shows the critical level of between-household transmission,
, required to sustain an epidemic (Plot 1A); the early growth rate
(Plot 1B); the offspring distribution (Plot 1C); the endemic prevalence proportion
(Plot 1D); and the transient dynamics of infection (Plot 1E; see 1F and 1G also), all for the case of exponentially distributed infectious period (
). The precise parameter values are provided in the Figure caption; other rate parameter values are presented in the Supporting Information File S1
Figure 1 Epidemiological quantities as defined in the main text for the exponential () infectious period distribution.
Examining the critical level of between-household transmission (Plot 1A) indicates that two factors contribute to the success of an infection. For large household sizes, the main determinant of epidemic success is whether within-household transmission (governed by
) can produce an epidemic within the household – if it can, then the final size within households is relatively large and so relatively small levels of between-household transmission,
, can sustain an epidemic. For smaller households of size
, this is not seen, and appreciable between- and within-household transmission is always necessary to sustain an epidemic.
The endemic prevalence of infection (when between-household transmission rate
; Plot 1D) is again predominately determined by within-household transmission,
, for larger households, with household size,
, only having a major impact when it is below size
. However, unlike the critical level of transmission (Plot 1A) and early growth rate (Plot 1C), varying the rate of waning immunity,
, has a significant effect in terms of absolute prevalence
, which reduces as the rate of loss of immunity
is reduced, and vanishes in the absence of waning immunity (
); see Supporting Information File S1
The early growth rate
(with between-household transmission
; Plot 1B) varies with both within-household transmission,
, and household size,
, and is only weakly affected by waning immunity,
(see Supporting Information File S1
). We observe that the critical between-household transmission rate (Plot 1A) and the endemic prevalence of infection (Plot 1D) show far greater saturation with both household size,
, and within household transmission rate,
, compared to the early growth rate (Plot 1B). This is because the critical transmission rate,
, and the endemic prevalence of infection,
, depend on the number of cases produced over one generation (which rapidly saturates as susceptibles within the household get infected), whereas the early growth rate,
, depends on the instantaneous transmission rate from infected individuals (which is less influenced by households reaching saturation).
The offspring distribution (Plot 1C) shows a significant probability that a newly infected household will fail to infect any further households; often because the infection fails to spread within the household. This failure probability is relatively unaffected by the waning immunity rate,
; whereas increasing
leads to an increased probability of generating very large numbers of secondary cases. The bimodality of these distributions is a qualitative difference from distributions considered in detail at the individual level 
and therefore can be expected to lead to very different stochastic invasion and persistence properties.
The complete model dynamics at a given parameter set (Plot 1E) demonstrates that our methods for calculating early growth (green line) and the endemic state (blue line) are sound, and also that the proportion of households with a given prevalence assumes a unimodal distribution around the mean, with significant variance.
and essentially repeat the evaluation of each of the epidemiologically relevant quantities for the case of gamma distributed infectious period of order
held constant, respectively. In panels (A), (B), (C) and (D) the change in the quantity (respectively,
, probability of offspring number, and
) with respect to the assumption of an exponential infectious period has been presented; likewise, in (E), the mean prevalence curve and, in (F) and (G), the full distribution of prevalence, under the assumption of an exponential infectious period have been superimposed for reference.
With respect to the critical transmission rate,
, the early growth rate,
, and the endemic prevalence of infection,
, it can be seen that both the magnitude and direction of change can differ depending upon whether the transmission parameter
or the probability of transmission,
, is held constant. With respect to these quantities, holding
constant generally results in more significant changes. Furthermore, in general, a higher level of between-household transmission,
, is required to sustain an epidemic, and thus the endemic prevalence of infection,
is generally reduced. The early growth rate increases if
is held constant (Plot 2B) and generally decreases if
is held constant (Plot 3B).
With respect to the offspring distributions (C), once again the incorporation of gamma-distributed infectious period, and choice of what is held constant between epidemics, has a significant impact. In both cases the probability of no secondary households infected decreases, and by a substantial margin in the case of
held constant. In the
constant case the probability of a small number of secondary infections also decreases, whilst in the
constant case this is not seen. In both cases there is a decrease in the tail of the distribution, corresponding to reduced probability of a large number of secondary infections, and the major increase in probability mass occurs in the vicinity of the second peak of the exponential offspring distribution case. Despite these changes, for this parameter set, the bimodal feature of these distributions remains.
Finally, we consider the influence of gamma-distributed infectious period, and what is held constant, by studying the full dynamics of infection. Holding
constant (Plot 2E) results in an earlier epidemic with a larger peak infection when compared to the exponential case, whilst holding
constant (Plot 3E) results in an epidemic with similar, but slightly reduced, peak incidence but slower take-off and hence delayed peak. Also interestingly, the incorporation of gamma-distributed infectious period results in a slight oscillatory approach to endemicity, with mean infection dropping below the endemic prevalence following the peak, and slightly overshooting the endemic level again before converging to equilibrium.