Dibyajyoti Mallick, Aniruddha Ray, Ankita Das and Sayantari Ghosh⋆
Department of Physics, National Institute of Technology Durgapur, Durgapur, West Bengal, India
The vaccines for preventing COVID-19 are being considered as the most effective way to reduce the high pandemic burden on the global health infrastructure. However, public hesitancy towards vaccination is a crucial and pressing problem. Our study has been designed to determine the parameters affecting the vaccination decisions of common individuals. Using the platforms of the compartmental model and network simulation, we categorize people and observe their motivation towards vaccinations in a mathematical social contagion process. We consider peer influence as an important factor in this dynamic and study how individuals influence their peers regarding vaccination decisions. The efficiency of the vaccination process is estimated by the period of time required to vaccinate a substantial fraction of the total population. We discovered the major barriers and drivers of this dynamics and concluded that it is required to formulate specific strategies targeting specifically the undecided and vaccine hesitant people.
Keywords: Complex network, mathematical model, social epidemic, vaccination strategies
The novel coronavirus disease 2019 (COVID-19), caused by severe acute respiratory syndrome Coronavirus 2 (SARS-CoV-2), was first reported in Wuhan, Hubei Province of China. This disease is now a global pandemic which is spreading rapidly from person to person, causing major public health concerns and economic crisis [1, 14]. A variety of active intervention policies have been introduced to suppress the spreading of this disease such as hand sanitizing, social distancing, travel restrictions, partial or complete lockdown, wearing masks, quarantining, etc. After the declaration of the pandemic by WHO (World Health Organization) in March 2020, pharmaceutical companies and scientists have encountered a race against time to develop vaccines [6]. The recent availability of multiple vaccines against coronavirus has brought hope for prevention of the spreading and a rapid recovery of our badly affected economy with a promise of a sooner resumption of normal life. However, the widespread hesitancy about vaccines is becoming a major obstacle for global health. Many people have strong hesitation towards vaccination, which is defined as the confusion about safety and effectiveness of the vaccine. The origin of this usually lies in some rumors regarding vaccines concerning side effects. This has become a huge challenge for governments and public health authorities for reaching the expected and required vaccination coverage [3, 9, 19]. The key priority now is ensuring vaccine acceptance because the lag in vaccination may provide a window for spreading the new variants and can also be a major obstacle in developing society-wide herd immunity. Until now, many researchers and scientists have tried to conduct different surveys to understand the behavior towards vaccine acceptance and hesitancy [2, 8, 10]. Some studies [7] have applied different theoretical models to explain vaccine acceptance, hesitancy, and willingness of individuals towards vaccines as well as refusal to vaccinate, which may vary depending upon personal decisions and epidemiological conditions [18]. Many advanced countries are trying to conduct country specific surveys [11], while several reports are being prepared by different countries such as the United Kingdom (UK), United States of America (USA), China, India, and Saudi Arabia to understand this vaccine acceptance behaviour [5, 12, 16]. However, most of these studies are based on heuristic arguments rather than mathematical analysis. A computational framework having a mathematical foundation helps to draw quantitative conclusions and is much more effective for predictive modeling purposes. This paper represents a new mathematical model regarding the vaccination process which incorporates behavioral changes of every individual in a society, driven by global and local factors. The main assumption here is that the vaccination dynamics can be considered as a social contagion process. This study aimed to identify the acceptability of the Covid-19 vaccine and information in support or opposing vaccination that is flowing in a society like a viral infection. We take into account refusal and hesitancy factors as well as the positive attitude of people towards vaccination [17]. In our study, we focused on identifying the effects of the factors that could increase the vaccination coverage through numerical simulations on an artificial society. Moreover, we could justify several results found by survey based studies where this kind of question has already been explored using survey results [4, 15]. Our results address the vaccination acceptance problem using a comprehensive mathematical and computational framework that would help us figure out correct strategies to encourage community for vaccine uptake and to stop further spreading of this pandemic.
To understand this kind of problem, we have considered a set of differential equations to depict the possible transitions. To describe the vaccine dynamics, we have a compartmental model, as shown in Figure 8.1. At any time t, the total population N(t) is subdivided into four states: Ignorant (I), Hesitant (H), Unwilling (U), and Vaccinated (V).
In this model, the key variable we have is the vaccinated population who are influencing other populations to get vaccinated. This is also the target population which has to be maximized over a certain period of time. Here, this group of people is playing an effective role to control the epidemic and spread the infodemic. Considering the total population as 1 (normalized form),
Now, we will discuss the following possibilities of transition of people from one sub-population to another over a given time period ‘t’. Let us discuss all the possible transitions considered in the model categorically.
Hence, all the rate equations of this model are compiled as follows:
In ODE-based models, one of the major issues is homogeneous mixing, which indicates every individual in a population has the same probability of having contact with each other [20]. Our society is highly heterogeneous and to accommodate that fact into our findings, we study the model on the heterogeneous setting of a complex network. The simulations are performed on a random network having 10000 nodes with an average degree of 5. Here, we choose the EoN module in Networkx from python [13] and run these following simulations in Google Colaboratory. Varying the transition rates of each parameter we see the effects of that particular parameter on the time evolution of the dynamics through our model. The general dynamics have been depicted in Figure 8.2. It shows that eventually most of the people in the population get vaccinated, however the time of coverage might be different depending on the parameter values. Thus, to quantify this growth curve, we define vaccination coverage time as ζV:
This means that we consider a timescale, ζV, within which 90% of maximum vaccination, Vmax has been achieved. This will give us an estimate of the vaccination coverage speed.
Vaccination rate is a major parameter of this dynamic. If you carefully observe different sub-populations, the effect could be prominently observed, as reported in Figure 8.3.
As shown in Figure 8.3(d), we can see that the saturation time decreases with increasing vaccination rate, that is λ, which states that this particular parameter will make the system reach the saturation point earlier.
Negative rumors often sway people towards some risky and dangerous activity. In this vaccination process, some negative rumors about the vaccine exist that concern matters like several side effects, cost effectiveness, fear, and misinformation regarding the vaccine. Hesitant people are often influenced by them. The parameter that takes into account this phenomena is β. The results related to this are shown in Figure 8.4.
Peer influence occurs when people are motivated or influenced towards something by seeing their neighbor or friends. When someone’s peers influence them to do something positive, it is considered as positive peer influence. In the vaccination context, positive peer influence occurs when someone who is vaccinated is motivating others who are not sure about vaccination. Here, the vaccinated people influence the hesitant people to get vaccinated. Here we are considering two parameters α and γ that take into account this phenomenon. The results related to this are shown in Figure 8.5 and Figure 8.6.
So, we can draw the conclusion that if more people are well informed about the success rate of vaccination, then that would speed up the vaccination process and also decrease the amount of unwilling people. As shown in Figures 8.5(d) and 8.6(d), for both parameters of positive peer influence, that is, α and γ, the saturation time decreases with increasing parameters, which means positive peer influence will make the system reach the saturation point earlier.
Along with the infectious spread of SARS-COV2, the information and misinformation regarding available vaccines is also spreading person-to-person, causing a large-scale effect on vaccination coverage. Through computational analysis and network simulations, we have explored the contagion dynamics and proposed a framework to analyze this decision process. From the model it has been observed that the presence of the efficient vaccination systems can run the entire dynamic with positive feedback. This is a significant result, as we can see a lot of people not only in India but around the world are hesitant about getting vaccinated. In our model, we have shown that if the peer influence is overall positive, then the saturation point shifts leftwards, so we can conclude that the spreading of awareness about the positive effects of vaccination should be carried out on the ground level. We have also shown in our model that if vaccination rate increases, then the saturation point of V class shift leftwards. In real life we can explain it like this: if the vaccination rate in a particular area can be increased by awareness and smooth execution of the vaccination process by a competent authority, then the system would reach its saturation point sooner. A strong peer influence also might signify a strongly connected society. The shift of the saturation point leftwards for high positive peer influence means that physically and virtually everyone is well connected. This may also indicate high vaccination rates in urban areas. In the future we will focus on implementing further realistic terms in our model. For example, if someone from a hesitant class goes to the unwilling class, it can be considered that the person was under the influence of someone who has been vaccinated, thus we can bring the non-linearity. Mathematical explorations are used to find out the fixed points and bifurcations if there are any. Considering coupled dynamics or delayed dynamics of disease along with the infodemic might be another interesting future study.