# The Effects of Imitation Dynamics on Vaccination Behaviours in SIR-Network Model

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Technical Background

#### 2.1.1. Basic Reproduction Number ${R}_{0}$

#### 2.1.2. Vaccination Model with Imitation Dynamics

#### 2.1.3. Multi-City Epidemic Model

#### 2.2. Methods

#### 2.2.1. Integrated Model

- Individual’s responsiveness to changes in disease prevalence, $\omega $
- Adjusted imitation rate, $\kappa $
- Vaccination failure rate, $\zeta $

#### 2.2.2. Vaccination Available to Newborns Only

**Proposition**

**1.**

**Proposition**

**2.**

#### 2.2.3. Vaccination Available to the Entire Susceptible Class

#### 2.2.4. Vaccination Available to the Entire Susceptible Class with Committed Vaccine Recipients

**Proposition**

**3.**

#### 2.2.5. Model Parameterisation

- a pilot case of a network with 3 nodes (suburbs),
- an Erdös-Rényi random network [38] with 3000 nodes (suburbs), and

## 3. Results

#### 3.1. 3-Node Network

#### 3.1.1. Vaccinating Newborns Only

#### 3.1.2. Vaccinating the Entire Susceptible Class

#### 3.2. Erdös-Rényi Random Network of 3000 Nodes

#### 3.2.1. Vaccinating Newborns Only

#### 3.2.2. Vaccinating Entire Susceptible Class

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Next Generation Operator Approach

## Appendix B. Propositions and Proofs

**Proposition**

**A1.**

**Proof.**

**Proposition**

**A2.**

**Proof.**

**Proposition**

**A3.**

**Proof.**

## Appendix C. Preliminary Analysis of the Oscillatory Behaviour of Epidemic and Vaccination Dynamics When Vaccinating Newborns

**Figure A1.**The function $f(x)=a{e}^{bx}+c{e}^{dx}$ is fitted for the period of epidemic and vaccination dynamics for three values of $\omega $. (

**a**) Period of the disease prevalence, ${T}_{I}$. (

**b**) Period of the relative proportion of vaccinated individuals, ${T}_{x}$. Circle: original data point. Solid line: fitted curve. Results of goodness-of-fit test are summarised in Table A1. Coefficients of fitting functions are summarised in Table A2.

**Table A1.**Goodness-of-fit test of curves shown in Figure A1. Results rounded to 4 significant figures. SSE: sum of squared errors of prediction. RMSE: Root Mean Square Error.

${\mathit{T}}_{\mathit{I}}$ | SSE | R-Square | Adjusted R-Square | RMSE |

$\omega =3500$ | 0.03564 | 0.9980 | 0.9977 | 0.04450 |

$\omega =2500$ | 0.002712 | 0.9998 | 0.9997 | 0.01260 |

$\omega =1000$ | 0.0008870 | 0.9980 | 0.9997 | 0.009418 |

${\mathit{T}}_{\mathit{x}}$ | SSE | R-Square | Adjusted R-Square | RMSE |

$\omega =3500$ | 0.03076 | 0.9982 | 0.9979 | 0.04385 |

$\omega =2500$ | 0.001248 | 0.9999 | 0.9998 | 0.01020 |

**Table A2.**Coefficients of two-term exponential fitted functions. Results are with 95% confidence bounds, shown in bracket.

${\mathit{T}}_{\mathit{I}}$ | a | b | c | d |

$\omega =3500$ | 4.593 (4.406,4.78) | −0.3882 (−0.4247,−0.3516) | 2.983 (2.832,3.134) | −0.008774 (−0.01176,−0.005788) |

$\omega =2500$ | 3.951 (3.9,4.003) | −0.3616 (−0.3732,−0.35) | 2.399 (2.353,2.446) | −0.00239 (−0.003547,−0.001234) |

$\omega =1000$ | 3.468 (3.398,3.538) | −0.6031 (−0.628,−0.5782) | 2.237 (2.195,2.28) | −0.006735 (−0.008434,−0.005035) |

${\mathit{T}}_{\mathit{x}}$ | a | b | c | d |

$\omega =3500$ | 4.535 (4.341,4.73) | −0.3971 (−0.4385,−0.3557) | 3.066 (2.889,3.244) | −0.01052 (−0.0142,−0.006853) |

$\omega =2500$ | 3.882 (3.827,3.937) | −0.3717 (−0.3857,−0.3577) | 2.492 (2.422,2.561) | −0.004952 (−0.006928,−0.002976) |

**Figure A2.**Oscillation properties of epidemic and vaccination dynamics for three values of $\omega $ when vaccinating newborns at low $\beta $ ($\beta =0.75$, or ${R}_{0}=7.5$). Note that for the results reported in the main body of the paper, the transmission rate $\beta $ was 1.5. (

**a**) Disease prevalence (i.e., proportion of infected individuals), I, (

**b**) Relative proportion of vaccinated individuals, x, (

**c**) comparison of the period of the disease prevalence, ${T}_{I}$, at different $\beta $, and (

**d**) comparison of period of the relative proportion of vaccinated individuals, ${T}_{x}$, at different $\beta $. Circle: $\beta =0.75$. Cross: $\beta =1.5$. A peak, for the purpose of measuring period, is defined by a peak threshold $\theta $: ${\theta}_{I}=0.0001$ for I, and ${\theta}_{x}=0.01$ for x.

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**Figure 1.**Schematic of daily population travel dynamics across different suburbs (nodes): a 4-node example. Solid line: network connectivity. Dashed line: volume of population flux (influx and outflux). Non-connected nodes have zero population flux (e.g., ${\varphi}_{34}={\varphi}_{43}=0$). For each node, the daily outflux proportions (including travel to the considered node itself) sum up to unity, however, the daily influx proportions do not.

**Figure 2.**Schematic of the 3-node case: population mobility across nodes. (

**a**) No population mobility. $i=j=\{1,2,3\},{\varphi}_{ij}=0$ where $i\ne j$. Otherwise ${\varphi}_{ij}=1$. (

**b**) Equal population mobility. $i=j=\{1,2,3\},{\varphi}_{ij}=\frac{1}{3}$.

**Figure 3.**Epidemic dynamics of a 3-node case for three values of $\omega $ when vaccinating newborns. Time series of (

**a**) the relative proportion of vaccinated individuals, x, and (

**b**–

**e**) Infection prevalence, I. Solid line: Symmetric uniform population mobility. Dotted line: No population mobility. Commuting suppresses prevalence peaks over time at high $\omega $, but may produce higher prevalence peaks over time at mid and low $\omega $.

**Figure 4.**Comparison of vaccination failure rates: epidemic dynamics of a 3-node case for three values of $\omega $ (which measures the responsiveness of individuals to prevalence) when vaccinating newborns. (

**a**) Relative proportion of vaccinated individuals, x, and (

**b**–

**d**) Disease prevalence (i.e., Proportion of infected individuals), I. Solid line: $\zeta =0$. Dashed line: $\zeta =0.5$.

**Figure 5.**Comparison of vaccination failure rates: epidemic dynamics of a 3-node case for three values of $\omega $ (which measures the responsiveness of individuals to prevalence) when vaccinating newborns and adults. (

**a**,

**b**) Relative proportion of vaccinated individuals, x, and (

**c**,

**d**) Disease prevalence (i.e., proportion of infected individuals), I. Solid line: $\zeta =0$. Dashed line: $\zeta =0.5$.

**Figure 6.**Comparison of vaccination failure rates: epidemic dynamics of a Erdös-Rényi random network of 3000 nodes for three values of $\omega $ (which measures the responsiveness of individuals to prevalence) when vaccinating newborns. (

**a**) Relative proportion of vaccinated individuals, x, and (

**b**–

**d**) Disease prevalence (i.e., proportion of infected individuals), I. Solid line: $\zeta =0$. Dashed line: $\zeta =0.5$.

**Figure 7.**Oscillation properties of epidemic and vaccination dynamics for three values of $\omega $ when vaccinating newborns. (

**a**) Period of the disease prevalence, ${T}_{I}$, and (

**b**) Period of the relative proportion of vaccinated individuals, ${T}_{x}$. Circle: $\zeta =0$; plus sign: $\zeta =0.5$. A peak, for the purpose of measuring period, is defined by a peak threshold $\theta $: ${\theta}_{I}=0.0001$ for I, and ${\theta}_{x}=0.01$ for x.

**Figure 8.**Epidemic and vaccination dynamics of a Erdös-Rényi random network of 3000 nodes for various values of basic reproduction number ${R}_{0}$, for three values of $\omega $. (

**a**) Cumulative prevalence, ${I}_{tot}$. (

**b**) Relative proportion of vaccinated individuals, x. ${R}_{0}$ is varied by varying the infection rate $\beta $. Cumulative prevalence ${I}_{tot}$ is obtained by integrating the prevalence over the simulated time frame. Note that different $\omega $ settings correspond to different ranges for ${R}_{0}$ due to the different vaccination coverage, x, at their respective endemic equilibria. Note that in (

**b**) the case for $\omega =1000$ is not shown because it is trivially zero for all values of ${R}_{0}$.

**Figure 9.**Epidemic dynamics of a Erdös-Rényi random network of 3000 nodes, varying the value of $\kappa $ for three values of $\omega $ (vaccinating newborns). Time series of relative proportion of vaccinated individuals, x, and disease prevalence (i.e., proportion of infected individuals), I. (

**a**) Solid line: $\kappa =0.001$. (

**b**) Dotted line: $\kappa =0.00025$.

**Figure 10.**The relationship between node degree and proportion of people who vaccinate voluntarily (vaccinating newborns only) for an Erdös-Rényi random network of 3000 nodes. (

**a**) The fraction of vaccinated individuals as a function of node degree (which is the number of neighbouring suburbs for each suburb considered) has for three values of $\omega $. (

**b**) The degree distribution of the Erdös-Rényi random network. The inset figure shows the population influx per node (sum of flux fractions from each source node) as a function of the node degree.

**Figure 11.**Simulated dynamics (vaccinating newborns only) of the commuting network in Greater Sydney generated from the 2016 Australian census data, for three values of $\omega $. Time series of (

**a**) disease prevalence, I, (

**b**) relative proportion of vaccinated individuals, x, (

**c**) out-degree distribution of the network (representing population outflux), and (

**d**) in-degree distribution of the network (representing population influx). The inset figure shows the population influx per node as a function of the node degree. Other network properties: $M=311,\langle k\rangle \approx 150$.

**Figure 12.**Epidemic dynamics of a Erdös-Rényi random network of 3000 nodes for three values of $\omega $ (vaccinating susceptible class regardless of age). Relative proportion of vaccinated individuals, x, and disease prevalence (i.e., the proportion of infected individuals), I, are shown against time. (

**a**) $\zeta =0$ (

**b**) $\zeta =0.5$. The inset figure in each figure is a magnified section to show small oscillations.

**Figure 13.**Epidemic dynamics of a Erdös-Rényi random network of 3000 nodes with committed vaccine recipients for three values of $\omega $ (vaccinating the entire susceptible class). Relative proportion of vaccinated individuals, x, against time, and disease prevalence (i.e., the proportion of infected individuals), I, against time. (

**a**) $\omega =1000$ (

**b**) $\omega =2500$ (

**c**) $\omega =3500$. Solid line: without committed vaccine recipients. Dotted line: with committed vaccine recipients. The proportion of committed vaccine recipients, ${x}^{c}=0.0002$. The existence of committed vaccine recipients delays the predominant peaks and reduces the magnitude of oscillation in the proportion of vaccine recipients in later stages.

Parameter | Interpretation | Baseline Value | References |
---|---|---|---|

$1/\gamma $ | Average length of recovery period (days) | 10 | [37] |

${R}_{0}$ | Basic reproduction number | 15 | [37] |

$\mu $ | Mean birth and death rate (days${}^{-1}$) | 0.000055 | [3] |

$\zeta $ | Vaccination failure rate | [0,1] | Assumed |

$\kappa $ | Imitation rate | 0.001 | [3] |

$\omega $ | Responsiveness to changes in disease prevalence | [1000,3500] | [3] |

${\varphi}_{ij}$ | Fraction of residents from node i travelling to j | [0,1] | Network connectivity |

(See Figures 10b and 11d | |||

for degree distribution.) | |||

I | Initial condition | 0.001 | [3] |

S | Initial condition | 0.05 | [3] |

x | Initial condition | 0.95 | [3] |

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## Share and Cite

**MDPI and ACS Style**

Chang, S.L.; Piraveenan, M.; Prokopenko, M.
The Effects of Imitation Dynamics on Vaccination Behaviours in SIR-Network Model. *Int. J. Environ. Res. Public Health* **2019**, *16*, 2477.
https://doi.org/10.3390/ijerph16142477

**AMA Style**

Chang SL, Piraveenan M, Prokopenko M.
The Effects of Imitation Dynamics on Vaccination Behaviours in SIR-Network Model. *International Journal of Environmental Research and Public Health*. 2019; 16(14):2477.
https://doi.org/10.3390/ijerph16142477

**Chicago/Turabian Style**

Chang, Sheryl Le, Mahendra Piraveenan, and Mikhail Prokopenko.
2019. "The Effects of Imitation Dynamics on Vaccination Behaviours in SIR-Network Model" *International Journal of Environmental Research and Public Health* 16, no. 14: 2477.
https://doi.org/10.3390/ijerph16142477