the serosurvey data was taken during the epidemic
and not at the end of it. Since individuals might take up to several weeks to
seroconvert, we most probably underestimated the level of immunity in the
population*. These results were used by the Los Angeles County,
California, Department of Public Health as part of their response to the pandemic
[32]. *Then, we
repeated the analysis for predicting the second wave of pandemic H1N1 in London
and the West Midlands. Using the estimates given in [**30**], our
method suggested that a second wave in London and the West Midlands was unlikely
if R*_{0} = 1.2, *but likely for all other values
of R*_{0}. *This shows that our method accurately
predicted the occurrence of the second wave of pandemic influenza in London and
the West Midlands during the fall of 2009*.

We used an influenza model capable of producing multiple epidemic waves. In
order to make our results as general as possible, we used the simplest structure
possible. In particular, we assumed that the probability of infection given contact
was constant. Modifying this structure for specific situations would involve more
complicated functions for the probability of infection, an interesting direction for
future work. The approach proposed here has a number of limitations. First of all,
this approach was based on either being able to explicitly find the largest
eigenvalue in closed form or on graphically determining the largest eigenvalue of
the next generation matrix. Obtaining closed forms for a more complicated system
might be impossible, even with the help of symbolic software. While analyzing models
with three age groups is still possible, working with more than three groups is
impossible graphically and some other analytical technique would need to be
employed. Also, the model proposed here is very simple. Even without changing the
number of age groups, one can make it more realistic. For example, we could allow
different probabilities of infection for children and adults or we could assume
different recovery rates for asymptomatic and symptomatic individuals. Given that
asymptomatic people will not reduce their level of activity, one could think of
having different contact rates for the two infected groups. The approach used here
to predict the recurrence of epidemic waves depends heavily on the next generation
matrix, which in turn depends heavily in the contact pattern. Our sensitivity
analysis showed that accurately estimating the contact rates is important.
*Our approach ignores the possibility that other factors, such as
time-dependent contact rates, or the dates of school openings might predict new
epidemic waves. We did not consider prior immunity to H1N1, which proved to be
enough for protecting older populations* [*33**]. We did not consider
future epidemic waves of H1N1*. Finally, we did not take into account
the possible antigenic drift of the virus that would make previous infected people
again susceptible to some degree and would decrease vaccine efficacy.
*Incorporating this feature in our method would allow the possibility of
investigating further H1N1 epidemics, like the ones that occurred in the UK and
the rest of Europe during the winter of 2010–2011*.

In the model of Hill and Longini in [16], the authors established thresholds for a model that does
not include asymptomatics or vaccine efficacy for pathogenicity. In this sense, the
current work is a natural extension of their model. The *SIR* model
proposed here is similar to the one proposed by Brauer in [34], but we omitted the latent period and
considered vaccination instead of treatment. They established useful final size
relations and we established threshold conditions. Thus, these results complement
each other. While our model was tailored for influenza, the methods used here can be
easily adapted for other acute infectious diseases. For example, this method could
be applied to cholera. Cholera is an infectious disease with strong seasonality in
some countries [35], and
immunity due to natural infection and vaccines wane over a short period of time
[36, 37].

We believe that our approach is novel in that we were able to predict the occurrence (or not) of multiple epidemic waves by incorporating information of both the vaccine induced immunity and the naturally-induced immunity. Our scheme relies in the use of standard methods to compute the next generation matrix and effective reproduction number which makes it to adapt to other diseases or more complicated models. The method developed here suggests yet another effect of vaccination. Vaccination not only directly protects the vaccinated and indirectly protects the unvaccinated during the current wave, but it can also help in preventing subsequent waves of influenza.

- Influenza pandemics occur in epidemic waves.
- We use mathematical epidemiology to develop thresholds for subsequent epidemic waves.
- We combine the thresholds with known herd immunity to predict subsequent waves.
- For the 2009 influenza pandemic, we predicted the absence of the third wave in the US.
- For the 2009 influenza pandemic, we predicted the occurrence of the second wave in England.

This work was partially supported by National Institute of General Medical Sciences MIDAS grant U01-GM070749 and National Institute of Allergy and Infectious Diseases grant R01-AI32042.

LM was partially supported by Consejo Nacional de Ciencia y Tecnologia, Mexico, scholarship 196221. LM thanks Nicholas Cain for his help with the computational implementation of the method, Dennis Chao for providing the estimates used to plot the box in Figure 4, Pauline van den Driessche for providing useful comments.

Here, we give the details of the computation of the effective reproduction number used above. The code can be provided upon request. We use the approach given in [20] and [21, 22]. Let

*S*_{10}(0) =**S**_{10},*A*_{10}(0) =**A**_{10},*I*_{10}(0) =**I**_{10},*R*_{10}(0) = 0,*S*_{11}(0) =**S**_{11},*A*_{11}(0) =**A**_{11},*I*_{11}(0) =**I**_{11},*R*_{11}(0) = 0,*S*_{20}(0) =**S**_{20},*A*_{20}(0) =**A**_{20},*I*_{20}(0) =**I**_{20},*R*_{20}(0) = 0,*S*_{21}(0) =**S**_{21}*A*_{21}(0) =**A**_{21}*I*_{21}(0) =**I**_{21}*R*_{21}(0) = 0

be the initial conditions for the system (1)–(8) where

$$\begin{array}{c}{S}_{10}+{S}_{11}+{A}_{10}+{A}_{11}+{I}_{10}+{I}_{11}={N}_{1},\\ {S}_{20}+{S}_{21}+{A}_{20}+{A}_{21}+{I}_{21}+{I}_{21}={N}_{2},\end{array}$$

and **A**_{10},
**A**_{11}, **I**_{10},
**I**_{11}, **A**_{20},
**A**_{21}, **I**_{21},
**I**_{21} are very small positive numbers, each close to
0. Define

$${E}_{0}=\left({S}_{10},{S}_{11},{S}_{20},{S}_{21},{A}_{10},{A}_{11},{I}_{10},{I}_{11},{A}_{20},{A}_{21},{I}_{21},{I}_{21},0,0,0,0\right).$$

If we set **A**_{10} =
**A**_{11} = **I**_{01} =
**I**_{11} = **A**_{20} =
**A**_{21} = **I**_{21} =
**I**_{21} = 0, and **S**_{10}
+ **S**_{11} = *N*_{1},
**S**_{20} + **S**_{21} =
*N*_{2} the model (1)–(8) has an infinite number of disease free
equilibria, namely, one per each initial condition given. We linearize the
system for the infectious equations
(3)–(6)
around the disease free equilibrium *E*_{0}. This gives
us the matrices (as given in [21]) *F* and *V* defined as
follows.

$$F=pA(\begin{array}{ccc}\frac{{c}_{11}}{{N}_{1}}m& \frac{{c}_{12}}{{N}_{2}}m& \frac{{c}_{11}}{{N}_{1}}m\frac{{c}_{12}}{{N}_{2}}m\frac{{c}_{11}}{{N}_{1}}& \frac{{c}_{12}}{{N}_{2}}& \frac{{c}_{11}}{{N}_{1}}\frac{{c}_{12}}{{N}_{2}}& \frac{{c}_{21}}{{N}_{1}}m& \frac{{c}_{22}}{{N}_{2}}m& \frac{{c}_{21}}{{N}_{1}}m\frac{{c}_{22}}{{N}_{2}}m\frac{{c}_{21}}{{N}_{1}}& \frac{{c}_{22}}{{N}_{2}}& \frac{{c}_{21}}{{N}_{1}}\frac{{c}_{22}}{{N}_{2}}& \frac{{c}_{11}}{{N}_{1}}m& \frac{{c}_{12}}{{N}_{2}}m& \frac{{c}_{11}}{{N}_{1}}m\frac{{c}_{12}}{{N}_{2}}m\frac{{c}_{11}}{{N}_{1}}& \frac{{c}_{12}}{{N}_{2}}& \frac{{c}_{11}}{{N}_{1}}\frac{{c}_{12}}{{N}_{2}}& \frac{{c}_{21}}{{N}_{1}}m& \frac{{c}_{22}}{{N}_{2}}m& \frac{{c}_{21}}{{N}_{1}}m\frac{{c}_{22}}{{N}_{2}}m\frac{{c}_{21}}{{N}_{1}}& \frac{{c}_{22}}{{N}_{2}}& \frac{{c}_{21}}{{N}_{1}}\frac{{c}_{22}}{{N}_{2}}& \frac{{c}_{11}}{{N}_{1}}m& \frac{{c}_{12}}{{N}_{2}}m& \frac{{c}_{11}}{{N}_{1}}m\frac{{c}_{12}}{{N}_{2}}m\frac{{c}_{11}}{{N}_{1}}& \frac{{c}_{12}}{{N}_{2}}& \frac{{c}_{11}}{{N}_{1}}\frac{{c}_{12}}{{N}_{2}}& \frac{{c}_{21}}{{N}_{1}}m& \frac{{c}_{22}}{{N}_{2}}m& \frac{{c}_{21}}{{N}_{1}}m\frac{{c}_{22}}{{N}_{2}}m\frac{{c}_{21}}{{N}_{1}}& \frac{{c}_{22}}{{N}_{2}}& \frac{{c}_{21}}{{N}_{1}}\frac{{c}_{22}}{{N}_{2}}& \frac{{c}_{11}}{{N}_{1}}m& \frac{{c}_{12}}{{N}_{2}}m& \frac{{c}_{11}}{{N}_{1}}m\frac{{c}_{12}}{{N}_{2}}m\frac{{c}_{11}}{{N}_{1}}& \frac{{c}_{12}}{{N}_{2}}& \frac{{c}_{11}}{{N}_{1}}\frac{{c}_{12}}{{N}_{2}}& \frac{{c}_{21}}{{N}_{1}}m& \frac{{c}_{22}}{{N}_{2}}m& \frac{{c}_{21}}{{N}_{1}}m\frac{{c}_{22}}{{N}_{2}}m\frac{{c}_{21}}{{N}_{1}}& \frac{{c}_{22}}{{N}_{2}}& \frac{{c}_{21}}{{N}_{1}}\frac{{c}_{22}}{{N}_{2}}& )\end{array}$$

where *A* is given by

$$A<\mathrm{diag}({S}_{10}\left(1-\rho \right),{S}_{20}\left(1-\rho \right),{S}_{11}(1-\rho \psi )\theta ,{S}_{21}(1-\rho \psi )\theta ,{S}_{10}\rho ,{S}_{20}\rho ,{S}_{11}\rho \psi \theta ,{S}_{21}\rho \psi \theta ),$$

and *V* is given by

$$V=\mathrm{diag}({\gamma}_{1},{\gamma}_{2},{\gamma}_{1},{\gamma}_{2},{\gamma}_{1},{\gamma}_{2},{\gamma}_{1},{\gamma}_{2})$$

here *diag*(*a*,
…, *b*) denotes a diagonal matrix with elements
*a*, …, *b* on the diagonal. The
matrix *K* = *FV*^{−1} is
called the next generation matrix. The effective reproduction number
*R _{f}* is then given by

$${R}_{f}=\rho (K)$$

where
*ρ*(*K*) is the spectral radius of
*K*, that is, the largest eigenvalue of *K* in
absolute value.

Recall that *f*_{1} was defined to be the
fraction of vaccinated children and *f*_{2} was defined
to be the fraction of vaccinated adults, where we assume that vaccination
occurred prior to the beginning of the epidemic. If the number of initial
infections is small, we have

$$\begin{array}{ll}{S}_{10}\approx (1-{f}_{1}){N}_{1}\hfill & {\mathrm{S}}_{11}\approx {f}_{1}{N}_{1}\hfill \\ {S}_{20}\approx (1-{f}_{2}){N}_{2}\hfill & {S}_{21}\approx {f}_{2}{N}_{2}.\hfill \end{array}$$

The matrix *K* for this particular model has rank 2,
allowing us to conclude that the eigenvalue 0 has multiplicity 6 and that the
remainder eigenvalues can be calculated as the root of the characteristic
polynomial *P*(λ) given by

$$\begin{array}{c}P\left(\lambda \right)={\lambda}^{2}+\frac{1}{{\gamma}_{1}{\gamma}_{2}{N}_{1}{N}_{2}}({c}_{11}{\gamma}_{2}{N}_{2}p(-\rho \left(\left(1-{f}_{1}\right){N}_{1}+{f}_{1}{N}_{1}\varphi \psi \theta \right)+\\ m\left(\left(1-{f}_{1}\right){N}_{1}\left(\rho -1\right)+{f}_{1}{N}_{1}\varphi \left(\psi \rho -1\right)\theta \right))\\ +{c}_{22}{\gamma}_{1}{N}_{1}p(-\rho \left(\left(1-{f}_{2}\right){N}_{2}+{f}_{2}{N}_{2}\varphi \psi \theta \right)+\\ m\left(\left(1-{f}_{2}\right){N}_{2}\left(\rho -1\right)+{f}_{2}{N}_{2}\varphi \left(\psi \rho -1\right)\theta \right)))\lambda \\ +\frac{1}{{\gamma}_{1}{\gamma}_{2}{N}_{1}{N}_{2}}(-{c}_{12}{c}_{21}{p}^{2}(-\rho \left(\left(1-{f}_{1}\right){N}_{1}+{f}_{1}{N}_{1}\varphi \psi \theta \right)+\\ m\left(\left(1-{f}_{1}\right){N}_{1}\left(\rho -1\right)+{f}_{1}{N}_{1}\varphi \left(\psi \rho -1\right)\theta \right))(-\rho \left(\left(1-{f}_{2}\right){N}_{2}+{f}_{2}{N}_{2}\varphi \psi \theta \right)+\\ m\left(\left(1-{f}_{2}\right){N}_{2}\left(\rho -1\right)+{f}_{2}{N}_{2}\varphi \left(\psi \rho -1\right)\theta \right))+{c}_{11}{c}_{22}{p}^{2}(-\rho \left(\left(1-{f}_{1}\right){N}_{1}+{f}_{1}{N}_{1}\varphi \psi \theta \right)+\\ m\left((1-{f}_{1}){N}_{1}(\rho -1)+{f}_{1}{N}_{1}\varphi (\psi \rho -1)\theta \right)\left)\right(-\rho \left(\right(1-{f}_{2}){N}_{2}+{f}_{2}{N}_{2}\varphi \psi \theta )+\\ m\left((1-{f}_{2}){N}_{2}\left(\rho -1\right)+{f}_{2}{N}_{2}\varphi \left(\psi \rho -1\right)\theta \right)\left)\right)\end{array}$$

The roots of *P*(λ) were calculated to be

$$\begin{array}{c}{\lambda}_{1,2}=\frac{1}{2{\gamma}_{1}{\gamma}_{2}{N}_{1}{N}_{2}}\{{c}_{22}{\gamma}_{1}{N}_{1}{N}_{2}p({f}_{2}[m(1-\rho -\varphi \theta +\rho \psi \theta )+\rho (1-\varphi \psi \theta )]-m-\rho +m\rho )\\ +{c}_{11}{\gamma}_{2}{N}_{1}{N}_{2}p({f}_{1}[m(1-\rho -\varphi \theta +\varphi \psi \rho \theta )]-m-\rho +m\rho )\\ \pm [{p}^{2}4({c}_{12}{c}_{21}-{c}_{11}{c}_{22}){\gamma}_{1}{\gamma}_{2}{N}_{1}{N}_{2}(-\rho {N}_{1}[(1-{f}_{1})+{f}_{1}\varphi \psi \theta ]+\\ m{N}_{1}[(1-{f}_{1})(\rho -1)+{f}_{1}\varphi (\psi \rho -1)\theta ])& (-\rho {N}_{2}[(1-{f}_{2})+{f}_{2}\varphi \psi \theta ]+m{N}_{2}[(1-{f}_{2})(\rho -1)+{f}_{2}\varphi (\psi \rho -1)\theta ])\\ +({c}_{11}{\gamma}_{2}{N}_{1}{N}_{2}[-\rho ((1-{f}_{1})+{f}_{1}\varphi \psi \theta )+m((1-{f}_{1})(\rho -1)+{f}_{1}\varphi (\psi \rho -1)\theta )]\\ {+{c}_{22}{\gamma}_{1}{N}_{1}{N}_{2}{[-\rho ((1-{f}_{2})+{f}_{2}\varphi \psi \theta )+m((1-{f}_{2})(\rho -1)+{f}_{2}\varphi \theta (\psi \rho -1))]}^{2}]}^{\frac{1}{2}}\}.\end{array}$$

(11)

The effective reproduction number *R _{f}* is the
largest of these roots,

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