The stochasticity in adherence to nonpharmaceutical interventions and booster doses and the mitigation of COVID-19

Yi Tan; Pei Yuan; Iain Moyles; Jane Heffernan; James Watmough; Sanyi Tang; Huaiping Zhu

doi:10.3934/dcdss.2023044

Article Contents

2023, Volume 16, Issue 3&4: 602-626. Doi: 10.3934/dcdss.2023044

This issue Previous Article Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential Next Article On solution manifolds of differential systems with discrete state-dependent delays

The stochasticity in adherence to nonpharmaceutical interventions and booster doses and the mitigation of COVID-19

1.
Laboratory of Mathematical Parallel Systems (LAMPS), Department of Mathematics and Statistics, York University, Toronto, Canada
2.
Canadian Centre for Diseases Modeling (CCDM), York University, Toronto, Canada
3.
Department of Mathematics and Statistics, York University, Toronto, Canada
4.
Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, Canada
5.
School of Mathematics and Statistics, Shaanxi Normal University, Xi'an, China

^*Corresponding author: Huaiping Zhu
^*Corresponding author: Huaiping Zhu

Dedicate to Professor Jibin Li on the occasion of his 80th birthday.
†These authors contributed equally to this work.

Received: November 2022

Revised: February 2023

Early access: March 2023

Published: March 2023

The first author is supported by [Chinese Scholarship Council]

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Abstract

Facing the more contagious COVID-19 variant, Omicron, nonpharmaceutical interventions (NPIs) were still in place and booster doses were proposed to mitigate the epidemic. However, the uncertainty and stochasticity in individuals' behaviours toward the NPIs and booster dose increase, and how this randomness affects the transmission remains poorly understood. We present a model framework to incorporate demographic stochasticity and two kinds of environmental stochasticity (notably variations in adherence to NPIs and booster dose acceptance) to analyze the effects of different forms of stochasticity on transmission. The model is calibrated using the data from December 31, 2021, to March 8, 2022, on daily reported cases and hospitalizations, cumulative cases, deaths and vaccinations for booster doses in Toronto, Canada. An approximate Bayesian computational (ABC) method is used for calibration. We observe that demographic stochasticity could dramatically worsen the outbreak with more incidence compared with the results of the corresponding deterministic model. We found that large variations in adherence to NPIs increase infections. The randomness in booster dose acceptance will not affect the number of reported cases significantly and it is acceptable in the mitigation of COVID-19. The stochasticity in adherence to NPIs needs more attention compared to booster dose hesitancy.

Keywords:

COVID-19,
stochastic differential equation models,
nonpharmaceutical interventions,
vaccines,
booster doses,
approximate Bayesian computational method.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

Full Text(HTML)

Figure 1. Flow diagram of COVID-19 transmission dynamics with third dose vaccination processes. All individuals are in different states, $ S $ (Susceptible), $ E $ (Latently infected), $ I_1 $ (Unreported infected), $ I_2 $ (Reported mild infected), $ H $ (Hospitalized), $ D $ (Deceased), $ R $ (Recovered), $ V $ (Vaccinated with booster doses)

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Figure 2. Fitting results of daily infections and hospitalizations, cumulative cases, deaths and vaccinations for booster doses in Toronto from December 31, 2021, to March 8, 2022. Circles represent observed data. Lines represent calibration results which are the average of 1,000 runs. Shaded areas with color represent the 95% confidence interval of predictions

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Figure S1. MCMC simulation. Chains of 1,000,000 samples are drawn with first 20,000 burn-in samples in every chain. We can see that the chains are well-mixed, implying the convergence of the sampling

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Figure S2. MCMC simulation. Posterior distributions of the parameters

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Figure 3. Effects of stochasticity in NPI adherence on the outbreak. Average results of reported cases and total cases under different volatility of the transmission rate with 1,000 simulations are shown, given two different means of transmission rate (0.17 and 0.27)

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Figure 4. Comparisons of the effect of demographic stochasticity and environmental stochasticity in adherence to NPIs on the outbreak under a reopening or closing strategy. Average results of daily reported cases under stepwise volatility of transmission rate are shown when lifting control measures with time (a) or intensifying control measures with time (b) with 1,000 simulations. The inner plots show the cumulative reported cases. Results for ODE model, stochastic model with demographic stochasticity, and stochastic model with demographic and environmental stochasticity are shown. For the model with demographic and environmental stochasticity, we assume volatility of transmission rate is time-dependent, and volatility of vaccination rate takes the estimated value. Used values of mean and volatility of transmission rate in figures see Table 5. The volatility in the figure legend refers to volatility of transmission rate, and DS represents demographic stochasticity

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Figure 5. Effects of stochasticity in vaccine acceptance on the outbreak. Average results of vaccinations and total cases under different volatility (a) and the speed of variation (b) of vaccination rate with 1,000 simulations are shown

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Figure 6. Comparisons of the effect of demographic stochasticity and environmental stochasticity in vaccines on the outbreak under a reopening strategy. Average results of daily reported cases under stepwise volatility of vaccination rate when lifting control measures with 1,000 simulations are shown. The inner plot shows cumulative reported cases. Results for ODE model, stochastic model with demographic stochasticity, and stochastic model with demographic and environmental stochasticity are shown. For the model with demographic and environmental stochasticity, we assume volatility of vaccine rate is time-dependent, and volatility of transmission rate takes the estimated value. Used values of mean and volatility of vaccination rate in the figure see Table 5. The volatility in the figure legend refers to volatility of vaccination rate, and DS represents demographic stochasticity

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Figure 7. Comparisons of the effect of variations in vaccination rate and transmission rate on the outbreak. Average results of cumulative vaccinations (a) and total cases (b) are shown under different mean and volatility of transmission rate, as well as the speed of variation and volatility of vaccination rate with 1,000 simulations

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Table 1. Model assumptions

Notation	Description
Testing/Reporting	A proportion of symptomatic infections get a PCR test and confirmed, then included in the report.
Severe Outcomes	Only severe infected individuals will be admitted to hospitals; Only hospitalized individuals die in ICUs will count as death from the infection.
Vaccination	Due to low efficacy of the first two doses and waning immunity, the vaccine compartment reflects the individuals protected by dose 3. We only include the booster dose efficacy against infections.

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Table 2. Model variables and fixed parameters

Variables and their initial values
Descriptions		Values
Notation	Description	Initial Value
$ S(t) $	Number of susceptible individuals at day $ t $	Estimated
$ E(t) $	Number of exposed individuals at day $ t $	Estimated
$ I_1(t) $	Number of unreported individuals who are asymptomatic or symptomatic mild infectious at day $ t $	Estimated
$ I_2(t) $	Number of reported individuals who are symptomatic mild infectious at day $ t $	Estimated
$ H(t) $	Number of severe patients that need hospital care	230
$ D(t) $	Number of deceased individuals at day $ t $	3,730
$ R(t) $	Number of recovered individuals at day $ t $	222,279
$ V(t) $	Number of effectively protected individuals from booster doses at day $ t $	552,800
$ log(\lambda(t)) $	Logarithm of the daily vaccination rate for booster dose at day $ t $	Estimated
Fixed parameters for Omicron transmission in Toronto
Stages		Values in stages
$ N $	Population size in Toronto	2,974,293 [7]
$ 1/\alpha $	Length of days between exposure and case report	5 [41,48]
$ \tau $	Latent period	1.5 [48,22]
$ \rho $	Proportion of infected people who will develop symptoms	0.632 [23]
$ \gamma_1 $	Recovery rate of unreported infectious individuals	1/8.5 [51,41,48,22]
$ \gamma_2 $	Recovery rate of reported infectious individuals	1/5 [51]
$ 1/ \gamma_h $	Length of days for mildly infected individuals developing severe symptoms and being admitted into hospital	8 [41,49]
$ \gamma_{hr} $	Recovery rate of hospitalized individuals	1/5.5 [11]
$ 1/\gamma_{d} $	Length of stay for patients in hospital before died	8 [15]
$ \epsilon $	Efficacy booster dose	70% [36]

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Table 3. Estimated initial values and parameters

Notation	Description	Prior distributions	Estimated mean value	95% CI
$ E(t_0) $	Number of exposed individuals at day $ t_0 $	$ \mathbb{N}(18000, 0.5) $	19545	$ [8133, 44304] $
$ I_1(t_0) $	Number of unreported individuals who are asymptomatic or symptomatic mild infectious at day $ t_0 $	$ \mathbb{N}(42233, 0.5) $	47043	$ [26351, 50883] $
$ I_2(t_0) $	Number of reported individuals who are symptomatic mild infectious at day $ t_0 $	$ \mathbb{N}(18670, 0.5) $	17160	$ [8368, 20081] $
$ log(\lambda(t_0) $	Logarithm of daily vaccination rate for booster dose at day $ t_0 $	$ \mathbb{N}(log(0.0093), 0.5) $	$ log(0.0097) $	[$ log(0.0047), log(0.0214) $]
$ p_2 $	Proportion of infected people who will conduct PCR test	$ \mathbb{N}(0.8, 0.5) $	0.7803	$ [0.3475, 1.2846] $
$ p_h $	Proportion of mildly infected individuals develop severe symptoms and need hospital care in stages	Stage1: $ \mathbb{N}(0.032, 0.5) $; Stage2: $ \mathbb{N}(0.097, 0.5) $	Stage1: 0.0399; Stage2: 0.1059	Stage1: [0.0254, 0.0672]; Stage2: [0.0605, 0.2114]
$ p_d $	Proportion of hospitalized patients died $ t_0 $	$ \mathbb{N}(0.159, 0.5) $	0.1555	$ [0.0860, 0.2591] $
$ \bar{\beta} $	Mean transmission rate	$ \mathbb{N}(0.17, 0.5) $	0.1654	$ [0.0915, 0.3662] $
$ \sigma_{\beta} $	Volatility of transmission rate	$ \mathbb{N}(0.2, 0.5) $	0.2	$ [0.0936, 0.3948] $
$ log(\bar{\lambda}) $	Logarithm of mean vaccination rate	$ \mathbb{N}(log(0.00056), 0.5) $	$ log(0.0005) $	[$ log(0.00024), log(0.001) $]
$ r_{\lambda} $	Speed of variation of the vaccination rate for third doses	$ \mathbb{N}(0.01, 0.5) $	0.0081	$ [0.0040, 0.0271] $
$ \sigma_{\lambda} $	Volatility of vaccination rate for booster doses	$ \mathbb{N}(0.153, 0.5) $	0.1517	$ [0.0916, 0.1828] $

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Table 4. Summary of possible changes on the process for the epidemic model when $ \Delta t $ is small

Possible change	Probability
$ [-1, 0, 0, 0, 0, 0, 0, +1] $	$ \lambda \epsilon S \Delta t = P_{c1} \Delta t $
$ [-1, +1, 0, 0, 0, 0, 0, 0] $	$ \beta S I_1 \Delta t /N = P_{c2} \Delta t $
$ [0, -1, +1, 0, 0, 0, 0, 0] $	$ \frac{1-\rho} {\tau} E\Delta t + \frac{\rho (1-p_2)}{ \tau} E \Delta t = P_{c3} \Delta t $
$ [0, 0, -1, 0, 0, 0, +1, 0] $	$ \gamma_1 I_1 \Delta t = P_{c4} \Delta t $
$ [0, -1, 0, +1, 0, 0, 0, 0] $	$ \rho p_2 \alpha E \Delta t = P_{c5} \Delta t $
$ [0, 0, 0, -1, 0, 0, 0, 0] $	$ (1-p_h) \gamma_2 I_2 \Delta t = P_{c6} \Delta t $
$ [0, 0, 0, -1, +1, 0, 0, 0] $	$ p_h \gamma_h I_2 \Delta t = P_{c7}\Delta t $
$ [0, 0, 0, 0, -1, 0, +1, 0] $	$ (1-p_d) \gamma_{hr} H \Delta t = P_{c8} \Delta t $
$ [0, 0, 0, 0, -1, +1, 0, 0] $	$ p_d \gamma_d H \Delta t = P_{c9} \Delta t $

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Table 5. List of scenarios related to partial/total reopening, and values of the mean transmission rate, the volatility of transmission rate, and the volatility of vaccination rate used to project cases in the investigation

Mean transmission rate
Descriptions		Values
Baseline	Limit capacity of social gatherings and indoor public settings & events venues	Estimated mean transmission rate using the data between December 31, 2021, and March 8, 2022
Partial opening	Lift mask mandates	Increase the baseline value by 127%$ ^{\circledast} $
Total reopening	Remove all control measures, directives and orders	Increase the partial opening value by 70%
Case 1: lift control measures over time
Stages		Mean transmission rate
Stage 1: Jan 1, 2022 - Mar 20, 2022[45]		Baseline value
Stage 2: Mar 21, 2022 - Apr 26, 2022[46]		Partial opening value
Stage 3: Apr 27, 2022 – Dec 31, 2022		Total reopening value
Case 2: strengthen control measures over time
Stages		Mean transmission rate
Stage 1: Jan 1, 2022 - Jan 10, 2022		Total reopening value
Stage 2: Jan 11, 2022 - Jan 20, 2022		Partial opening value
Stage 3: Jan 21, 2022 - Mar 29, 2022		Baseline value
Volatility of transmission rate
Stages		Values in stages
Gradually increased		$ 1.3 \quad \quad 1.6 \quad 1.9$
Gradually decreased		$1.3 \quad \quad 0.9 \quad 0.5$
Fixed		$1.3 \quad\quad 1.3 \quad 1.3$
Zero		$\;0 \quad\; \quad 0 \quad\quad 0$
Volatility of vaccination rate
Descriptions		Values in stages
Gradually increased		$0.1 \quad \quad 0.3 \quad 0.5$
Gradually decreased		$0.5 \quad \quad 0.3 \quad 0.1$
Fixed		$0.5 \quad\quad 0.5 \quad 0.5$
Zero		$0 \quad\; \quad 0 \quad\quad 0$
$ ^{\circledast}$ Assumed a 70% reduced risk of acquiring infection for persons wearing masks [47], and 80% of populations adhere to mask mandates. Then transmission rate for partial opening can be calculated by the formula: baseline /(1-0.7^*0.8) [35].

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