Prevention and Control of COVID-19 Risks for Long-Term Care Facilities Based on the Prospect Theory

Abstract

Given the complexity and uncertainty of the current COVID-19 risks, the elderly people in long-term care facilities are at the highest risk for infection. In order to study the prevention and control strategies of COVID-19 risks in long-term care facilities, this paper uses the prospect theory to construct the decision-making model of COVID-19 risk behavior of long-term care facilities, analyses the risk behavior strategies of the caregivers and managers, and reveals the impact of risk management cost, risk loss and external supervision on the risk behavior decision-making of the caregivers and managers. Furthermore, from the perspective of long-term care facilities, this paper analyzes the constraints that enable it to achieve optimal risk management strategy. Combined with the simulation analysis, it is found that the decision of risk behavior of the caregivers and managers is positively related to the risk behavior choice, risk loss, and supervision. Then, only when the incentives set by the supervision are within a reasonable range can the caregivers and managers be motivated to take proactive risk management strategies. The study has important theoretical and practical significance for the management of COVID-19 risks in long-term care facilities.

1 Introduction

According to the statistics of the World Health Organization (WHO), as of August 27, 2020, 168 countries and regions around the world had reported confirmed cases of coronavirus disease 2019 (COVID-19), with a total of more than 24.0212 million cases and 820,000 deaths [1]. Currently, the known ways of COVID-19 transmission include human-to-human transmission, mainly through droplets and contact. A basic reproduction number of 3.6 ~ 4.0, COVID-19 is an infectious disease with moderate to high infectivity [2, 3]. The elderly people, who especially suffering with asthma, diabetes and heart disease are at increased risk for virus infection, with the elderly being the most susceptible to the novel coronavirus and the most likely to progress to severe illness [4, 5]. In the context of COVID-19 pandemic, many countries have reported cases of cluster infection in the long-term care facilities [6]. According to an article on the website of The New York Times dated August 13, 2020, “At least 68,000 residents and workers have died from the coronavirus at nursing homes and other long-term care facilities for older adults in the United States, according to a New York Times database. As of August 13, the virus has infected more than 402,000 people at some 17,000 facilities.” Moreover, deaths related to Covid-19 in long-term care facilities accounted for more than 41% of the pandemic fatalities (The New York Times, 2020) [7]. The older people are, the more difficult it is for them to have COVID-19 treated in case of infection. Long-term care facilities take the responsibility of controlling the spread of COVID-19 and ensuring the physical and mental health of the elderly [8, 9].

The COVID-19 risks referred to in this paper are defined as the risk for the occurrence of COVID-19 among the elderly and caregivers that may cause illness, damage or death to the elderly due to improper prevention and control in long-term care facilities as well as incidents that may seriously affect the operation of long-term facilities. COVID-19 is characterized by suddenness, public coverage, severity, urgency, complexity and variability. Besides, it comes with high risk levels and serious consequences, and it is present throughout the work carried out in senior care facilities in the pandemic context. In fact, in the context of COVID-19 pandemic, any improper prevention and control may lead to COVID-19 risks and cluster infection in such facilities as community hospitals, clinics, home care settings, nursing homes and geracomium. And such incidents can be caused by factors such as facility management, caregivers’ behaviour and environmental facilities. The risk for the occurrence of COVID-19 among the elderly and caregivers that may cause illness, damage or death to the elderly due to improper prevention and control in long-term care facilities as well as incidents that may seriously affect the operation of long-term care facilities [10]. And such incidents can be caused by factors such as facility management, caregivers’ behaviour and environmental facilities [11,12,13,14]. The primary task of COVID-19 risk management is to identify high risks and coordinate the elimination and reduction of the occurrence of COVID-19 risk events. Achieving maximum safety at the lowest cost is the most scientific risk management method [15]. The key to COVID-19 risk management is to improve the risk awareness and response ability of caregivers in long-term care facilities. Using the risk system assessment scale, managers entered all relevant data such as the current management status and risk factors into the computer to establish the minimum data set for long-term care facilities [16, 17], so as to provide managers with information and decision-making support and improve the efficiency of COVID-19 risk management [18]. Using the MDS 2.0 assessment scale, Shaw et al. [19] analyzed the correlation between the occurrence of senior care risks and the characteristics of long-term care facilities, including management process and staffing, to prove the validity of the assessment system.

Through risk management, the risk of contracting COVID-19 in long-term care facilities was transformed, decomposed as well as effectively prevented and controlled to varying degrees [20, 21]. Nevertheless, long-term care facilities suffer from a serious shortage of staff [22]. Specifically, the proportion of facility to the number of caregivers in private long-term facilities has been staying at somewhere between 1:15 and 1:20, far lower than the international standard of 1:4.5 in nursing homes [23]. To save costs, some long-term facilities recruit less staff and give them low salaries and a heavy workload, leading to an increased turnover rate [24]. Caregivers in long-term facilities have heavy daily tasks, which are repetitive and tedious and cause them high pressure [25]. As a result, caregivers generally have low willingness to work and low job satisfaction. In China, there is a serious shortage of professional senior caregivers, and the turnover rate of caregivers in undeveloped areas is as high as 20% [26,27,28]. Caregivers find it difficult to properly cater to the nursing needs of the elderly if they are dissatisfied with their jobs. Under the impact of COVID-19 pandemic in the world, the risk of being exposed to COVID-19 is even more obviously. As a result, there is an inevitable link between risk occurrence and the caregivers and managers’ behaviour, which will pose even greater challenges to the long-term care facilities [29].

The prospect theory has made ground-breaking contributions to decision-making research in the context of uncertainty and is now widely applied in various decision-making analyse [30, 31]. In the process of COVID-19 risk management in long-term care facilities, there are two types of groups, namely, long-term care facilities managers and caregivers. Due to the uncertainty and complexity of long-term care facilities compared to general community service centres, the managers and caregivers in long-term care facilities differ in such aspects as COVID-19 risk perception, risk management knowledge, and the abilities of analysing COVID-19 risk information, and effectively response for the situation. And they tend to exhibit characteristics of bounded rationality [32,33,34]. In reality, caregivers in long-term care facilities are often not completely rational and are susceptible to the influences of their surroundings and their psychological factors. The analysis of value according to the prospect theory is based on the change in the gain of managers and caregivers instead of the final state of the gain, and this change is defined according to the degree of deviation from the reference standard. So, managers and caregivers often determine their behaviour based on their perspectives or reference standards. As their reference standards differ, their expected “gain” or “loss” also differs.

Based on this, this paper introduces the prospect theory to analyse the evolution process and mechanism of the risk responses of COVID-19 risk management subjects in long-term care facilities. It also explains the reasons for the behavioural tendencies of different COVID-19 risk subjects and the conditions for the selection of COVID-19 risk management strategies from the perspective of risk perception theory. The main research contribution of this paper lies in provide theoretical and decision-making support for COVID-19 risk management in long-term care facilities by analysing the interrelationship between risk management behaviour and nursing behaviour under different conditions as well as the consequences of its evolution.

The structure of the rest of the paper is organized as follows. Section two sets up the decision-making model for COVID-19 risk response in long-term care facilities. In section three, the decision-making on risk responses is simulated and analyzed. Section four puts forward the risk management and control strategies based on the prospect theory. Last, section five concludes the study and proposes the directions for future research.

2 Decision-making model for COVID-19 risk response in long-term care facilities

2.1 Main factors influencing the risk response of the caregivers and the managers

The risk response of caregivers and managers in long-term care facilities is related to the COVID-19 risk loss caused to long-term care facilities by risk management behaviour. Since the behaviour of caregivers and managers in long-term care facilities directly affect the intensity of the risk of COVID-19 infection among the elderly in these facilities, they also tend to significantly affect such risk loss as the loss of reputation and financial loss caused to long-term care facilities by the risk of COVID-19 infection.

Assume that facility caregivers can choose between “standard” and “non-standard” caregiving behaviour, and that facility managers can choose between “proactive” and “passive” risk management behaviour [22,23,24,25]. If the probability for the caregivers to choose “standard” caregiving behaviour is P_x, then the probability for them to choose “non-standard” caregiving behaviour is 1–P_x. Similarly, if the probability for the managers to choose “proactive” risk management is P_y, then the probability for them to choose “passive” risk management is 1–P_y. The gain the caregivers get for providing regular caregiving services is E_C, and the gain for the facility managers to carry out regular risk management is E_M. The cost the caregivers pay for choosing “standard” caregiving behaviour (reflecting the cost the caregivers incur for their increased efforts and more care time) is represented by C_C. The cost the facility managers pay for choosing “proactive” risk management behaviour (reflecting the cost the facility managers incur for such factors as their enhanced management ability, increased efforts and more management time) is represented by C_M. Therefore, 0 < C_C < E_C, 0 < C_M < E_M.

If the caregivers engage in “standard” caregiving behaviour and the managers engage in “proactive” risk management, the breakout of COVID-19 in long-term care facilities can be prevented, thereby preventing risk loss. Otherwise, long-term care facilities will suffer losses incurred by COVID-19 infection among elderly residents. Assume that the risk loss a long-term care facilities suffers when the caregivers engage in “non-standard” caregiving behaviour and the managers carry out passive risk management is L. In fact, the caregivers’ “standard” caregiving behaviour and the managers’ “proactive” risk management behaviour can both lower the infection rate of elderly residents in the long-term care facilities, thereby reducing relevant losses to the facility. γ_C represents the discount factor of the risk loss incurred to long-term care facilities (referred to as “risk loss discount factor” below) by COVID-19 when the caregivers adopt “standard” caregiving behaviour, regardless of whether or not the managers engage in “proactive” risk management. In this case, the loss the long-term care facilities suffers is (1 − γ_C)L, and γ_C ∈ (0, 1). γ_Mrepresents the risk loss discount factor by COVID-19 when the managers engage in “proactive” COVID-19 risk management, regardless of whether or not the caregivers choose “standard” caregiving behaviour. And in this case, the loss the long-term care facilities suffers is (1 − γ_M)L, and γ_M ∈ (0, 1. γ_C measures the positive effect of the caregivers’ “standard” caregiving behaviour on the risk loss incurred by COVID-19 in the facility, whereas γ_M measures the positive effect of the managers’ “proactive” risk manageemnt behaviour on the risk loss incurred by COVID-19 in the facility.

In essence, different risk responses cause long-term care facilities to face different discount rates of risk loss incurred by COVID-19 and reflect the intensity of COVID-19 infection among elderly people in long-term care facilities. In particular, γ_C + γ_M = 1. In other words, if both the caregivers and the facility managers choose to proactively respond to COVID-19 risks, effective risk management behaviour will prevent the long-term care facilities from suffering loss incurred by COVID-19 risks, at which point the risk discount rate reaches its maximum 1. Furthermore, assume that the risk loss suffered by a long-term care facilities is shared between its caregivers and managers. If β refers to the proportion of risk loss incurred by COVID-19 borne by the caregivers, then 1 − β refers to the proportion of risk loss incurred by COVID-19 borne by the facility managers, and β ∈ (0, 1).

To effectively regulate and encourage proactive COVID-19 risk management in long-term care facilities, the government and industry regulators will introduce a strict punishment and reward mechanism. When the caregivers choose to engage in “non-standard” caregiving behaviour, they will be punished by the government and industry regulators, and the punishment the caregivers face is represented by K_C; when the caregivers choose to engage in “standard” caregiving behaviour, they will be rewarded by the government and industry regulators, and the reward the caregivers receive is represented by S_C. Similarly, when the facility managers choose “passive” risk management behaviour, they will be punished by the government and industry regulators, and the punishment the managers face is indicated by K_M; and when the facility managers choose to engage in “proactive” caregiving behaviour, they will be rewarded by the government and industry regulators, and the reward the facility managers receive is represented by S_M. Without loss of generality, this paper assumes that S_C = K_C, S_M = K_M.

The main factors influencing the risk responses of the caregivers and the managers and the consequences of such responses are indicated in Table 1 below. To distinguish between different participants and risk response strategies, this paper has respectively added subscripts C and M to the corresponding variables, namely, the caregivers and the facility managers, indicated the “standard” and “non-standard” caregiving behaviour of the caregivers as C₁ and C₂ respectively, and represented the facility managers’ “proactive” and “passive” risk management behaviour by M₁ and M₂ respectively.

Table 1 The main factors influencing the risk responses of the caregivers and the managers and the consequences of such responses

In this table, T_C1, T_C2and T_C3 respectively represent the set reference standards for the caregivers’ gain, the discount factor of the loss incurred to the facilities by caregivers’ risk response and the punishment on the caregivers by the government and industry regulators. T_M1,T_M2,T_M3respectively stand for the set reference standards for the facility managers’ gain, the discount factor of the loss incurred to the facilities by the managers’ risk response, and the punishment on the managers by the government and industry regulators. Thus, T_C = W_C1T_C1 − W_C2T_C2L + W_C3T_C3 is the weighted reference standard of caregivers’ expected gain, while T_M = W_M1T_M1 − W_M2T_M2L + W_M3T_M3 is the weighted reference standard of managers’ expected gain.

2.2 The matrix of the risk responses of the caregivers and the managers

As mentioned earlier, the risk response strategies of the caregivers and the managers in long-term care facilities are mainly influenced by three factors, namely, their own gain, external regulation and the decisions of internal stakeholders. Therefore, when analysing the prospective value of the strategies under the caregivers’ different caregiving behaviour or the managers’ different risk management behaviour, it is imperative to include the influence of the risk responses of the facility managers (or the caregivers) on their own gain.

First, the caregivers’ net gain from different risk response strategies is analyzed. And the decision tree for the caregivers’ caregiving behaviour is shown in Fig. 1. If caregivers choose “standard” caregiving behaviour C₁, the managers will be faced with two decisions: (1) When the managers choose “proactive” risk management behaviour M₁, the risk loss discount factor” below is γ_C + γ_M = 1, and the net gain of the caregivers is ${\pi}_{C_{1,}{M}_1}^C={E}_C-{C}_C-\left(1-{\gamma}_C-{\gamma}_M\right)L={E}_C-{C}_C$. (2) When the managers choose “passive” risk management behaviour M₂, the risk loss discount factor is only γ_C, and the caregivers get the reward K_M for engaging in “standard” caregiving behaviour. Therefore, the net gain of the caregivers is ${\pi}_{C_{1,}{M}_2}^C={E}_C-{C}_C+{K}_M-\left(1-{\gamma}_C\right)\beta L$.

Fig. 1

Decision tree of the caregivers’ behaviour

If the caregivers choose “non-standard” caregiving behaviour C₂, the managers will also be faced with two decisions: (1) When the managers choose “proactive” risk management behaviour M₁, the risk loss discount factor is γ, and the punishment imposed on the caregivers is K_C. So the net gain of the caregivers is ${\pi}_{C_{2,}{M}_1}^C={E}_C-{K}_C-\left(1-{\gamma}_M\right)\beta L$; (2) When the managers choose “passive” risk management behaviour M₂, the risk loss discount factor is L, and the punishment imposed on the caregivers is K_C. So, the net gain of the caregivers is ${\pi}_{C_{2,}{M}_2}^C={E}_C-{K}_C-\beta L$.

Similarly, the facility managers’ net gain from different risk response strategies can also be analyzed. And the decision tree for the facility managers’ caregiving behaviour is shown in Fig. 2. If the managers choose “proactive” risk management behaviour M₁, the caregivers will be faced with two decisions: (2) When the caregivers choose “standard” caregiving behaviour C₁, the risk loss discount factor is γ_C + γ_M = 1, and the net gain of the managers is ${\pi}_{C_{1,}{M}_1}^M={E}_M-{C}_M-\left(1-{\gamma}_C-{\gamma}_M\right)L={E}_M-{C}_M$. (2) When the caregivers choose “non-standard” caregiving behaviour C₂, the risk loss discount factor is only γ_M, and the managers receive the reward K_C for engaging in proactive risk management behaviour. Therefore, the net gain of the managers is ${\pi}_{C_{2,}{M}_1}^M={E}_M-{C}_M+{K}_M-\left(1-{\gamma}_M\right)\left(1-\beta \right)L$.

Fig. 2

Decision tree of the managers’ risk management behaviour

If the managers choose “passive” risk management behaviour M₂, the caregivers will also be faced with two decisions: (1) When the caregivers choose “standard” caregiving behaviour C₁, the risk loss discount factor is γ_C, and the punishment imposed on the managers is K_M. So the net gain of the managers is ${\pi}_{C_{1,}{M}_2}^M={E}_M-{K}_M-\left(1-{\gamma}_C\right)\left(1-\beta \right)L$; (2) When the caregivers choose the “non-standard” risk management behaviour C₂, the risk loss discount factor is L, and the punishment imposed on the managers is K_M. So net gain of the managers is ${\pi}_{C_{2,}{M}_2}^M={E}_M-{K}_M-\left(1-\beta \right)L$.

To sum up, the matrix of the risk responses of the caregivers and the managers is indicated in Table 2 below. The strategies for the caregivers are “standard” caregiving behaviour C₁ or “non-standard” caregiving behaviourC₂. And the strategies for the managers are “proactive” risk management behaviour M₁ or the “passive” risk management M₂. For long-term care facilities, there are four strategy profiles for the caregivers and managers, namely, (C₁, M₁), (C₁, M₂), (C₂, M₁), (C₂, M₂).

Table 2 Matrix for the risk responses of the caregivers and the managers

Next, the decision-making models for the different risk responses of the caregivers and the managers are established respectively. And the optimal strategies will be formulated by comparing the prospective value of their gain from different risk responses. On this basis, the influences of factors such as the stakeholders’ behaviour, risk management cost, risk loss and external regulation on the two parties’ risk response decisions are discussed. Finally, the constraints for the caregivers and the managers to achieve the optimal risk management strategies are analyzed from the perspective of long-term care facilities.

2.3 The decision-making model for the caregivers’ risk responses

According to caregivers’ gain from different risk responses and the corresponding prospective value, the prospective value of the caregivers’ choice of “standard” caregiving behaviour C₁ can be arrived at as follows:

$$ V\left({C}_1\right)=W\left({P}_y\right)V\left({\pi}_{C_{1,}{M}_1}^C-{T}_C\right)+W\left(1-{P}_y\right)V\left({\pi}_{C_{1,}{M}_2}^C-{T}_C\right)=W\left({P}_y\right)V\left({E}_C-{C}_C-{T}_C\right)+W\left(1-{P}_y\right)V\left({E}_C-{C}_C+{K}_M-\left(1-{\gamma}_C\right)\beta L-{T}_C\right) $$

(2)

And the prospective value of the caregivers’ choice of “non-standard” caregiving behaviour C₂ is:

$$ V\left({C}_2\right)=W\left({P}_y\right)V\left({\pi}_{C_{2,}{M}_1}^C-{T}_C\right)+W\left(1-{P}_y\right)V\left({\pi}_{C_{2,}{M}_2}^C-{T}_C\right)=W\left({P}_y\right)V\left({E}_C-{C}_C-\left(1-{\gamma}_M\right)\beta L-{T}_C\right)+W\left(1-{P}_y\right)V\left({E}_C-{K}_C-\beta L-{T}_C\right) $$

(3)

∆V_C, the deviation of the prospective value of the caregivers’ different caregiving behaviour, is:

$$ \Delta {V}_C=V\left({C}_1\right)-V\left({C}_2\right)=W\left({P}_y\right)\left[V\left({E}_C-{C}_C-{T}_C\right)-V\left({E}_C-{C}_C-\left(1-{\gamma}_M\right)\beta L-{T}_C\right)\right]+W\left(1-{P}_y\right)\left[V\left({E}_C-{C}_C+{K}_M-\left(1-{\gamma}_C\right)\beta L-{T}_C\right)-V\left({E}_C-{K}_C-\beta L-{T}_C\right)\right] $$

(4)

Specifically, with reference to Gonzalez and Wu’s (1996,1999) [34, 35] setting of weight functions, assume that $W(x)=\frac{x^k}{{\left(1-x\right)}^k+{x}^k},0<k<1$. If${\pi}_{C_{i,}{M}_j}^C-{T}_C>0$, then the prospective value $V\left({\pi}_{C_{i,}{M}_j}^C\right)={\left({\pi}_{C_{i,}{M}_j}^C-{T}_C\right)}^{\alpha }$; and if ${\pi}_{C_{i,}{M}_j}^C-{T}_C\le 0$, then$\mathrm{the}\ \mathrm{prospective}\ \mathrm{value}\ V\left({\pi}_{C_{i,}{M}_j}^C\right)=-\lambda {\left(-{\pi}_{C_{i,}{M}_j}^C+{T}_C\right)}^{\beta }$, and i, j ∈ {1, 2}. Specifically, 0 < α, β < 1, λ > 1, V^′(x) > 0, α and β respectively indicate the concave and convex degrees of the gain area and the loss area of the value function, reflecting a senior care facility’s different risk attitudes towards gain and loss: The bigger α and β are, the more likely the senior care facility is to take risks; λ represents the degree of loss avoidance of the senior care facility: The bigger λ is, to a great degree the senior care facility avoids losses.

For the caregivers, if the prospective value V(C₁) of “standard” caregiving behaviour C₁ is greater than the prospective value V(C₂) of “non-standard” caregiving behaviour C₂, the caregivers will tend to choose “standard” caregiving behaviour; otherwise, they will prefer “non-standard” caregiving behaviour. It can be easily seen that ∆V_C, the deviation of the caregivers’ prospective value, can be used to judge how the caregivers will choose their caregiving behaviour. When ∆V_C > 0, the caregivers will choose “standard” caregiving behaviour, whereas when ∆V_C ≤ 0, they will choose “non-standard” caregiving behaviour.

•Proposition 1

The prospective value of the caregivers’ choice of “standard” caregiving behaviour is negatively correlated with the cost of their risk responses C_C, negatively correlated with their risk loss sharing ratio β, and positively correlated with their risk loss discount factor γ_C.

Proof: Because

$$ \frac{\partial V\left({C}_1\right)}{\partial {C}_C}=-W\left({P}_y\right){V}^{\prime}\left({E}_C-{C}_C-{T}_C\right)-W\left(1-{P}_y\right){V}^{\prime}\left({E}_C-{C}_C+{K}_M-\left(1-{\gamma}_C\right)\beta L-{T}_C\right)<0 $$

$$ \frac{\partial V\left({C}_1\right)}{\partial \beta }=-\left(1-{\gamma}_C\right) LW\left(1-{P}_y\right){V}^{\prime}\left({E}_C-{C}_C+{K}_M-\left(1-{\gamma}_C\right)\beta L-{T}_C\right)<0 $$

and

$$ \frac{\partial V\left({C}_1\right)}{\partial {\gamma}_C}=-\beta L W\left(1-{P}_y\right){V}^{\prime}\left({E}_C-{C}_C+{K}_M-\left(1-{\gamma}_C\right)\beta L-{T}_C\right)>0 $$

Proposition 1 stands. Proven.

Proposition 1 suggests that the greater the prospective value of the caregivers’ choice of “standard” caregiving behaviour, the more willing they are to choose “standard” caregiving behaviour. Therefore, the cost of risk responses negatively impacts the caregivers’ willingness to choose “standard” caregiving behaviour. Their risk loss sharing ratio negatively influences the caregivers’ willingness to choose “standard” caregiving behaviour. And their risk loss discount factor positively impacts the caregivers’ willingness to choose “standard” caregiving behaviour.

•Proposition 2

When K_M < (1 − γ_C)βL, the prospective value of the caregivers’ choice of “standard” caregiving behaviour is positively correlated with the facility managers’ choice of “proactive” risk management behaviour, whereas whenK_M > (1 − γ_C)βL, the prospective value of the caregivers’ choice of “standard” caregiving behaviour is negatively correlated with the facility managers’ choice of “proactive” risk management behaviour.

Proof: It can be known from Formula (2) that

$$ \frac{\partial V\left({C}_1\right)}{\partial {P}_y}={W}^{\prime}\left({P}_y\right)V\left({E}_C-{C}_C-{T}_C\right)+{W}^{\prime}\left(1-{P}_y\right)V\left({E}_C-{C}_C+{K}_M-\left(1-{\gamma}_C\right)\beta L-{T}_C\right)=\frac{k{\left({P}_y\right)}^{k-1}{\left(1-{P}_y\right)}^{k-1}}{{\left[{\left({P}_y\right)}^k+{\left(1-{P}_y\right)}^k\right]}^2}\left[V\left({E}_C-{C}_C-{T}_C\right)-V\left({E}_C-{C}_C+{K}_M-\left(1-{\gamma}_C\right)\beta L-{T}_C\right)\right]. $$

Since the prospective value function V(x) is a monotonically increasing function on (−∞, +∞),

whenK_M − (1 − γ_C)βL > 0, V(E_C − C_C − T_C) − V(E_C − C_C + K_M − (1 − γ_C)βL − T_C) < 0,and therefore $\frac{\partial V\left({C}_1\right)}{\partial {P}_y}<0$.

When K_M − (1 − γ_C)βL < 0, (E_C − C_C − T_C) − V(E_C − C_C + K_M − (1 − γ_C)βL − T_C) > 0, and therefore $\frac{\partial V\left({C}_1\right)}{\partial {P}_y}>0$. Proven.

Proposition 2 indicates that when the gain the caregivers receive from the government or industry regulators for choosing “standard” caregiving behaviour is greater than the risk loss they share, they might, driven by a speculative mentality, expect to benefit from the punishment on the managers’ “passive” risk management behaviour. As a result, their perceived prospective value of providing “standard” caregiving behaviour will decrease, and their willingness for choosing “standard” caregiving behaviour might also reduce. In comparison, when the gain the caregivers receive from the government or industry regulators for choosing “standard” caregiving behaviour is less than the risk loss they share, their perceived prospective value of providing “standard” caregiving behaviour will increase, leading to their increased willingness to choose “standard” caregiving behaviour. Therefore, when the government and industry regulators formulate reward strategies for the caregivers, they need to bear in mind that higher reward does not necessarily lead to better results. Instead, they must limit the reward to a reasonable range so as to effectively motivate the caregivers to adopt “standard” caregiving behaviour.

2.4 The decision-making model for the facility managers’ risk responses

According to the managers’ gain from different risk response strategies and their corresponding prospective value, the prospective value of the managers’ choice of “proactive” risk management behaviour M₁ is:

$$ V\left({M}_1\right)=W\left({P}_x\right)V\left({\pi}_{C_{1,}{M}_1}^M-{T}_M\right)+W\left(1-{P}_x\right)V\left({\pi}_{C_{1,}{M}_2}^M-{T}_M\right)=W\left({P}_x\right)V\left({E}_M-{C}_M-{T}_M\right)+W\left(1-{P}_x\right)V\left({E}_M-{C}_M+{K}_C-\left(1-{\gamma}_M\right)\beta L-{T}_M\right) $$

(5)

The prospective value of the managers’ choice of “passive” risk management behaviour M₂ is:

$$ V\left({M}_2\right)=W\left({P}_x\right)V\left({\pi}_{C_{2,}{M}_1}^M-{T}_M\right)+W\left(1-{P}_x\right)V\left({\pi}_{C_{2,}{M}_2}^M-{T}_M\right)=W\left({P}_x\right)V\left({E}_M-{K}_M-\left(1-{\gamma}_C\right)\left(1-\beta \right)L-{T}_M\right)+W\left(1-{P}_x\right)V\left({E}_M-{K}_M-\left(1-\beta \right)L-{T}_M\right) $$

(6)

∆V_M, the deviation of the prospective value of the managers’ different risk management behaviour, can be indicated as:

$$ \Delta {V}_M=V\left({M}_1\right)-V\left({M}_2\right)=W\left({P}_x\right)\left[V\left({E}_M-{C}_M-{T}_M\right)-V\left({E}_M-{K}_M-\left(1-{\gamma}_C\right)\left(1-\beta \right)L-{T}_M\right)\right]+W\left(1-{P}_x\right)\left[V\left({E}_M-{C}_M+{K}_C-\left(1-{\gamma}_M\right)\left(1-\beta \right)L-{T}_M\right)-V\left({E}_M-{K}_M-\left(1-\beta \right)L-{T}_M\right)\right] $$

(7)

In Formulas (5) to (7), if${\pi}_{C_{i,}{M}_j}^M-{T}_M>0$, then $V\left({\pi}_{C_{i,}{M}_j}^M\right)={\left({\pi}_{C_{i,}{M}_j}^M-{T}_M\right)}^{\alpha }$; and if${\pi}_{C_{i,}{M}_j}^M-{T}_M\le 0$, then $V\left({\pi}_{C_{i,}{M}_j}^M\right)=-\lambda {\left(-{\pi}_{C_{i,}{M}_j}^M+{T}_M\right)}^{\beta }$, and i, j ∈ {1, 2}.

This study uses ∆V_M, the deviation of the prospective value of the managers’ different risk management behaviour, to judge how managers will make risk management decisions. When ∆V_M > 0, the prospective value V(M₁) of “proactive” risk management behaviour M₁ is greater than the prospective value V(M₂) of “passive” risk management behaviour M₂, in which case the managers will choose “proactive” risk management behaviour; in comparison, when ∆V_M ≤ 0, the prospective value V(M₁) of “proactive” risk management behaviour M₁ is lower than the prospective value V(M₂) of “passive” risk management behaviourM₂, in which case the managers will choose “passive” risk management behaviour.

•Proposition 3

The prospective value of the managers choice of “proactive” risk management behaviour is negatively correlated with their risk response cost C_M, negatively correlated with their risk loss sharing ratio β, and positively correlated with their risk loss discount factor γ_M.

Proof: Because

$$ \frac{\partial V\left({M}_1\right)}{\partial {C}_M}=-W\left({P}_x\right){V}^{\prime}\left({E}_M-{C}_M-{T}_M\right)+W\left(1-{P}_x\right){V}^{\prime}\left({E}_M-{C}_M+{K}_C-\left(1-{\gamma}_M\right)\left(1-\beta \right)L-{T}_M\right)<0 $$

$$ \frac{\partial V\left({M}_1\right)}{\partial \left(1-\beta \right)}=-\left(1-{\gamma}_M\right) LW\left(1-{P}_x\right){V}^{\prime}\left({E}_M-{C}_M+{K}_C-\left(1-{\gamma}_M\right)\left(1-\beta \right)L-{T}_M\right)<0 $$

and

$$ \frac{\partial V\left({M}_1\right)}{\partial {\gamma}_M}=\left(1-\beta \right) LW\left(1-{P}_x\right){V}^{\prime}\left({E}_M-{C}_M+{K}_C-\left(1-{\gamma}_M\right)\left(1-\beta \right)L-{T}_M\right)>0 $$

Proposition 3 stands. Proven.

Proposition 3 suggests that the greater the prospective value of the managers’ choice of “proactive” risk management behaviour, the more willing they are to choose “proactive” risk management behaviour. Therefore, the risk response cost negatively impacts the managers’ willingness to choose “proactive” risk management behaviour. The managers’ risk loss sharing ratio negatively influences their willingness to choose “proactive” risk management behaviour. And the managers’ loss discount factor positively impacts their willingness to choose “proactive” risk management behaviour.

•Proposition 4

When K_C < (1 − γ_M)(1 − β)L, the managers’ willingness to choose “proactive” risk management behaviour is positively correlated with the caregivers’ probability of choosing “standard” caregiving behaviour, whereas when K_C > (1 − γ_M)(1 − β)L, the managers’ willingness to choose “proactive” risk management behaviour is negatively correlated with the caregivers’ probability of choosing “standard” caregiving behaviour.

Proof: The process of proof is similar to that of Proposition 2. It can be easily proven that Proposition 4 stands.

It can be known from Proposition 4 that once the gain the managers receive from their choice of “proactive” risk management behaviour exceeds the risk loss they share, they might develop the speculative mentality of benefiting from the punishment on the caregivers’ “non-standard” caregiving behaviour. As a result, their willingness to choose “proactive” risk management behaviour might reduce. In comparison, when the gain the managers receive from their choice of “proactive” risk management behaviour is less than the risk loss they share, their willingness to choose “proactive” risk management behaviour might increase.

Propositions 2 and 4 suggest that the government and industry regulators need to weigh and balance the pros and cons of the reward and punishment when formulating the reward and punishment mechanism for long-term care facilities. They must limit the reward and punishment within a proper range to truly give full play to the incentive and restrictive role of the optimal risk management behaviour on the staff in long-term care facilities.

2.5 The decision-making model for risk responses under centralized decisions

To analyze the optimal strategies for the risk responses of long-term care facilities and their constraints, this section compare the caregivers’ two strategies and the facility managers’ two strategies respectively based on the matrix of the risk responses of the caregivers and the managers, and underlines the best strategies. In the end, the strategy profile with all the strategies underlined is the equilibrium.

For the caregivers, if C_C < K_C + (1 − γ_M)βL, then ${\pi}_{C_{1,}{M}_1}^C={E}_C-{C}_C>{\pi}_{C_{2,}{M}_1}^C={E}_C-{K}_C-\left(1-{\gamma}_M\right)\beta L$. If C_C < K_C + K_M + γ_CβL, then ${\pi}_{C_{1,}{M}_2}^C={E}_C-{C}_C+{K}_M-\left(1-{\gamma}_C\right)\beta L>{\pi}_{C_{2,}{M}_2}^C={E}_C-{K}_C-\beta L$. Therefore, when the cost for the caregivers to engage in “standard” caregiving behaviour C_C <  min {K_C + (1 − γ_M)βL, K_C + K_M + γ_CβL}, “standard” caregiving behaviour C₁ is caregivers’ dominant strategy.

For the facility managers, if C_M < K_M + (1 − γ_C)(1 − β)L, then ${\pi}_{C_{1,}{M}_1}^M={E}_M-{C}_M>{\pi}_{C_{1,}{M}_2}^M={E}_M-{K}_M-\left(1-{\gamma}_C\right)\left(1-\beta \right)L$. If C_M < K_C + K_M + γ_M(1 − β)L, then ${\pi}_{C_{2,}{M}_1}^M={E}_M-{C}_M+{K}_C-\left(1-{\gamma}_M\right)\left(1-\beta \right)L>{\pi}_{C_{2,}{M}_2}^M={E}_M-{K}_M-\left(1-\beta \right)L$. Therefore, when the cost for the managers to engage in “proactive” risk management behaviour C_M <  min {K_M + (1 − γ_C)(1 − β)L, K_C + K_M + γ_M(1 − β)L}, “proactive” risk management behaviour M₁ is the managers’ dominant strategy.

The optimal state of risk management in long-term care facilities is one in which both the facility managers and the caregivers attach enough importance to and actively engage in COVID-19 risk management activities and “standard” caregiving behaviour respectively to minimize COVID-19 risks and related risk loss for the facilities. Therefore, if the cost of the risk responses of both the caregivers and the managers in long-term care facilities meets the following two conditions,

$$ {C}_C<\mathit{\min}\left\{{K}_C+\left(1-{\gamma}_M\right)\beta L,{K}_C+{K}_M+{\gamma}_C\beta L\right\} $$

(8)

$$ {C}_M<\mathit{\min}\left\{{K}_M+\left(1-{\gamma}_C\right)\left(1-\beta \right)L,{K}_C+{K}_M+{\gamma}_M\left(1-\beta \right)L\right\} $$

(9)

the system will converge to the equilibrium strategy (C_1,M₁) (see Table 3). At this point, the facility managers attach importance to and proactively carry out COVID-19 risk management activities; the caregivers proactively engage in “standard” caregiving behaviour; and the COVID-19 risk intensity in long-term care facilities is minimized, so is the risk loss.

Table 3 The equilibrium strategy between the caregivers and the managers under centralized decisions.

The C_C constraints indicate that, in general, the cost for the caregivers to proactively engage in “standard” caregiving behaviour should be lower than their perceived value of the punishment on their engagement in “non-standard” caregiving behaviour, their perceived value of their gain from the managers’ “passive” COVID-19 risk management behaviour, and their perceived value of the risk loss reduction. It should also be lower than their perceived value of the risk loss reduced by the managers’ “proactive” COVID-19 risk management behaviour.

The C_M constraints suggest that, in general, the cost for the long-term care facilities managers to proactively engage in COVID-19 risk management should be lower than their perceived value of the punishment on their engagement in “passive” risk management behaviour, their perceived value of the risk loss caused to long-term care facilities by the caregivers’ “non-standard” caregiving behaviour, and their perceived value of COVID-19 risk loss reduction. Besides, it should not exceed their perceived value of the risk loss reduced by the caregivers’ engagement in “standard” caregiving behaviour.

3 Simulation analysis of the decision-making on risk responses

In order to visualize the impact of such factors as industry personnel factors (the probability of risk response choices reflects the general characteristics of the risk responses of the industry personnel), risk response cost as well as risk loss incurred to the facilities on the risk response decisions of the caregivers and the managers, this section employs numerical simulation to simulate the evolution of the risk response decisions of the caregivers and the managers.

In the following simulation analysis, first, with reference to Gonzalez and Wu’s (1996, 1999) [35, 36] setting of weight function, assume the parameter of the weight function k = 0.61. And with reference to Kahneman and Tversky’s (1992) [37] setting of the parameter of value function in their study, assume that α = 0.89, β = 0.92, λ = 2.25. Second, fix the values of the parameters as follows: E_C = 10, E_M = 15, C_C = 5, C_M = 6, γ_C = 0.4, γ_M = 0.45, L = 50, K_C = K_M = 5, P_x = 0.5, P_y = 0.6, T_C = T_M = 0, (All these parameters are dimension-free. The dimensions of the parameters can be reasonably set according to the specific question.). Finally, when examining the change in one parameter (for example, the increase of P_x from 0 to 1), fix the rest of the parameters as indicated above, and only analyze the impact of the said parameter on the variable in question (such as V(C₁)).

3.1 The impact of the industry personnel on risk response decision-making

Respectively set the variable variables of the model as P_y, the probability for the managers to choose “proactive” risk management behaviour (increase from 0 to 1 with a step size of 0.1) and P_x, the probability for the caregivers to choose “standard” caregiving behaviour (increase from 0 to 1 with a step size of 0.1). The simulation results are shown in Fig. 3. It can be seen from Fig. 3 that as P_y, the probability for the managers to choose “proactive” risk management behaviour, increases, the prospective value of the caregivers’ “standard” and “non-standard” caregiving behaviour will both increase, and the prospective value of the caregivers’ “standard” caregiving behaviour is always greater than the prospective value of their “non-standard” caregiving behaviour. Similarly, it can be seen that the probability for the caregivers to choose “standard” caregiving behaviour is positively correlated with the managers’ willingness to choose “proactive” risk management. In other words, industry personnel (the probability of risk response choices) have a positive impact on the risk response decision-making of the caregivers and the managers.

Fig. 3

The impact of risk response probabilities P_x and P_y on risk response decision-making

3.2 The impact of management cost on risk response decision-making

Respectively set the variable variables of the model as C_C, the cost of the caregivers’ choice of “standard” caregiving behaviour (increase from 1 to 10 with a step size of 1) and C_M, the cost of the managers’ choice of “proactive” risk management behaviour (increase from 1 to 10 with a step size of 1). The simulation results are shown in Fig. 4. It can be seen from Fig. 4 that as C_C, the cost of the caregivers’ choice of “standard” caregiving behaviour, increases, the prospective value of the caregivers’ “standard” caregiving behaviour gradually decreases, the prospective value of their “non-standard” caregiving behaviour remains unchanged (the prospective value is always negative), and the willingness for the caregivers to choose “standard” caregiving behaviour gradually reduces. However, because the prospective value of the caregivers’ “standard” caregiving behaviour is always greater than the prospective value of their “non-standard” caregiving behaviour, choosing “standard” caregiving behaviour remains the optimal decision. Similarly, it can be seen that the greater the C_M, the cost of the managers’ choice of “proactive” risk management behaviour, the smaller the prospective value of the managers’ engagement in “proactive” risk management behaviour, and the lower their willingness to engage in “proactive” risk management. However, since the prospective value of the managers’ “proactive” risk management is always greater than the prospective value of their “passive” risk management, choosing “proactive” risk management remains the optimal decision for the managers. In other words, the management cost of risk responses has a negative impact on the risk response decision-making of the caregivers and the managers.

Fig. 4

The impact of risk management cost C_C and C_M on risk response decision-making

3.3 The impact of risk loss on risk response decisions

First, respectively set the variable variables of the model as γ_C, the risk loss discount factor of the caregivers’ choice of “standard” caregiving behaviour (increase from 0.1 to 1 with a step size of 0.1) and γ_M, the risk loss discount factor of the managers’ choice of “proactive” risk management behaviour (increase from 0.1 to 1 with a step size of 0.1). The simulation results are shown in Fig. 5. Then, set the variable of the model as L, the risk loss of long-term care facilities (increase from 20 to 70 with a step size of 5). Afterwards, analyze the impact of the risk loss of long-term care facilities on the risk response decision-making of the caregivers and the managers. The simulation results are shown in Fig. 6.

Fig. 5

The impact of the risk loss discount factors γ_C and γ_M on risk response decision-making

Fig. 6

The impact of the risk loss L on risk response decision-making

It can be seen from Fig. 5 that as γ_C, the risk loss discount factor of the caregivers’ choice of “standard” caregiving behaviour, increases, the prospective value of the caregivers’ “standard” caregiving behaviour also increases, and the prospective value of their “non-standard” caregiving behaviour remains unchanged (the prospective value is negative), causing their willingness to choose “standard” caregiving behaviour to gradually increase. For the managers, the larger the γ_M, the risk loss discount factor of their choice of “proactive” risk management behaviour, the greater the prospective value of their engagement in “proactive” risk management behaviour, and correspondingly the stronger their willingness to engage in “proactive” risk management. In other words, risk loss discount factor has a positive impact on the risk response decision-making of the caregivers and the managers.

It can be seen from Fig. 6 that the larger the L, the risk loss of long-term care facilities, the smaller the prospective values of both the caregivers’ “standard” caregiving behaviour and their “non-standard” caregiving behaviour. Nevertheless, since the prospective value of the caregivers’ “non-standard” caregiving behaviour reduces faster, their willingness to choose “standard” caregiving behaviour still gradually increases. A similar conclusion can be drawn for the managers. Therefore, both risk loss discount factor and the risk loss of long-term care facilities have a positive impact on the risk response decision-making of the caregivers and the managers.

4 Risk management and control strategies based on the prospect theory

(1) Noel et al. [38] found in their research that risk perception is the subjective judgment people make about the characteristics and severity of a particular risk and that it is correlated with individual behaviour. The two-factor theory of risk suggests that risk perception involves value judgments about the subjective probability of the occurrence of decision consequences and the severity of the consequences of wrong decisions. The lower the level of risk perception, the greater the tendency to ignore the risk instead of proactively taking precautions to reduce loss when faced with a potential risk threat. On the contrary, the higher the level of risk perception, the greater the tendency to choose to proactively avoid COVID-19 risks or take precautions to reduce loss caused by COVID-19 risks when faced with potential COVID-19 risks. Factors influencing the perception of COVID-19 risks include the perception bias towards COVID-19 risks and the acceptable level of risks.

To enable the caregivers and the managers of long-term care facilities to converge to the equilibrium strategy (C₁, M₁)with the largest probability, it is necessary to have the two conditions in Formulas (8) and (9) met. However, the existence of risk perception differences and prospect theory effect may make it difficult for the system to converge to the equilibrium strategy (C₁, M₁).

The actual COVID-19 risk management cost is less than the perceived COVID-19 risk management cost. In other words, V(C_C) ≥ C_Cand V(C_M) ≥ C_M. When the facility managers or the caregivers choose to proactively carry out COVID-19 risk management, the probability of occurrence of the said behaviour p_x = p_y = 1 According to the prospect theory, it can be known that:

$$ V\left({C}_C\right)=W\left({p}_x\right)V\left({C}_C\right)+W\left(1-{p}_x\right)V(0)=\mathrm{W}(1)V\left({C}_C\right)+\mathrm{W}(0)V(0)\ge {C}_C $$

$$ V\left({C}_M\right)=W\left({p}_y\right)V\left({C}_M\right)+W\left(1-{p}_y\right)V(0)=\mathrm{W}(1)V\left({C}_M\right)+\mathrm{W}(0)V(0)\ge {C}_M $$

In the above formulas, C_C and C_M respectively refer to the actual cost for the facility managers and the caregivers to proactively carry out risk management.

(2) The facility managers and the caregivers are prone to a fluke mentality and optimism bias when making judgments about the value of strategy choices. Optimism bias manifests as a form of unrealistic optimism. People are more prone to optimism bias when considering their own risks, believing that they are less likely to experience negative events and more likely to experience positive events than ordinary people [39], and they tend to believe that things will turn out well. In general, optimism bias lowers the level of risk perception among managers and staff, thus causing them to underestimate the existence of COVID-19 risks, reduce problem-solving efforts, and reduce the willingness to prevent COVID-19 risk. The optimism biases of the facility managers and caregivers cause them to hope for occasional non-occurrence of risk accidents and loss and be more prone to underestimating the probability of occurrence of risky events and the probability of punishment that may result from passive risk management practices, while knowing that passive risk management may result in risk loss. Besides, according to the prospect theory, except for events of minimal probability, weighting function has the characteristic of W(p_i) ≤ p_i. And when ∆ϖ_i > 0, V(∆ϖ_i) has the features of a convex function. It can thus be seen that W(p₁) ≤ p₁, W(p₂) ≤ p₂, W(p₃) ≤ p₃, V(d_C) ≤ d_C, V(d_M) ≤ d_M and V(l) ≤ l. Specifically, P₁ refers to the objective probability of occurrence of COVID-19 risk accidents: P₂ and P₃ respectively refer to the objective probability for facility managers and that for caregivers to be punished due to the occurrence of COVID-19 risk; d_C and d_M refer to the actual punishment cost paid by the facility managers and caregivers respectively for adopting passive risk management strategies; and l represents the actual loss caused by the occurrence of COVID-19 risk. Thus:

$$ L=W\left({p}_1\right)V(l)+W\left(1-{p}_1\right)\mathrm{V}(0)=W\left({p}_1\right)V(l)\le {p}_1l $$

$$ {D}_C=W\left({p}_2\right)V\left({d}_C\right)+W\left(1-{p}_2\right)\mathrm{V}(0)=W\left({p}_2\right)V\left({d}_C\right)\le {p}_2{d}_C $$

$$ {D}_M=W\left({p}_3\right)V\left({d}_M\right)+W\left(1-{p}_3\right)\mathrm{V}(0)=W\left({p}_3\right)V\left({d}_M\right)\le {p}_3{d}_M $$

(3) According to the reflection effect put forward in the prospect theory, people’s preferences for gains and losses are asymmetric. And when faced with the prospect of gain (or profits), people tend to be risk averse, whereas when faced with the prospect of possible loss, they tend to be risk seeking. When faced with the certainty of loss, the managers and the caregivers in long-term care facilities, as decision-makers with bounded rationality, tend to have a high appetite for risks. Specifically, they prefer to give up the certain cost incurred by a proactive risk management strategy in favour of taking the risk of choosing a passive risk management strategy and facing the uncertainty of loss and punishment caused by risk accidents, making it difficult for the system to converge at the equilibrium strategy (C₁, M₁). As a result, the cost of COVID-19 risks of the facility managers and caregivers tends to be overestimated and the risk loss underestimated, causing C_C and C_M to be larger than necessary while D_C, D_M and L smaller than needed, and both parties have the tendency to take the risk of choosing passive risk management strategies. In reality, there are inevitable reasons as to why the facility managers and caregivers attach insufficient importance to risk management and adopt passive risk management strategies. When long-term care facilities meet the above conditions, the optimal equilibrium strategy (C₁, M₁) will be realized, where both the facility managers and the caregivers attach important to and proactively carry out risk management activities or engage in “standard” caregiving behaviour, thereby reducing the probability of occurrence of COVID-19 risks and minimize the loss incurred by COVID-19 risks to long-term care facilities.

5 Conclusions

This study uses the prospect theory to establish the decision-making models for the different risk responses of the caregivers and the managers, formulates the optimal strategies by comparing the prospective value of their gain from different risk responses. Furthermore, the influences of factors such as the stakeholders’ behaviour, risk management cost, risk loss and external regulation on the two parties’ risk response decisions are discussed. Finally, the constraints for the caregivers and the managers to achieve the optimal risk management strategies are analyzed from the perspective of long-term care facilities. This study reveals the internal mechanism of risk behavior decision-making by the caregivers and managers in long-term care facilities, deepens the cognition of COVID-19 risks by caregivers and managers, and contributes to the transformation, decomposition and effective prevention and control of COVID-19 risks in long-term care facilities.