Table 1.
A list of factors that compromise vaccine effectiveness incorporated in our model.
Vaccine responsiveness and administration are both captured by the same parameter, p, which is the vaccine effective coverage. An incomplete coverage is either due to the vaccine only being administrated to a proportion of pigs or because not all vaccinated pigs have developed protective immunity. A detailed description of the model parameters is given in Table 2.
Fig 1.
Flow diagrams of the heterogeneous vaccine SIR model given by (1).
Table 2.
Description of the model parameters together with their assumed value ranges considered in this study.
Fig 2.
Tree diagram showing modelled scenarios.
We refer to the text for definitions of the parameters and further explanations. Here, the vaccine coverage p represents either the proportion of individuals to which the vaccine is administered or the proportion of vaccine responders. Also, delay in the onset of vaccine induced immunity is not explicitly included as vaccine property, as it is equivalent to modelling a reactive vaccination strategy.
Fig 3.
The dependence of R0 on the vaccine-induced sterilizing immunity, ϵs, vaccine-induced reduction in host infectivity, ϵi, vaccine-induced increase in recovery rate, ϵγ, and vaccine coverage, p.
A. 3D surface plot corresponding to R0 = 1 in Eq (12), for a high average transmission potential () and full immunization coverage (p = 1). B. 2D contour plot showing the dependence of R0 on ϵi and ϵγ, for different values of ϵs and p = 1. C. 3D surface plot corresponding to R0 = 1 in Eq (12), for a high average transmission potential (
) and ϵi = 0.5. D. 2D contour plot showing the dependence of R0 on the effective vaccine coverage, p, and ϵγ, for different values of ϵs and ϵi = 0.5. The coloured continuous curves in 2D plots correspond to R0 = 1 in Eq (12) for a high average transmission potential (
) and dotted curves correspond to R0 = 1 for a low virulent PRRSv strain (
). Areas above these curves correspond to R0 < 1 where the infection cannot invade the herd. Here d = 0.001, μ = 0.0017. The transmission and recovery rates are values within the ranges given in Table 2, that satisfy
and
for the continuous and dotted curves, respectively.
Fig 4.
The effect of prophylactic mass vaccination for different ϵ = ϵs,i,γ values on the infection dynamics for continuous vaccination, i.e. λN = 0, λV = 0.0017 (left column), and one-off vaccination, i.e. λN = 0.0017, λV = 0 (right column).
A.-B. The vaccine only affects host susceptibility, i.e. sterilizing immunity ϵ = ϵs, ϵγ = ϵi = 0 (see relation (7)); C.-D. The vaccine only affects recovery rate, i.e. ϵ = ϵγ, ϵs = ϵi = 0 (see relation (9)); E.-F. The vaccine equally affects host susceptibility, infectivity and recovery rate, ϵ = ϵs,i,γ (see relations (7)–(9)). Other chosen parameter values were βNN = 0.12, γN = 0.01785, d = 0.001, μ = 0.0017, and the initial conditions are SV (0) = 0.999, IV (0) = 0.001 (see relation (11) for p = 1 and I0 = 0.001).
Fig 5.
The effect of fully effective mass vaccination (i.e. p = 1 at the time of vaccination) with vaccines of different properties on the infection dynamics in the case of prophylactic (left column) and reactive (right column) vaccination in a closed herd.
Profiles of different colours correspond to different ϵ = ϵs,i,γ values on the infection dynamics (left column; see caption in Fig 4), or different times when vaccination is applied (right column; see caption in Fig 6). A.-B. The vaccine only affects host susceptibility, i.e. sterilizing immunity, but no effect on infectivity or recovery (see Eqs (7) and (10)); C.-D. The vaccine only speeds up recovery, but offers no protection from infection and no reduction in infectivity (see Eqs (9) and (10)); E.-F. The vaccine equally affects host susceptibility, infectivity and recovery rate (see Eqs (7)–(9) and (10)). Here βNN = 0.12, γN = 0.01785, p = 1, d = 0.001, μ = λN,V = 0, and the initial conditions are SV (0) = 0.999, IV (0) = 0.001 (left column) and SN (0) = 0.999, IN (0) = 0.001 (right column).
Fig 6.
The effect of complete mass vaccination (i.e. p = 1) with vaccines of different properties on the infection dynamics in the case of continuous (left column) and one-off (right column) reactive vaccination.
Profiles of different colours correspond to different times when vaccination is applied, i.e. (i) at the start of the epidemic, equivalent to prophylactic vaccination (red continuous curves), (ii) at one quarter of the time of peak prevalence (green dotted curves), (iii) at one half of the time of peak prevalence (blue dotted curves), (iv) at the time of peak prevalence (red dotted curves), (v) at double the time until peak prevalence (gold dotted curves) and when (vi) vaccination is not applied at all (black continuous curves). A.-B. The vaccine only affects host susceptibility, i.e. sterilizing immunity, but no effect on infectivity or recovery (ϵs = 0.5, ϵγ = ϵi = 0) (see relations (7) and (10)); C.-D. The vaccine only speeds up recovery, but offers no protection from infection and no reduction in infectivity (ϵγ = 0.5, ϵs = ϵi = 0) (see Eqs (9) and (10)); E.-F. The vaccine equally affects host susceptibility, infectivity and recovery rate, ϵ = ϵs,i,γ = 0.5 (see Eqs (7)–(9) and (10)). Here βNN = 0.12, γN = 0.01785, p = 1, d = 0.001, μ = 0.0017, and the initial conditions are SN(0) = 0.999, IN(0) = 0.001 (see relation (11) for p = 0 and I0 = 0.001).