Mathematical assessment of the impact of cohort vaccination on pneumococcal carriage and serotype replacement

Figures & data

Figure 1. Flow diagram of the model (A.1(A.1) $\begin{array}{rl}(N_u{)}^{\prime }& =\mathrm{\Lambda }(1-\varphi )+\omega N_v-({\lambda }_i+{\lambda }_j+\mu )N_u,\\ (N_v{)}^{\prime }& =\mathrm{\Lambda }\varphi -\left[\right(1-ϵ){\lambda }_i+{\lambda }_j+\omega +\mu ]N_v,\\ (C_u^i{)}^{\prime }& ={\lambda }_i{N}_u+{\eta }_i{\lambda }_i{R}_{uni}-({\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_u^i,\\ (C_v^i{)}^{\prime }& =(1-ϵ){\lambda }_i{N}_v+{\eta }_i{\lambda }_i{R}_{vni}-({\theta }_v{\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_v^i,\\ (C_u^j{)}^{\prime }& ={\lambda }_j{N}_u+{\eta }_j{\lambda }_j{R}_{unj}+\omega C_v^j-({\gamma }_j+{\theta }_j{\lambda }_i+\mu )C_u^j,\\ (C_v^j{)}^{\prime }& ={\lambda }_j{N}_v+{\eta }_j{\lambda }_j{R}_{vnj}-[{\gamma }_j+{\theta }_j(1-ϵ){\lambda }_i+\omega +\mu ]C_v^j,\\ (C_u^{ij}{)}^{\prime }& ={\theta }_i{\lambda }_j{C}_u^i+{\theta }_j{\lambda }_i{C}_u^j+{\theta }_j{\eta }_i{\lambda }_i{C}_{uni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{unj}^i-({\gamma }_{ij}+\mu )C_u^{ij},\\ (C_v^{ij}{)}^{\prime }& ={\theta }_j(1-ϵ){\lambda }_i{C}_v^j+{\theta }_i{\lambda }_j{C}_v^i+{\theta }_j{\eta }_i{\lambda }_i{C}_{vni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{vnj}^i\\ & -[\left({\theta }_v\right(1-r_2)+r_2){\gamma }_{ij}+\mu ]C_v^{ij},\\ (R_{uni}{)}^{\prime }& ={\gamma }_i{C}_u^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{uni},\\ (R_{vni}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_v^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{vni},\\ (R_{unj}{)}^{\prime }& ={\gamma }_j{C}_u^j+\omega R_{vnj}-({\eta }_j{\lambda }_j+{\lambda }_i+\mu )R_{unj},\\ (R_{vnj}{)}^{\prime }& ={\gamma }_j{C}_v^j-[{\eta }_j{\lambda }_j+(1-ϵ){\lambda }_i+\omega +\mu ]R_{vnj},\\ (R_{uninj}{)}^{\prime }& ={\gamma }_i{C}_{unj}^i+{\gamma }_j{C}_{uni}^j+(1-r_1-r_2){\gamma }_{ij}{C}_u^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{uninj},\\ (R_{vninj}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_{vnj}^i+{\gamma }_j{C}_{vni}^j+{\theta }_v(1-r_2)(1-r_1){\gamma }_{ij}{C}_v^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{vninj},\\ (C_{unj}^i{)}^{\prime }& ={\lambda }_i{R}_{unj}+{\eta }_i{\lambda }_i{R}_{uninj}+r_2{\gamma }_{ij}{C}_u^{ij}-({\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{unj}^i,\\ (C_{vnj}^i{)}^{\prime }& =(1-ϵ){\lambda }_i{R}_{vnj}+{\eta }_i{\lambda }_i{R}_{vninj}+r_2{\gamma }_{ij}{C}_v^{ij}-({\theta }_v{\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{vnj}^i,\\ (C_{uni}^j{)}^{\prime }& ={\lambda }_j{R}_{uni}+{\eta }_j{\lambda }_j{R}_{uninj}+r_1{\gamma }_{ij}{C}_u^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{uni}^j,\\ (C_{vni}^j{)}^{\prime }& ={\lambda }_j{R}_{vni}+{\eta }_j{\lambda }_j{R}_{vninj}+{\theta }_v(1-r_2)r_1{\gamma }_{ij}{C}_v^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{vni}^j.\end{array}$(A.1) ), showing the transitions of the unvaccinated (left) and vaccinated (right) state variables (or components) of the model.

Table 1. Description of the state variables of the model (A.1(A.1) $\begin{array}{rl}(N_u{)}^{\prime }& =\mathrm{\Lambda }(1-\varphi )+\omega N_v-({\lambda }_i+{\lambda }_j+\mu )N_u,\\ (N_v{)}^{\prime }& =\mathrm{\Lambda }\varphi -\left[\right(1-ϵ){\lambda }_i+{\lambda }_j+\omega +\mu ]N_v,\\ (C_u^i{)}^{\prime }& ={\lambda }_i{N}_u+{\eta }_i{\lambda }_i{R}_{uni}-({\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_u^i,\\ (C_v^i{)}^{\prime }& =(1-ϵ){\lambda }_i{N}_v+{\eta }_i{\lambda }_i{R}_{vni}-({\theta }_v{\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_v^i,\\ (C_u^j{)}^{\prime }& ={\lambda }_j{N}_u+{\eta }_j{\lambda }_j{R}_{unj}+\omega C_v^j-({\gamma }_j+{\theta }_j{\lambda }_i+\mu )C_u^j,\\ (C_v^j{)}^{\prime }& ={\lambda }_j{N}_v+{\eta }_j{\lambda }_j{R}_{vnj}-[{\gamma }_j+{\theta }_j(1-ϵ){\lambda }_i+\omega +\mu ]C_v^j,\\ (C_u^{ij}{)}^{\prime }& ={\theta }_i{\lambda }_j{C}_u^i+{\theta }_j{\lambda }_i{C}_u^j+{\theta }_j{\eta }_i{\lambda }_i{C}_{uni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{unj}^i-({\gamma }_{ij}+\mu )C_u^{ij},\\ (C_v^{ij}{)}^{\prime }& ={\theta }_j(1-ϵ){\lambda }_i{C}_v^j+{\theta }_i{\lambda }_j{C}_v^i+{\theta }_j{\eta }_i{\lambda }_i{C}_{vni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{vnj}^i\\ & -[\left({\theta }_v\right(1-r_2)+r_2){\gamma }_{ij}+\mu ]C_v^{ij},\\ (R_{uni}{)}^{\prime }& ={\gamma }_i{C}_u^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{uni},\\ (R_{vni}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_v^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{vni},\\ (R_{unj}{)}^{\prime }& ={\gamma }_j{C}_u^j+\omega R_{vnj}-({\eta }_j{\lambda }_j+{\lambda }_i+\mu )R_{unj},\\ (R_{vnj}{)}^{\prime }& ={\gamma }_j{C}_v^j-[{\eta }_j{\lambda }_j+(1-ϵ){\lambda }_i+\omega +\mu ]R_{vnj},\\ (R_{uninj}{)}^{\prime }& ={\gamma }_i{C}_{unj}^i+{\gamma }_j{C}_{uni}^j+(1-r_1-r_2){\gamma }_{ij}{C}_u^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{uninj},\\ (R_{vninj}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_{vnj}^i+{\gamma }_j{C}_{vni}^j+{\theta }_v(1-r_2)(1-r_1){\gamma }_{ij}{C}_v^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{vninj},\\ (C_{unj}^i{)}^{\prime }& ={\lambda }_i{R}_{unj}+{\eta }_i{\lambda }_i{R}_{uninj}+r_2{\gamma }_{ij}{C}_u^{ij}-({\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{unj}^i,\\ (C_{vnj}^i{)}^{\prime }& =(1-ϵ){\lambda }_i{R}_{vnj}+{\eta }_i{\lambda }_i{R}_{vninj}+r_2{\gamma }_{ij}{C}_v^{ij}-({\theta }_v{\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{vnj}^i,\\ (C_{uni}^j{)}^{\prime }& ={\lambda }_j{R}_{uni}+{\eta }_j{\lambda }_j{R}_{uninj}+r_1{\gamma }_{ij}{C}_u^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{uni}^j,\\ (C_{vni}^j{)}^{\prime }& ={\lambda }_j{R}_{vni}+{\eta }_j{\lambda }_j{R}_{vninj}+{\theta }_v(1-r_2)r_1{\gamma }_{ij}{C}_v^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{vni}^j.\end{array}$(A.1) ). $k\in \{i,j\}$.

Variable	Description
P	Total population
$N_u\left(N_v\right)$	Number of unvaccinated(vaccinated) non-colonized individuals
$C_u^k$	Number of unvaccinated individuals colonized with serotype k
$C_v^i\left(C_v^j\right)$	Number of vaccinated individuals acquiring breakthrough carriage with serotype i (acquiring carriage with j)
$C_u^{ij}\left(C_v^{ij}\right)$	Number of co-colonized unvaccinated(vaccinated) individuals
$R_{unk}\left(R_{vnk}\right)$	Number of unvaccinated(vaccinated) individuals who have recovered from carriage, possessing natural immunity against serotype k
$R_{uninj}\left(R_{vninj}\right)$	Number of unvaccinated (vaccinated) recovered individuals who possess natural immunity against serotypes i and j
$C_{unj}^i\left(C_{uni}^j\right)$	Number of unvaccinated individuals colonized with serotype $i\left(j\right)$ possessing natural immunity against serotype $j\left(i\right)$ due to a history of carriage with serotype $j\left(i\right)$
$C_{vnj}^i\left(C_{vni}^j\right)$	Number of vaccinated individuals colonized with serotype $i\left(j\right)$ and possessing natural immunity against serotype $j\left(i\right)$

Table 2. Description of parameters of the model (A.1(A.1) $\begin{array}{rl}(N_u{)}^{\prime }& =\mathrm{\Lambda }(1-\varphi )+\omega N_v-({\lambda }_i+{\lambda }_j+\mu )N_u,\\ (N_v{)}^{\prime }& =\mathrm{\Lambda }\varphi -\left[\right(1-ϵ){\lambda }_i+{\lambda }_j+\omega +\mu ]N_v,\\ (C_u^i{)}^{\prime }& ={\lambda }_i{N}_u+{\eta }_i{\lambda }_i{R}_{uni}-({\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_u^i,\\ (C_v^i{)}^{\prime }& =(1-ϵ){\lambda }_i{N}_v+{\eta }_i{\lambda }_i{R}_{vni}-({\theta }_v{\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_v^i,\\ (C_u^j{)}^{\prime }& ={\lambda }_j{N}_u+{\eta }_j{\lambda }_j{R}_{unj}+\omega C_v^j-({\gamma }_j+{\theta }_j{\lambda }_i+\mu )C_u^j,\\ (C_v^j{)}^{\prime }& ={\lambda }_j{N}_v+{\eta }_j{\lambda }_j{R}_{vnj}-[{\gamma }_j+{\theta }_j(1-ϵ){\lambda }_i+\omega +\mu ]C_v^j,\\ (C_u^{ij}{)}^{\prime }& ={\theta }_i{\lambda }_j{C}_u^i+{\theta }_j{\lambda }_i{C}_u^j+{\theta }_j{\eta }_i{\lambda }_i{C}_{uni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{unj}^i-({\gamma }_{ij}+\mu )C_u^{ij},\\ (C_v^{ij}{)}^{\prime }& ={\theta }_j(1-ϵ){\lambda }_i{C}_v^j+{\theta }_i{\lambda }_j{C}_v^i+{\theta }_j{\eta }_i{\lambda }_i{C}_{vni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{vnj}^i\\ & -[\left({\theta }_v\right(1-r_2)+r_2){\gamma }_{ij}+\mu ]C_v^{ij},\\ (R_{uni}{)}^{\prime }& ={\gamma }_i{C}_u^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{uni},\\ (R_{vni}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_v^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{vni},\\ (R_{unj}{)}^{\prime }& ={\gamma }_j{C}_u^j+\omega R_{vnj}-({\eta }_j{\lambda }_j+{\lambda }_i+\mu )R_{unj},\\ (R_{vnj}{)}^{\prime }& ={\gamma }_j{C}_v^j-[{\eta }_j{\lambda }_j+(1-ϵ){\lambda }_i+\omega +\mu ]R_{vnj},\\ (R_{uninj}{)}^{\prime }& ={\gamma }_i{C}_{unj}^i+{\gamma }_j{C}_{uni}^j+(1-r_1-r_2){\gamma }_{ij}{C}_u^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{uninj},\\ (R_{vninj}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_{vnj}^i+{\gamma }_j{C}_{vni}^j+{\theta }_v(1-r_2)(1-r_1){\gamma }_{ij}{C}_v^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{vninj},\\ (C_{unj}^i{)}^{\prime }& ={\lambda }_i{R}_{unj}+{\eta }_i{\lambda }_i{R}_{uninj}+r_2{\gamma }_{ij}{C}_u^{ij}-({\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{unj}^i,\\ (C_{vnj}^i{)}^{\prime }& =(1-ϵ){\lambda }_i{R}_{vnj}+{\eta }_i{\lambda }_i{R}_{vninj}+r_2{\gamma }_{ij}{C}_v^{ij}-({\theta }_v{\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{vnj}^i,\\ (C_{uni}^j{)}^{\prime }& ={\lambda }_j{R}_{uni}+{\eta }_j{\lambda }_j{R}_{uninj}+r_1{\gamma }_{ij}{C}_u^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{uni}^j,\\ (C_{vni}^j{)}^{\prime }& ={\lambda }_j{R}_{vni}+{\eta }_j{\lambda }_j{R}_{vninj}+{\theta }_v(1-r_2)r_1{\gamma }_{ij}{C}_v^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{vni}^j.\end{array}$(A.1) ).

Parameters	Interpretation	Baseline (year${}^{-1}$)	Reference
	Demographic
Λ	Recruitment rate into the population	$3.7\times {10}^6$	[9]
μ	Natural death rate	$\frac{1}{78.6}$	[10]
	Natural history
${\gamma }_i$, ${\gamma }_j$, ${\gamma }_{ij}$	Recovery rate from carriage of serotype i, j or co-colonization ij	9.9, 8.9, 7.0${}^{†}$	[8, 13, 41]
$r_1\left(r_2\right)$	Proportion of co-colonized individuals who recovered from carriage of serotype $i\left(j\right)$	0.4 (0.3)${}^{‡}$	a
c	Effective contact rate	4,509${}^{†}$	[51]
${\text{β}}_i\left({\text{β}}_j\right)$	Probability of carriage of serotype $i\left(j\right)$ per contact	0.0041 (0.0034)${}^{‡}$	a
${\eta }_i$,${\eta }_j$	Modification parameter for the reduction of carriage rate with serotype k due to prior carriage of serotype k	0.6($0.39-0.90{)}^{‡}$	[27]
	Vaccine
ε	Vaccine efficacy against the (vaccine type) serotype i	0.7${}^{†}‡$	[8, 41, 52]
ω	Vaccine waning rate	$\frac{1}{8.3}$	[41]
ϕ	Proportion of the population vaccinated	varied${}^{‡}$
${\alpha }_v$	Modification parameter for the reduction of infectiousness of vaccinated colonized individuals, in relation to unvaccinated colonized individuals	0.5${}^{‡}$	a
${\theta }_v$	Modification parameter for the increase in recovery rate of vaccinated colonized individuals, in relation to unvaccinated colonized individuals	2.0${}^{‡}$	a
${\theta }_i$	Competition parameter for serotype-i; risk reduction factor for carriage of serotype-j due to current carriage of serotype-i	0.1${}^{‡}$	[5]
${\theta }_j$	Competition parameter for serotype-j; risk reduction factor for carriage of serotype-i due to current carriage with serotype-j	0.15${}^{‡}$	[5]

Notation: †: derived. ‡: dimensionless. a: assumed.

Figure 2. Simulation of the simplified model, showing convergence of the initial solutions (given by $N_u\left(0\right)=79,760,C_u^i\left(0\right)=335,860,C_u^j\left(0\right)=130,710,C_u^{ij}\left(0\right)=43,076,000,C_{unj}^i\left(0\right)=110,700,000,C_{uni}^j\left(0\right)=71,044,000,R_{uni}\left(0\right)=95,210,R_{unj}\left(0\right)=29,597,R_{uninj}\left(0\right)=6,533,000$) to carriage-free equilibrium. Parameter values used are: $\mathrm{\Lambda }=3,700,000$, $\mu =\frac{1}{78.6}$, c = 4508, ${\gamma }_1={\gamma }_2={\gamma }_{12}=9.9$, $r_1=r_2=0$, ${\eta }_i={\eta }_j=1$, ${\theta }_i=0.1$, ${\theta }_j=0.15$ and (a) ${\text{β}}_i=0.00041$, ${\text{β}}_j=0.00034$ (so that, ${\mathcal{R}}_0^i=0.186$, ${\mathcal{R}}_0^j=0.153$ and ${\mathcal{R}}_0=0.186<1$), (b) ${\text{β}}_i=0.0041$, ${\text{β}}_j=0.00034$ (so that, ${\mathcal{R}}_{0s}^i=1.86$, ${\mathcal{R}}_{0s}^j=0.153$ and $\frac{{\mathcal{R}}_{0s}^i}{({\mathcal{R}}_{0s}^i-1){\theta }_i+1}=1.72$), (c) ${\text{β}}_i=0.00041$, ${\text{β}}_j=0.0034$ (so that ${\mathcal{R}}_{0s}^i=0.186$, ${\mathcal{R}}_{0s}^j=1.53$ and $\frac{{\mathcal{R}}_{0s}^j}{({\mathcal{R}}_{0s}^j-1){\theta }_j+1}=1.43$), (d) ${\text{β}}_i=0.0041$, ${\text{β}}_j=0.0043$ (so that, ${\mathcal{R}}_{0s}^i=1.86$, ${\mathcal{R}}_{0s}^j=1.96$, $\frac{{\mathcal{R}}_{0s}^i}{({\mathcal{R}}_{0s}^i-1){\theta }_i+1}=0.2$ and $\frac{{\mathcal{R}}_{0s}^j}{({\mathcal{R}}_{0s}^j-1){\theta }_j+1}=1.43$) and (e) ${\text{β}}_i={\text{β}}_j=0.0041$, ${\theta }_i={\theta }_j=0$ (so that, ${\mathcal{R}}_{0s}^i={\mathcal{R}}_{0s}^j=1.86$). All the rate parameters are in years.

Figure 3. Illustration of the results in Theorems 3.4, 3.5 and Conjecture 3.1, showing stability regions of the equilibria of the simplified model in the ${\mathcal{R}}_0^i-{\mathcal{R}}_0^j$ parameter space for various values of the serotype-specific competition parameters, ${\theta }_i$ and ${\theta }_j$. The values of ${\theta }_i$ and ${\theta }_j$ used to compute the thresholds given in Theorems 3.4, 3.5 and Conjecture 3.1 are: (a) ${\theta }_i={\theta }_j=1$, (b) ${\theta }_i=0.1$ and ${\theta }_j=1.0$, (c) ${\theta }_i=1.0$ and ${\theta }_j=0.15$, (d) ${\theta }_i=0.1$ and ${\theta }_j=0.15$, and (e) ${\theta }_i={\theta }_j=0$. In this case, a boundary equilibrium containing serotype-i and serotype-j (without co-colonization) exists and is stable. The rate parameters are in years.

Figure 4. Effect of cohort vaccination coverage (ϕ) on the prevalence of the two SP serotyes and co-colonization. Simulations of the vaccination model (A.1(A.1) $\begin{array}{rl}(N_u{)}^{\prime }& =\mathrm{\Lambda }(1-\varphi )+\omega N_v-({\lambda }_i+{\lambda }_j+\mu )N_u,\\ (N_v{)}^{\prime }& =\mathrm{\Lambda }\varphi -\left[\right(1-ϵ){\lambda }_i+{\lambda }_j+\omega +\mu ]N_v,\\ (C_u^i{)}^{\prime }& ={\lambda }_i{N}_u+{\eta }_i{\lambda }_i{R}_{uni}-({\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_u^i,\\ (C_v^i{)}^{\prime }& =(1-ϵ){\lambda }_i{N}_v+{\eta }_i{\lambda }_i{R}_{vni}-({\theta }_v{\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_v^i,\\ (C_u^j{)}^{\prime }& ={\lambda }_j{N}_u+{\eta }_j{\lambda }_j{R}_{unj}+\omega C_v^j-({\gamma }_j+{\theta }_j{\lambda }_i+\mu )C_u^j,\\ (C_v^j{)}^{\prime }& ={\lambda }_j{N}_v+{\eta }_j{\lambda }_j{R}_{vnj}-[{\gamma }_j+{\theta }_j(1-ϵ){\lambda }_i+\omega +\mu ]C_v^j,\\ (C_u^{ij}{)}^{\prime }& ={\theta }_i{\lambda }_j{C}_u^i+{\theta }_j{\lambda }_i{C}_u^j+{\theta }_j{\eta }_i{\lambda }_i{C}_{uni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{unj}^i-({\gamma }_{ij}+\mu )C_u^{ij},\\ (C_v^{ij}{)}^{\prime }& ={\theta }_j(1-ϵ){\lambda }_i{C}_v^j+{\theta }_i{\lambda }_j{C}_v^i+{\theta }_j{\eta }_i{\lambda }_i{C}_{vni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{vnj}^i\\ & -[\left({\theta }_v\right(1-r_2)+r_2){\gamma }_{ij}+\mu ]C_v^{ij},\\ (R_{uni}{)}^{\prime }& ={\gamma }_i{C}_u^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{uni},\\ (R_{vni}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_v^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{vni},\\ (R_{unj}{)}^{\prime }& ={\gamma }_j{C}_u^j+\omega R_{vnj}-({\eta }_j{\lambda }_j+{\lambda }_i+\mu )R_{unj},\\ (R_{vnj}{)}^{\prime }& ={\gamma }_j{C}_v^j-[{\eta }_j{\lambda }_j+(1-ϵ){\lambda }_i+\omega +\mu ]R_{vnj},\\ (R_{uninj}{)}^{\prime }& ={\gamma }_i{C}_{unj}^i+{\gamma }_j{C}_{uni}^j+(1-r_1-r_2){\gamma }_{ij}{C}_u^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{uninj},\\ (R_{vninj}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_{vnj}^i+{\gamma }_j{C}_{vni}^j+{\theta }_v(1-r_2)(1-r_1){\gamma }_{ij}{C}_v^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{vninj},\\ (C_{unj}^i{)}^{\prime }& ={\lambda }_i{R}_{unj}+{\eta }_i{\lambda }_i{R}_{uninj}+r_2{\gamma }_{ij}{C}_u^{ij}-({\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{unj}^i,\\ (C_{vnj}^i{)}^{\prime }& =(1-ϵ){\lambda }_i{R}_{vnj}+{\eta }_i{\lambda }_i{R}_{vninj}+r_2{\gamma }_{ij}{C}_v^{ij}-({\theta }_v{\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{vnj}^i,\\ (C_{uni}^j{)}^{\prime }& ={\lambda }_j{R}_{uni}+{\eta }_j{\lambda }_j{R}_{uninj}+r_1{\gamma }_{ij}{C}_u^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{uni}^j,\\ (C_{vni}^j{)}^{\prime }& ={\lambda }_j{R}_{vni}+{\eta }_j{\lambda }_j{R}_{vninj}+{\theta }_v(1-r_2)r_1{\gamma }_{ij}{C}_v^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{vni}^j.\end{array}$(A.1) ), showing the prevalence of VT, NVT and co-colonization and overall community-wide vaccination coverage, as a function of time for (a) no cohort vaccination($\varphi =0$), (b) low coverage level of cohort vaccination ($\varphi =0.4$) and (c) high coverage level of cohort vaccination ($\varphi =0.8$). Other parameter values used are as given in Table 2. The constituent reproduction numbers of the vaccination model for each of the settings (a)–(d) above are given by (a) ${\mathcal{R}}_v^i=1.865$, ${\mathcal{R}}_v^j=1.72$ (so that, ${\mathcal{R}}_v=1.865$), (b) ${\mathcal{R}}_v^i=1.799$, ${\mathcal{R}}_v^j=1.72$ (so that, ${\mathcal{R}}_v=1.799$), and (c) ${\mathcal{R}}_v^i=1.73$, ${\mathcal{R}}_v^j=1.72$ (so that, ${\mathcal{R}}_v=1.73$). The rate parameters are in years.

Figure 5. Effect of serotype-specific competition parameters, ${\theta }_i$ and ${\theta }_j$, and cohort vaccination (ϕ) on the prevalence of VT, NVT and co-colonization. Simulations of the vaccination model (A.1(A.1) $\begin{array}{rl}(N_u{)}^{\prime }& =\mathrm{\Lambda }(1-\varphi )+\omega N_v-({\lambda }_i+{\lambda }_j+\mu )N_u,\\ (N_v{)}^{\prime }& =\mathrm{\Lambda }\varphi -\left[\right(1-ϵ){\lambda }_i+{\lambda }_j+\omega +\mu ]N_v,\\ (C_u^i{)}^{\prime }& ={\lambda }_i{N}_u+{\eta }_i{\lambda }_i{R}_{uni}-({\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_u^i,\\ (C_v^i{)}^{\prime }& =(1-ϵ){\lambda }_i{N}_v+{\eta }_i{\lambda }_i{R}_{vni}-({\theta }_v{\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_v^i,\\ (C_u^j{)}^{\prime }& ={\lambda }_j{N}_u+{\eta }_j{\lambda }_j{R}_{unj}+\omega C_v^j-({\gamma }_j+{\theta }_j{\lambda }_i+\mu )C_u^j,\\ (C_v^j{)}^{\prime }& ={\lambda }_j{N}_v+{\eta }_j{\lambda }_j{R}_{vnj}-[{\gamma }_j+{\theta }_j(1-ϵ){\lambda }_i+\omega +\mu ]C_v^j,\\ (C_u^{ij}{)}^{\prime }& ={\theta }_i{\lambda }_j{C}_u^i+{\theta }_j{\lambda }_i{C}_u^j+{\theta }_j{\eta }_i{\lambda }_i{C}_{uni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{unj}^i-({\gamma }_{ij}+\mu )C_u^{ij},\\ (C_v^{ij}{)}^{\prime }& ={\theta }_j(1-ϵ){\lambda }_i{C}_v^j+{\theta }_i{\lambda }_j{C}_v^i+{\theta }_j{\eta }_i{\lambda }_i{C}_{vni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{vnj}^i\\ & -[\left({\theta }_v\right(1-r_2)+r_2){\gamma }_{ij}+\mu ]C_v^{ij},\\ (R_{uni}{)}^{\prime }& ={\gamma }_i{C}_u^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{uni},\\ (R_{vni}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_v^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{vni},\\ (R_{unj}{)}^{\prime }& ={\gamma }_j{C}_u^j+\omega R_{vnj}-({\eta }_j{\lambda }_j+{\lambda }_i+\mu )R_{unj},\\ (R_{vnj}{)}^{\prime }& ={\gamma }_j{C}_v^j-[{\eta }_j{\lambda }_j+(1-ϵ){\lambda }_i+\omega +\mu ]R_{vnj},\\ (R_{uninj}{)}^{\prime }& ={\gamma }_i{C}_{unj}^i+{\gamma }_j{C}_{uni}^j+(1-r_1-r_2){\gamma }_{ij}{C}_u^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{uninj},\\ (R_{vninj}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_{vnj}^i+{\gamma }_j{C}_{vni}^j+{\theta }_v(1-r_2)(1-r_1){\gamma }_{ij}{C}_v^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{vninj},\\ (C_{unj}^i{)}^{\prime }& ={\lambda }_i{R}_{unj}+{\eta }_i{\lambda }_i{R}_{uninj}+r_2{\gamma }_{ij}{C}_u^{ij}-({\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{unj}^i,\\ (C_{vnj}^i{)}^{\prime }& =(1-ϵ){\lambda }_i{R}_{vnj}+{\eta }_i{\lambda }_i{R}_{vninj}+r_2{\gamma }_{ij}{C}_v^{ij}-({\theta }_v{\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{vnj}^i,\\ (C_{uni}^j{)}^{\prime }& ={\lambda }_j{R}_{uni}+{\eta }_j{\lambda }_j{R}_{uninj}+r_1{\gamma }_{ij}{C}_u^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{uni}^j,\\ (C_{vni}^j{)}^{\prime }& ={\lambda }_j{R}_{vni}+{\eta }_j{\lambda }_j{R}_{vninj}+{\theta }_v(1-r_2)r_1{\gamma }_{ij}{C}_v^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{vni}^j.\end{array}$(A.1) ), showing the prevalence of VT, NVT and co-colonization as a function of time for (i) low serotype-specific competition: ${\theta }_i={\theta }_j=0.8$ and (a) no cohort vaccination ($\varphi =0$), (b) low coverage level of cohort vaccination ($\varphi =0.4$), and (c) high coverage level of cohort vaccination ($\varphi =0.8$), (ii) moderate serotype-specific competition: ${\theta }_i={\theta }_j=0.5$ and (d) no cohort vaccination ($\varphi =0$), (e) low coverage level of cohort vaccination ($\varphi =0.4$), and (f) high coverage level of cohort vaccination ($\varphi =0.8$), and (iii) low serotype-specific competition ${\theta }_i={\theta }_j=0.2$ and (g) no cohort vaccination ($\varphi =0$), (h) low coverage level of cohort vaccination ($\varphi =0.4$), (i) high coverage level of cohort vaccination ($\varphi =0.8$). Other parameter values used are as given in Table 2. The reproduction numbers for these simulations are given as follows: (i) no cohort vaccination ($\varphi =0$): ${\mathcal{R}}_v^i=1.865$, ${\mathcal{R}}_v^j=1.72$ (so that, ${\mathcal{R}}_v=1.865$); (ii) low coverage level of cohort vaccination ($\varphi =0.4$): ${\mathcal{R}}_v^i=1.799$, ${\mathcal{R}}_v^j=1.72$ (so that, ${\mathcal{R}}_v=1.799$); (iii) high coverage level of cohort vaccination ($\varphi =0.8$): ${\mathcal{R}}_v^i=1.73$, ${\mathcal{R}}_v^j=1.72$ (so that, ${\mathcal{R}}_v=1.73$). The rate parameters are in years.

Figure A1. Data fitting of the model (A.1(A.1) $\begin{array}{rl}(N_u{)}^{\prime }& =\mathrm{\Lambda }(1-\varphi )+\omega N_v-({\lambda }_i+{\lambda }_j+\mu )N_u,\\ (N_v{)}^{\prime }& =\mathrm{\Lambda }\varphi -\left[\right(1-ϵ){\lambda }_i+{\lambda }_j+\omega +\mu ]N_v,\\ (C_u^i{)}^{\prime }& ={\lambda }_i{N}_u+{\eta }_i{\lambda }_i{R}_{uni}-({\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_u^i,\\ (C_v^i{)}^{\prime }& =(1-ϵ){\lambda }_i{N}_v+{\eta }_i{\lambda }_i{R}_{vni}-({\theta }_v{\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_v^i,\\ (C_u^j{)}^{\prime }& ={\lambda }_j{N}_u+{\eta }_j{\lambda }_j{R}_{unj}+\omega C_v^j-({\gamma }_j+{\theta }_j{\lambda }_i+\mu )C_u^j,\\ (C_v^j{)}^{\prime }& ={\lambda }_j{N}_v+{\eta }_j{\lambda }_j{R}_{vnj}-[{\gamma }_j+{\theta }_j(1-ϵ){\lambda }_i+\omega +\mu ]C_v^j,\\ (C_u^{ij}{)}^{\prime }& ={\theta }_i{\lambda }_j{C}_u^i+{\theta }_j{\lambda }_i{C}_u^j+{\theta }_j{\eta }_i{\lambda }_i{C}_{uni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{unj}^i-({\gamma }_{ij}+\mu )C_u^{ij},\\ (C_v^{ij}{)}^{\prime }& ={\theta }_j(1-ϵ){\lambda }_i{C}_v^j+{\theta }_i{\lambda }_j{C}_v^i+{\theta }_j{\eta }_i{\lambda }_i{C}_{vni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{vnj}^i\\ & -[\left({\theta }_v\right(1-r_2)+r_2){\gamma }_{ij}+\mu ]C_v^{ij},\\ (R_{uni}{)}^{\prime }& ={\gamma }_i{C}_u^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{uni},\\ (R_{vni}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_v^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{vni},\\ (R_{unj}{)}^{\prime }& ={\gamma }_j{C}_u^j+\omega R_{vnj}-({\eta }_j{\lambda }_j+{\lambda }_i+\mu )R_{unj},\\ (R_{vnj}{)}^{\prime }& ={\gamma }_j{C}_v^j-[{\eta }_j{\lambda }_j+(1-ϵ){\lambda }_i+\omega +\mu ]R_{vnj},\\ (R_{uninj}{)}^{\prime }& ={\gamma }_i{C}_{unj}^i+{\gamma }_j{C}_{uni}^j+(1-r_1-r_2){\gamma }_{ij}{C}_u^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{uninj},\\ (R_{vninj}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_{vnj}^i+{\gamma }_j{C}_{vni}^j+{\theta }_v(1-r_2)(1-r_1){\gamma }_{ij}{C}_v^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{vninj},\\ (C_{unj}^i{)}^{\prime }& ={\lambda }_i{R}_{unj}+{\eta }_i{\lambda }_i{R}_{uninj}+r_2{\gamma }_{ij}{C}_u^{ij}-({\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{unj}^i,\\ (C_{vnj}^i{)}^{\prime }& =(1-ϵ){\lambda }_i{R}_{vnj}+{\eta }_i{\lambda }_i{R}_{vninj}+r_2{\gamma }_{ij}{C}_v^{ij}-({\theta }_v{\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{vnj}^i,\\ (C_{uni}^j{)}^{\prime }& ={\lambda }_j{R}_{uni}+{\eta }_j{\lambda }_j{R}_{uninj}+r_1{\gamma }_{ij}{C}_u^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{uni}^j,\\ (C_{vni}^j{)}^{\prime }& ={\lambda }_j{R}_{vni}+{\eta }_j{\lambda }_j{R}_{vninj}+{\theta }_v(1-r_2)r_1{\gamma }_{ij}{C}_v^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{vni}^j.\end{array}$(A.1) ) using the prevalence data for serotypes 19A (VT) and 35B (NVT) generated from a cohort of 6–59 months children in Atlanta, GA, in 6 study periods, 6 months apart, from July 2010 to June 2013 (the end of the first study period is considered as the initial time for the start of the simulations), as reported in [16].

Figure A2. Effect of cohort vaccination coverage (ϕ) on the prevalence of the two SP serotyes and co-colonization. Simulations of the vaccination model (A.1(A.1) $\begin{array}{rl}(N_u{)}^{\prime }& =\mathrm{\Lambda }(1-\varphi )+\omega N_v-({\lambda }_i+{\lambda }_j+\mu )N_u,\\ (N_v{)}^{\prime }& =\mathrm{\Lambda }\varphi -\left[\right(1-ϵ){\lambda }_i+{\lambda }_j+\omega +\mu ]N_v,\\ (C_u^i{)}^{\prime }& ={\lambda }_i{N}_u+{\eta }_i{\lambda }_i{R}_{uni}-({\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_u^i,\\ (C_v^i{)}^{\prime }& =(1-ϵ){\lambda }_i{N}_v+{\eta }_i{\lambda }_i{R}_{vni}-({\theta }_v{\gamma }_i+{\theta }_i{\lambda }_j+\mu )C_v^i,\\ (C_u^j{)}^{\prime }& ={\lambda }_j{N}_u+{\eta }_j{\lambda }_j{R}_{unj}+\omega C_v^j-({\gamma }_j+{\theta }_j{\lambda }_i+\mu )C_u^j,\\ (C_v^j{)}^{\prime }& ={\lambda }_j{N}_v+{\eta }_j{\lambda }_j{R}_{vnj}-[{\gamma }_j+{\theta }_j(1-ϵ){\lambda }_i+\omega +\mu ]C_v^j,\\ (C_u^{ij}{)}^{\prime }& ={\theta }_i{\lambda }_j{C}_u^i+{\theta }_j{\lambda }_i{C}_u^j+{\theta }_j{\eta }_i{\lambda }_i{C}_{uni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{unj}^i-({\gamma }_{ij}+\mu )C_u^{ij},\\ (C_v^{ij}{)}^{\prime }& ={\theta }_j(1-ϵ){\lambda }_i{C}_v^j+{\theta }_i{\lambda }_j{C}_v^i+{\theta }_j{\eta }_i{\lambda }_i{C}_{vni}^j+{\theta }_i{\eta }_j{\lambda }_j{C}_{vnj}^i\\ & -[\left({\theta }_v\right(1-r_2)+r_2){\gamma }_{ij}+\mu ]C_v^{ij},\\ (R_{uni}{)}^{\prime }& ={\gamma }_i{C}_u^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{uni},\\ (R_{vni}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_v^i-({\eta }_i{\lambda }_i+{\lambda }_j+\mu )R_{vni},\\ (R_{unj}{)}^{\prime }& ={\gamma }_j{C}_u^j+\omega R_{vnj}-({\eta }_j{\lambda }_j+{\lambda }_i+\mu )R_{unj},\\ (R_{vnj}{)}^{\prime }& ={\gamma }_j{C}_v^j-[{\eta }_j{\lambda }_j+(1-ϵ){\lambda }_i+\omega +\mu ]R_{vnj},\\ (R_{uninj}{)}^{\prime }& ={\gamma }_i{C}_{unj}^i+{\gamma }_j{C}_{uni}^j+(1-r_1-r_2){\gamma }_{ij}{C}_u^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{uninj},\\ (R_{vninj}{)}^{\prime }& ={\theta }_v{\gamma }_i{C}_{vnj}^i+{\gamma }_j{C}_{vni}^j+{\theta }_v(1-r_2)(1-r_1){\gamma }_{ij}{C}_v^{ij}-({\eta }_i{\lambda }_i+{\eta }_j{\lambda }_j+\mu )R_{vninj},\\ (C_{unj}^i{)}^{\prime }& ={\lambda }_i{R}_{unj}+{\eta }_i{\lambda }_i{R}_{uninj}+r_2{\gamma }_{ij}{C}_u^{ij}-({\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{unj}^i,\\ (C_{vnj}^i{)}^{\prime }& =(1-ϵ){\lambda }_i{R}_{vnj}+{\eta }_i{\lambda }_i{R}_{vninj}+r_2{\gamma }_{ij}{C}_v^{ij}-({\theta }_v{\gamma }_i+{\theta }_i{\eta }_j{\lambda }_j+\mu )C_{vnj}^i,\\ (C_{uni}^j{)}^{\prime }& ={\lambda }_j{R}_{uni}+{\eta }_j{\lambda }_j{R}_{uninj}+r_1{\gamma }_{ij}{C}_u^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{uni}^j,\\ (C_{vni}^j{)}^{\prime }& ={\lambda }_j{R}_{vni}+{\eta }_j{\lambda }_j{R}_{vninj}+{\theta }_v(1-r_2)r_1{\gamma }_{ij}{C}_v^{ij}-({\gamma }_j+{\theta }_j{\eta }_i{\lambda }_i+\mu )C_{vni}^j.\end{array}$(A.1) ), showing the prevalence of 19A, 35B and co-colonization and overall cohort-wide vaccination coverage, as a function of time for (a) no cohort vaccination($\varphi =0$), (b) low coverage level of cohort vaccination ($\varphi =0.4$), (c) high coverage level of cohort vaccination ($\varphi =0.8$). Other parameter values used are as given in Table 2. The initial condition for no ($\varphi =0$), low ($\varphi =0.4$) and high ($\varphi =0.8$) coverage level of cohort vaccination are listed in items (i), (ii) and (iii), respectively, in Section D. The constituent reproduction numbers of the vaccination model for each of the settings (a)–(c) above are given by (a) ${\mathcal{R}}_v^i=3.214$, ${\mathcal{R}}_v^j=1.1$ (so that, ${\mathcal{R}}_v=3.214$), (b) ${\mathcal{R}}_v^i=2.222$, ${\mathcal{R}}_v^j=1.1$ (so that, ${\mathcal{R}}_v=2.222$) and (c) ${\mathcal{R}}_v^i=1.23$, ${\mathcal{R}}_v^j=1.1$ (so that, ${\mathcal{R}}_v=1.23$). The rate parameters are in years.

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