Impact of opinion dynamics on the public health damage inflicted by COVID-19 in the presence of societal heterogeneities

Introduction Certain behavioral practices, such as wearing masks, practicing social distancing, and accepting vaccines, play a crucial role in impeding the spread of COVID-19 and reducing the severity of symptoms. Opinions regarding whether to observe such behavioral practices evolve over time through interactions via networks that overlap with but are not identical to the physical interaction networks over which the disease progresses. This necessitates the joint study of the dynamics of COVID-19 and opinion evolution. Methods We develop a mathematical model that can be easily adapted to a wide range of behavioral practices and captures in a computationally tractable manner the joint evolution of the disease and relevant opinions in populations of large sizes. Populations of large sizes are typically heterogeneous in that they comprise individuals of different age groups, genders, races, and underlying health conditions. Such groups have different propensities to imbibe severe forms of the disease, different physical contact, and social interaction patterns and rates. These lead to different disease and opinion dynamics in them. Our model is designed to effectively capture such diversities. Results Computations using our model reveal that opinion dynamics have a strong impact on fatality and hospitalization counts and the number of man-days lost due to symptoms both in the regular form of the disease and the extended forms, more commonly known as long COVID. We show that opinion dynamics in certain groups have a disproportionate impact on the overall public health attributes because they have high physical interaction rates, even when they have the lowest propensity to imbibe severe forms of the disease. This identifies a social vulnerability that malactors can utilize to inflict heavy public health damages through opinion campaigns targeting specific segments. Once such vulnerabilities are identified, which we accomplish, adequate precautions may be designed to enhance resilience to such targeted attacks and better protect public health. Discussion By recognizing and understanding the vulnerabilities, appropriate precautions can be developed to enhance resilience against targeted attacks and safeguard public health. Our study underscores the importance of considering opinion evolution alongside disease dynamics, providing insights into the interplay between behavioral practices, opinions, and disease outcomes. We believe that our model is a valuable tool for understanding the joint dynamics of COVID-19 and opinions. We hope that our findings will help to inform public health policy and facilitate evidence-based decision-making for public health interventions.

between individuals of the same race and age group (youngest and middle age groups). Older people have smaller, more family-centric networks, and spend less time with others [16]. Thus, their rate of interaction is lower compared to the other groups. Therefore, we denote the disease spread rate between the oldest group of the same race with m where m < 1. Similarly, we denote the physical contact rate between the oldest group and the other groups with  where  < 1. In addition, we represent the physical contact rate between the youngest age group of different races with a , middle age group of different races as b , and oldest age group of different races with c . We denote any other physical interaction rate with x , where a < 1, b < 1, c < 1, x < 1. Similarly, we use ↵ to represent the opinion spread rate. In this case, ideas are exchanged during interactions -which might lead to a change in opinion. Meanwhile, one may be converted by exposure to public awareness campaigns (which may have a stronger impact because the individual might be infected and experiences the symptoms acutely). We denote the rate of such opinion change as .
Interaction between susceptible individuals i.e. S c (t), S n (t) and Infectious individuals i.e. P c (t), P n (t), I cs (t), I ns (t), I ca (t), I na (t) may spread the disease to the susceptibles and transform them to early incubators -refer to the yellow arrows in Fig 4. We assumed that those that are hospitalized, H c (t), H n (t), are isolated from the general public (quarantined) and hence cannot infect other susceptible nor can they convert the opinions of others. Natural progression of the disease change E c (t) to P c (t) to I cs (t) to either H c (t) or R c (t) or D(t); also, E c (t) to I ca (t) to R c (t) -refer to the blue arrows in  We model the evolution of the states through a set of meta-population epidemiological differential equations.
Each differential equation captures the evolution of a particular variable. Thus, the solution of the system of differential equations provides the fraction of individuals in different states at given times, that is, the spatio-temporal distribution of the disease and opinion spread. The terms in the differential equations are either quadratic or linear. The quadratic ones represent the transitions brought on by interactions between two individuals, specifically physical proximity, and exchange of ideas (refer to the yellow arrows and the black arrows in Fig 4) and the linear ones represent the transitions that happen otherwise, specifically natural progression of the disease, conversion through reading, media, etc. (refer to the blue and black arrows in The terms in green color are quadratic terms that represent the spread of the SARS-CoV-2 to the susceptibles due to physical interaction with the infectious individuals and the subsequent transformation of the susceptibles to the exposed states -refer to the yellow arrows in Note that (1) is similar to (2) as transitions are similar for the cooperative and non-cooperative individuals.
The first terms in (1), (3), (the terms in green color) are quadratic terms that represent the spread of the disease to the susceptibles due to interaction with the infectious individuals and the subsequent transformation of the susceptibles to early incubators -refer to the yellow arrows in These products lead to the quadratic terms. The overall proportionality constants are the products of the two proportionality constants mentioned above, and give us , the disease spread rates, which have been summarized in Table 8. Since the spread of the disease reduces the number of susceptibles and increases the number of early incubators, the first term in (1) has a positive sign and that in (3) has a negative sign.
The second terms in (1)  group at time t ((P ci + P hi + C ci + C hi + E ci + E hi )(t)); (5) are contagious and are in cluster i at time t ((P c + P n + I cs + I ns + I ca + I na )(t)).
Meanwhile, equations (1) -(15) represent the basic compartments in our model. We now generalize the states in our model. Recall that an individual is characterized by his cooperativity, health conditions, gender, ancestry, age, stage of the disease, as well as symptomatic or asymptomatic manifestation. Thus, we use the suffixes c to denote cooperativity and n to denote non-cooperativity. Similarly, h denotes being healthy (immunocompetent), d denotes immunodeficient (not healthy, immunocompromised), m represents male, while f represents female. In addition, we use the subscripts x, y, and z respectively to represent individuals of African American, Hispanic, and White-American ancestry. Furthermore, we classify the age of an individual into three groups with subscript 1 denoting 0 24 years, 2 denoting 25 49 years, and 3 denoting 50 years and above. Finally, we use the suffixes s and a to respectively denote symptomatic and asymptomatic individuals. Therefore, the state S chmx1 denotes cooperative, healthy, male susceptible African American in the 0 24 years age group. Similarly, E ndf x3 denotes non-cooperative, not healthy (immunocompromised), exposed female Hispanic/Latino who is 50 years and above, I chmzs2 denotes cooperative, healthy, infected symptomatic male White-American in the 25 49 years age group while I ndf za1 denotes non-cooperative, not healthy (immunocompromised), infected asymptomatic female African American in the 0 24 years age group.

Supporting information II
Parameter estimation = 25.6%. In addition, the gender distribution in Pennsylvania is 51.1% female and 48.9% male [35]. The total number of people we considered in our model is 10 million whereas our choice for the initial number of infected individuals is 10000. Comorbidity refers to the existence of more than one disease or condition in the same person at the same time. The Centers for Disease Control and Prevention (CDC) and the PA Department of Health (PaDOH) have highlighted concern for individuals of all ages who suffer from underlining medical conditions such as obesity, as well as those who are immunocompromised due to conditions like cancer treatment and being HIV/AIDS positive [12]. Furthermore, according to [36], about 40% of the US population are immunocompromised. According to the Centers for Disease Control and Prevention (CDC), the current best estimate of the basic reproduction number, R 0 for COVID-19 is 2.5 [11]. Recall that R 0 is the average number of secondary infections caused by a single typical infected individual among a completely susceptible population. If R 0 > 1, epidemic takes off. On the other hand, if R 0 < 1, no major epidemic occurs. In addition, following the procedure outlined in (Chapter 6, [38]), we obtain the expression for R 0 for our model as shown in equation 16. Furthermore, 30% of infected individuals are asymptomatic [11]. The transmission of SARS-CoV-2 from an infected person to a secondary patient before the source patient developed symptoms is known as presymptomatic transmission [39]. The presymptomatic stage lasts for 2 days on average [40], [39]. The mean time from exposure to symptom onset is 6 days [11], i.e., 1/ + 1/ = 6 days. But 1/ = 2 days on average [40]. Therefore, 1/ = 4 days, and 1/ = 2 days. The median number of days from symptom onset to hospitalization is 5 days [11]. That is expected time 1/ + 1/ = 5 days. Therefore, 1/ = 2 days. The median number of days from symptom onset to death is 15 days [11]. Therefore, 1/ = 15 -1/µ 1/ = 15 -3 -3 = 9 days. Thus, 1/ = 9 days. Furthermore, the median duration of hospital stays among survivors = 9.3 days [41]. Similarly, according to CDC, the persons who never develop symptoms, isolation, and other precautions can be discontinued 10 days after the date of their first positive RT-PCR test for SARS-CoV-2 RNA. Therefore, 1/ = 10 days [42,43]. The percent that dies among those hospitalized is 0.7% (0 -17 years old), 2.1% (18 -49 years old), 7.9% (50 -64 years old), 18.8% ( 65 years old) [11]. Therefore, fatality p Probability that an infected person is asymptomatic 0.3 [11] ⌘ Probability that an infected person is hospitalized Table 7 [45] COVID-19 fatality rate see p. 40 [11] 1 Expected time an individual is in exposed stage 4 days p. 40 1 Expected time an individual is pre-symptomatic 2 days [39,40]  Similar to [24], we define the fractions of individuals who fully comply with COVID-19 prevention measures at the initial time as the initial cooperativity. According to [36], approximately 40% of the populace is not cooperative. Thus, our default choice for the initial cooperativity is 0.6, but we also consider other values of initial cooperativity. We assume that interaction between a pair of non-cooperatives is twice likely to result in infection. In addition, we assume that the disease spread rate between a pair of individuals such that only one is non-cooperative equals that when both are non-cooperatives. COVID-19 vaccines have been shown to be effective and help to reduce hospitalizations, intensive care unit admissions, and deaths [7,8]. However, the secondary attack rate among household contacts exposed to fully vaccinated index cases was similar to household contacts exposed to unvaccinated index cases [44]. Meanwhile, vaccination against COVID-19 is now seamless and vaccines are readily available to receive across the U.S. Thus, we also considered cases in which non-cooperatives are assumed to be twice likely to be hospitalized compared with cooperatives. The parameters we use in this research alongside their estimated values are outlined in Table 6.
Next, we use data from [45] to estimate the probability of hospitalization for African Americans, Hispanics/Latinos, and White Americans for the various age groups. We assumed that each individual averages 1 -4 minutes of exercise per day. We also assumed that a healthy individual weighs 147 lbs on average while an obese individual weighs 203 lbs or more. We use the subscripts x, y, and z to respectively represent people of African American, Hispanic, and White American ancestry. We also classify the age of an individual into three groups with subscript 1 denoting 0 24 years, 2 denoting 25 49 years, and 3 denoting 50 years and above. We use to denote the physical rate of interaction between individuals of the same race and age group (youngest and middle age groups).
Older people have smaller, more family-centric networks, and spend less time with others [16]. Thus, their rate of interaction is lower compared to the other groups. Therefore, we denote the disease spread rate between the oldest group of the same race with m where m < 1. Similarly, we denote the physical contact rate between the oldest group and the other groups with  where  < 1. In addition, we represent the physical contact rate between the youngest age group of different races with a , middle age group of different races as b , and oldest age group of different races with c . We denote any other physical interaction rate with x , where a < 1, b < 1, c < 1, and x < 1. As shown above, = 2.4122 ⇥ 10 8 per person per day, a = 0.9, b = 0.8, c = 0.2, k = 0.6, x = 0.3, and m = 0.4. Similarly, we use ↵ to represent the virtual interaction rate also known as the opinion spread rate. We assumed that ↵ = 10 9 and = 2 ⇥ 10 9 (default values).