Supplementary material Expanding vaccine efficacy estimation with dynamic models fitted to cross-sectional prevalence data post-licensure

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1 Supplementary material Expanding vaccine efficacy estimation ith dynamic models fitted to cross-sectional prevalence data post-licensure Erida Gjini a, M. Gabriela M. Gomes b,c,d a Instituto Gulbenkian de Ciência, Apartado 4, Oeiras, Portugal. b CIBIO-InBIO, Centro de Investigação em Biodiversidade e Recursos Genéticos, Universidade de Porto, Portugal. c Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil. d Liverpool School of Tropical Medicine, Liverpool, United Kingdom Text S. Coexistence endemic equilibria in the competition models prior to vaccination SI-2 model The equilibria for this model are calculated by setting all equations ( 2.2 in the main text equal to zero, ith ρ =. Besides the type only and type 2 only endemic equilibria, the SI-2 model admits a coexistence equilibrium given by: S = R I ( 2 = R σ (R σ σ 2 + σ + σ 2 I2 ( 2 = R σ 2 (R σ σ 2 + σ + σ 2 I2 = (S + I + I 2 ( here R = β/µ in the SI-2 frameork. Stability of this pre-vaccine equilibrium requires: σ > σ 2 (R σ 2 + ; σ σ 2 > (R σ +. For an illustration of these criteria, depending on σ and σ 2 see Figure S. SIS-2 model We find the equilibria pre-vaccine by setting equations ( 2.3 in the main text equal to zero (assuming ρ =, and then simplify the expressions using R = β/(γ + µ. The endemic coexistence equilibrium is given by: S = R I R = I 2 = R [2 + (κσ + σ(r ] I = I22 = κσ(r 2 2R [2 + (κσ + σ(r ] I2 = (S + I + I 2 + I + I 22, (2 here the indices specify carriage: hether single or double carrier of pathogen types of group, 2 or both. Asymptotic analysis of the system reveals that stability of this equilibrium requires κ <. Preprint submitted to Epidemics November 2, 25

2 Supplementary figures.. 2 only Coexistence 2 only Coexistence Σ2 Σ2.2 only...2. Σ (a Lo transmission intensity.2 only...2. Σ (b Higher transmission intensity Figure S: Stability criteria on the competition coefficients σ and σ 2 in the SI-2 model that guarantee stable coexistence beteen the to pathogen types at the endemic equilibrium. The regions are computed by checking the eigenvalues of the Jacobian matrix evaluated at the endemic equilibria. In our vaccination models, e consider parameter regimes of stable coexistence prior to vaccine introduction (hite region, here the vaccine targets type. Parameter values: a β =.5, µ =.2 (R = 2.5; b β =., µ =.2 (R = 5. 2

3 Prevalence odds ratio of type, POR(t.2 Type 2 dominant (=σ >σ 2 β = Type dominant (σ <σ 2 = Time post vaccination (years Prevalence odds ratio of type, POR(t.2 Type 2 dominant (=σ >σ 2 β =.32 Type dominant (σ <σ 2 = Time post vaccination (years Prevalence odds ratio of type, POR(t Type 2 dominant (=σ >σ 2 β = Type dominant (σ <σ 2 = Time post vaccination (years Prevalence odds ratio of type, POR(t.2 β =.32 Type dominant (σ <σ 2 = Type 2 dominant (=σ >σ Time post vaccination (years Figure S2: Prevalence-odds-ratio (POR of target type in vaccinated and non-vaccinated hosts vs. true relative risk in the SI-2 model. Type can be dominant prior to vaccination (.2 σ /σ 2, or alternatively, type 2 can be dominant (.2 σ 2 /σ. The coloured lines from blue to red correspond to increasing values of the competition ratio, σ /σ 2 and σ 2 /σ from.2 to. Other parameter values: µ =.67, ρ =.5. Initial conditions at endemic equilibrium. Time is in units of years. The lo transmission cases (β =.32, correspond to R =.9. While the high transmission cases (β =.32 correspond to R = 9. POR(t is closer to than PR(t. 3

4 β = Prevalence ratio type, PR(t σ = Time (months Figure S3: Prevalence ratio can under-estimate relative risk for high β (SIS-2. Parameter values: σ =, γ =, µ =.2, ρ =.5 and β =, κ is varied beteen. and.9 (colored lines from blue to red to reflect different scenarios of ithin-group competition. The true risk ratio is =, hich implies VE =.2. The orst discrepancy beteen PR and is observed for κ close to, hich leads to a negative deviation, indicating an over-estimation of VE if PR ere to be used for this purpose. 4

5 β =.32 β =.32 PR(t, type PR(t, type Time post vaccination (years 2 4 Time post vaccination (years β =.32 β =.32 PR(t, type.2 PR(t, type Time post vaccination (years 2 4 Time post vaccination (years Figure S4: Prevalence ratio in the SI-2 model (counting for mixed carriage I 2. Type can be dominant prior to vaccination (.2 σ /σ 2, or alternatively, type 2 can be dominant (.2 σ 2 /σ. The coloured lines from blue to red correspond to increasing values of the competition ratio, σ /σ 2 and σ 2 /σ from.2 to. Parameters as in Figure 3 in the main text. The sensitivity to competition asymmetries decreases but the discrepancy ith increases overall. 5

6 β =2 β =6 PR(t, group σ = PR(t, group σ = PR(t, group Time (months β =2 σ =.5 PR(t, group Time (months β =6 σ = Time (months Time (months Figure S5: Prevalence ratio in the SIS-2 model (counting for mixed carriage I 2. Analogous to figure 4 in main text. We can see the sensitivity of PR(t to competition hierarchies (ithin/beteen group, represented by κ, reflected in the blue to red lines, increases, especially for large transmission intensity. This indicates that mixed multiple carriage holds important information about indirect vaccine effects, but may be unsuitable to include in the analyses aimed at extracting true relative risk. 6

7 Text S2. The SIR-2 model Model structure and assumptions In this model, e change the SI-2 model, assuming hosts recover ith life-long type-specific immunity, at rate γ. They enter these recovery classes: R, refers to those that are immune against type but can acquire type 2; R 2, refers to those that are immune against type 2 but can acquire type, and R 2, describes hosts immune to both circulating types. Singly infected hosts ith type, and immune to type 2 are denoted by I (2, and viceversa. Hosts infected ith type 2, but immune against type, are denoted by I 2(. Both these types of hosts contribute no to the forces of infection λ and λ 2. Vaccination (status denoted by subscript / acts as before, against type, ith partial protection, given by the factor (. After hosts experience multiple infection ith both types, or sequential infection by to types, they recover ith full immunity to both types R 2. Clearance of single and double carriage is assumed to occur at equal rates, as in the earlier models. We assume no cross-immunity. here Non-vaccinated hosts ds = µ( ρ (λ + λ 2 S µs di = λ S I σ λ 2 (µ + γi di2 = λ 2 S I 2 σ 2λ (µ + γi 2 di2 = σ λ 2 I + σ 2λ I 2 (µ + γi 2 dr = γi λ 2R µr dr 2 = γi 2 λ R 2 µr 2 dr 2 = γ(i 2 + I 2( + I (2 µr 2 di2( = λ 2 R (µ + γi 2( di(2 = λ R 2 (µ + γi (2 Vaccinated hosts ds = µρ (λ + λ 2 S µs di = λ S I σ λ 2 (µ + γi di2 = λ 2 S I2 σ 2λ (µ + γi2 di2 = σ λ 2 I + σ 2λ I2 (µ + γi 2 dr = γi λ 2R µr dr 2 = γi2 λ R 2 µr 2 dr 2 = γ(i2 + I 2( + I (2 µr 2 di2( = λ 2 R (µ + γi 2( di(2 = λ R 2 (µ + γi (2 λ = β(i + I 2 /2 + I (2 + I + I 2 /2 + I (2 λ 2 = β(i 2 + I 2 /2 + I 2( + I 2 + I 2 /2 + I 2(. 7

8 Inference under the SIR-2 model We generate hypothetical trajectories using the SIR-2 equations, for given coverage ρ, demographic parameter µ and clearance rate γ (e.g. Figure S3 a. The prevalence ratio of type pathogen in vaccinated and non-vaccinated hosts still shos significant deviation from the true vaccine protection parameter (Figure S3 b. For model-based inference, e vary the other four parameters: β representing the transmission rate,, describing vaccine protection against type here vaccine efficacy is VE =, and the direct competition parameters σ and σ 2 acting at co-infection. By sampling the pathogen type prevalences at different time points post-vaccination, and fitting the dynamic model to these data, e can recover back the original parameters. We group observations into pathogen-free hosts: S + R + R 2 + R 2, hosts carrying type : I + I (2, hosts carrying type 2: I 2 + I 2(, and hosts carrying both types I 2. We do this grouping both for vaccinated and non-vaccinated hosts. Given that all host proportions sum to, this results in 6 data points at each snapshot considered. If observations are perfect, the backard inference orks very ell: all four parameters can be estimated accurately from model trajectories, even hen the prevalence proportions are extremely lo (data not shon. As e realistically allo for sampling error, e superimpose a multinomial sampling process on the deterministic model proportions, and then apply nonlinear least squares optimization to the resulting synthetic sample proportions. In this case, larger sample sizes are needed to identify σ and σ 2, because the expected pathogen prevalence (and consequently of constituent subtypes is rather lo in a SIR-2 frameork. As γ decreases, the prevalence of infection increases (and there are less immune individuals, augmenting our statistical poer for the inference of competition parameters, for a given sample size (Figure S4 c-d. The inference of β and = VE is hoever accurate, even in the presence of observation error. Notice that here e have assumed that the serological status of sampled individuals is not knon, hich may not alays be the case ith ever advancing molecular technologies. Thus model fitting as performed only on aggregated carriage status variables. It is likely that if the serological status (naive/immune to /immune to 2/ immune to both is knon, e ill have much more poer to extract competition parameters σ and σ 2 by fitting all the ODE s to the complete data, even hen γ is large. 8

9 Prevalence All population Type 2 Type Prevalence Prevalence.2 5 Time (years Vaccinated hosts Type 2 Type.2 5 Time (years Non vaccinated hosts (a Dynamics in the years post-vaccination Type Type Time (years.9 Prevalence ratio type, PR(t Time (years (b Prevalence ratio and direct competition Figure S6: SIR-2 model frameork. a The lines sho dynamics post-vaccination for β = 2, γ =.2, µ =.67, and different competition coefficients σ (<=> σ 2 =.25. The endemic pre-vaccine equilibrium is assumed as initial conditions. Vaccination is implemented at time t = ith coverage ρ =.5 and vaccine efficacy VE =.5. Average life-expectancy of any individual in the population is /µ, assumed 6 years. The panels illustrate oscillatory prevalences of type and type 2 in the entire population (left, and in the vaccinated and non-vaccinated hosts (right. Such oscillations are typical of an SIR frameork, making the sensitivity of dynamic prevalences to σ and σ 2 only moderate. b Prevalence ratio in the years post-vaccination, does not reflect accurately the true relative risk in the SIR-2 model, and there is minimal sensitivity to direct competition parameters. 9

10 .2 Bias Parameters (β,, σ, σ 2 (a Bias (γ =.2 (b Range (γ =.2.2 Bias Parameters (β,, σ, σ 2 (c Bias (γ =. (d Range(γ =. Figure S7: Dynamic model fitting in the SIR-2 frameork tested for random parameter combinations. a-b Bias and parameter range in the high clearance rate scenario, thus smaller prevalence at endemic equilibrium pre-vaccination. c-d Bias and parameter range in the smaller clearance rate case, thus loer prevalence of carriage at endemic equilibrium pre-vaccination. The parameters are ordered as β,, σ, σ 2. We simulated the model for 5 different parameter combinations. Fixed parameter values: µ =.67, ρ =.5, sample size N = 5. Initial conditions are alays fixed at pre-vaccine endemic equilibrium (solved numerically for each parameter combination. The time points (years in this model post-vaccination used are t i = 5,, 2, 3, here the population is sampled multinomially for infection status (uninfected, infected ith type, ith type, ith both. The performance of the method to estimate the competition coefficients σ and σ 2 in the SIR-2 model depends on the prevalence that can be observed in a population. Loer rate of recovery γ leads to higher prevalence of infection, thus enables better estimation of σ, σ 2, for same N, due to less sampling error. The inference of β and vaccine efficacy remains remarkably stable.

11 .9 Target types prevalence at t=36 A.35.3 Within/beteen group competition, κ B B.2 A Vaccine efficacy, VE Figure S8: Parameter correlation in the SIS-2 model. Parameters κ and VE may appear correlated at one scale, namely, if only global prevalence of target types at a given time post-vaccination is available: Prev (type = I + I + I 2 /2 (contour plot on the left. Hoever, hen one zooms further into each scenario (marked hite dots, by using the information contained in the finer-scale epidemiological variables: S, I, I 2, I, I 22, I 2 across vaccinated (listed -6 and non-vaccinated individuals (listed 7-2, differing beteen A and B (right sub-panels, the parameters κ and VE should be suitably detangled. Fixed parameter values β = 6, µ =.2, γ =, ρ =.5, σ =. Initial conditions at pre-vaccine endemic equilibrium.