Minimizing Societal Cost of Epidemics: Optimal Catastrophe Insurance Compensation Allocation

ABSTRACT

This study investigates optimal strategies for allocating catastrophe insurance compensation to contain infectious disease outbreaks. We extend a classic SIR model by allowing insurance payouts to fund quarantine of infected individuals, thereby slowing transmission dynamics. We derive the basic reproduction number $R_0$ and analyze the existence and stability of both disease‐free and endemic equilibria. A key innovation is the linkage of real‐time $R_0$ with the allocation of compensation funds. We obtain a closed‐form expression for the optimal usage strategy ${\alpha }^{\ast }$ that minimizes the real‐time $R_0$ during an epidemic. Our results show that compensation should be prioritized toward subpopulations with high transmission potential or high prevalence, and toward those that can be quarantined cost‐effectively. Moreover, enhancing the quarantine efficiency of compensation significantly strengthens epidemic control. The optimal strategy reduces real‐time $R_0$, which in turn limits outbreak damage, lowers insurance claims, and thereby cuts premiums, ultimately minimizing the total societal cost of epidemic mitigation. This framework offers actionable insights for policymakers and advances the literature at the intersection of epidemic management and catastrophe insurance.

1. Introduction

1.1. Background

In December 2019, the COVID‐19 pandemic emerged and spread rapidly worldwide. As of 2023, data from the World Health Organization indicate over 760 million confirmed cases and 6.9 million deaths globally.¹ The outbreak placed unprecedented strain on public health systems across many countries and regions. As a novel infectious disease, COVID‐19 proved difficult for individuals and public health authorities to identify and contain in the early stages of transmission. As the epidemic worsened, the rapid surge in cases generated substantial control‐related costs, leading to resource shortages that hindered subsequent prevention efforts. Beyond its direct threat to global health, the pandemic also caused significant economic losses and triggered a series of secondary crises.

These realities underscore that effective epidemic control depends not only on medical advances but also on the robustness of public health system design. Implementing containment measures as effectively as possible during the early stages, despite limited resources, has become an increasingly urgent challenge. Beyond the efficient allocation of medical resources, there is growing recognition of the need for more effective epidemic risk transfer mechanisms to alleviate financial pressures.

Epidemic insurance has long been recognized as a key risk transfer tool for infectious disease outbreaks. The compensation paid upon a covered event can serve as a critical source of funding for epidemic prevention, particularly in the initial stage of an outbreak. This study explores the optimal strategy for using catastrophe insurance compensation based on an epidemiological model.

1.2. Literature Review

This section reviews the literature on epidemic insurance and its role in risk transfer. Early work by Loue (1993) examined insurance products designed to address epidemic risks. Feng and Garrido (2011) bridged epidemiological and actuarial modeling by introducing insurance frameworks that quantify infection risk and structure financial arrangements between insurers and the insured. Subsequent research expanded this line of inquiry. Lefèvre and Picard (2015) linked the finite‐time ruin probability in insurance with the final outcome distribution in epidemic models. Building on an extended SIR framework, Lefèvre et al. (2017) proposed an insurance scheme in which premiums during outbreaks are based on the expected duration of susceptibility.

More recent contributions include Chernov et al. (2021), who examined fair premium rates within an SEIR model, Tran (2024), who introduced a Markov multi‐state model for epidemic insurance, and Zhai et al. (2024), who explored the role of vaccination in epidemic insurance design. Beyond pricing, the issue of insurer insolvency has received sustained attention. Feng and Garrido (2011) proposed a premium collection strategy to prevent insolvency, while Lefèvre and Picard (2018) discussed final outcomes in controlled epidemic models. The ruin problem under the SIS model was subsequently analyzed by Lefèvre and Simon (2021);;2022).

Despite the existence of insurance‐based risk transfer mechanisms for epidemic risks, fundamental barriers to insurability may constrain their capacity to absorb the economic losses caused by pandemics. Nell et al. (2009) argue that the potential for large loss accumulation and the presence of external moral hazard deter private insurers from underwriting epidemic risk. Richter and Wilson (2020) emphasize that COVID‐19 highlighted the need for enhanced solvency, liquidity, and improved business interruption coverage, noting that pandemic‐related business interruptions may be inherently uninsurable. Kraut and de Kuiper (2021) observe that while epidemic risk transfer instruments exist, insurability challenges limit their capacity to cover large‐scale economic losses.

In response to these limitations, Hartwig et al. (2020) propose that pooling risks over time could reduce required capital and lower insurance costs, though such an approach would likely require government involvement given its capacity for borrowing and taxation. Bi and Liu (2023) analyze Germany's statutory health insurance fund during COVID‐19, underscoring the role of public insurance mechanisms in epidemic prevention and control.

Beyond these approaches, catastrophe insurance plays an important role in mitigating epidemic risks. Gründl and Regele (2020) propose the use of insurance‐linked bonds for pandemic risks. Liu et al. (2021) examine biotechnology investment portfolios structured with pandemic bonds as liabilities. Medders and Schwarcz (2022) explore pandemic catastrophe bonds as a securitization tool for pandemic risk, addressing unmet capital needs in the reinsurance market. Schmitt and Spaeter (2023) show how catastrophe bonds can provide pandemic business interruption coverage within a public–private insurance framework. Chen et al. (2023a) explore catastrophe mortality bonds for managing large‐scale mortality risks in the post‐pandemic context.

For the pricing of such instruments, Gründl et al. (2021) note that premium markups for pandemic insurance are comparable to the top 20% observed in natural catastrophe markets. Peng et al. (2022) develop a pricing model for pandemic European options using the infected population as the underlying variable. Zheng and Mamon (2023) evaluate the World Bank's pricing framework for pandemic bonds. Manathunga and Deng (2023) apply the Hull–White and stochastic logistic growth models to price pandemic bonds. Chatoro et al. (2023) investigate key pricing factors in the catastrophe bond market. Abdikerimova et al. (2024) introduce pandemic‐linked securities that combine financial derivatives with a peer‐to‐peer structure. Chen et al. (2024) employ a machine learning approach for bond pricing.

Beyond pricing, the literature also examines the broader role of insurance in mitigating pandemic risks. Salanié and Treich (2020) analyze the externalities of individual preventive actions, exploring when social planners should promote or discourage self‐protection during pandemics. Ho et al. (2023) develop an integrated health insurance model that optimizes vaccination rewards and cost‐sharing to reduce healthcare costs and prevent seasonal flu outbreaks. Cui et al. (2024) investigate optimal subsidy policies within an insurance framework for effective epidemic control. Huang et al. (2024) propose two capital market–based financing mechanisms: pandemic bonds to hedge against severe outbreaks and endemic swaps to manage recurrent risks. Hong and Seog (2023) evaluate the efficiency of health insurance systems in managing infectious diseases, concluding that public insurance systems outperform market‐based alternatives due to their capacity to implement tracking technologies and enforce social distancing.

1.3. Motivation and Contributions

The literature shows that existing studies on epidemic insurance have focused on pricing, solvency, and product design. Less attention has been paid to how insurance compensation can be used for epidemic intervention. In particular, the link between insurance mechanisms and disease transmission remains underexplored. This gap is important because during the early phase of an outbreak, a rise in infections coincides with shortages in public healthcare resources, and insurance compensation, if allocated strategically, could fund prevention and containment efforts. This study explores how insurance‐based financial mechanisms can mitigate disease transmission.

This work makes three distinct contributions to the existing literature. First, we focus on the early phase of an epidemic and develop an insurance scheme grounded in epidemiological models. Unlike existing literature, we conceptualize insurance compensation as a resource managed by policymakers, establishing a link between the insurance scheme and the basic reproduction number, $R_0$. Second, we account for different categories of infected individuals and types of healthcare resources to determine the optimal allocation of insurance compensation that minimizes $R_0$. This approach highlights the role of insurance in managing financial risks and public health outcomes. Third, we show that this optimal compensation strategy reduces premium costs. Efficient use of insurance compensation lowers the insurance sector's risk exposure and decreases the overall societal cost of epidemic management.

This paper is structured as follows. Section 2 constructs an epidemic model with multiple infected populations, derives the basic reproduction number $R_0$, and analyzes the existence and stability of equilibria and the persistence of the epidemic. Section 3 develops an insurance model linked to $R_0$ and formulates an optimization problem for the use of insurance compensation. Section 4 provides the analytical solution to this optimization and discusses its relationship with key epidemiological parameters. Section 5 presents numerical simulations to validate the theoretical results. Section 6 compares our findings with existing literature. Section 7 concludes the paper.

2. Epidemiology Model

2.1. Transmission Dynamics

Based on the early characteristics of the COVID‐19 outbreak in Wuhan, China, we use the SIR model to describe the interaction among three population compartments. Let $N(t)$ denote the total population at time $t$. The SIR model divides the population into susceptible individuals $S(t)$, infected individuals $I(t)$, and recovered individuals $R(t)$. Susceptible individuals have no immunity to the disease. Infected individuals can transmit the disease through contact with susceptible individuals. Recovered individuals are assumed to have immunity post‐recovery, and we follow the common assumption in the literature that individuals who die from the disease are excluded from the recovered group (Sun et al. 2020; Gatto et al. 2020).

We extend the basic SIR model by incorporating two key features: infection severity and quarantine status.

2.1.1. Infection Severity: Mildly and Severely Infected

COVID‐19 manifests in varying degrees of severity. According to a report from the Chinese Center for Disease Control and Prevention, among approximately 44,500 confirmed cases, 81% were mild (no or mild pneumonia) and 19% were severe (dyspnea, hypoxia, or significant lung involvement) or critical (respiratory failure, shock, or multiple organ dysfunction) (Wang et al. 2020; Guan et al. 2020; Chan et al. 2020; Zhou et al. 2020). In our model, we assume that individuals initially experience mild symptoms and may progress to severe illness due to health deterioration or insufficient medical intervention. The infected population $I(t)$ is therefore divided into mildly infected individuals $I_M(t)$ and severely infected individuals $I_S(t)$.

These two groups play different roles in epidemic transmission. Mildly infected individuals, due to less severe symptoms, are more likely to maintain direct contact with others. Severely infected individuals, while generally less able to engage in direct contact, may be more contagious due to symptoms such as frequent coughing and higher viral loads. We do not impose specific assumptions on the relative magnitudes of the effective contact rates for these two groups, allowing flexibility in modeling their impact.

2.1.2. Quarantine Status: Quarantined and Non‐Quarantined Infected

In addition to infection severity, we distinguish between quarantined and non‐quarantined infected individuals. Initially, both mildly and severely infected individuals are non‐quarantined and can transmit the virus. Over time, infected individuals may be quarantined, which limits their ability to spread the disease. We assume that quarantined individuals are isolated and receiving treatment, rendering them non‐infectious.

Combining infection severity and quarantine status, the infected population $I(t)$ is divided into four subgroups: quarantined mildly infected individuals $I_M^Q(t)$ and quarantined severely infected individuals $I_S^Q(t)$, both non‐infectious, and non‐quarantined mildly infected individuals $I_M^U(t)$ and non‐quarantined severely infected individuals $I_S^U(t)$, both infectious. This classification allows for a more nuanced understanding of transmission dynamics, accounting for both infection severity and the impact of quarantine measures.

2.1.3. Fatality and Recovery Pathways

In our model, only mildly infected individuals $I_M(t)$ can recover directly from infection. The recovery pathways are as follows:

$$\text{Susceptible}\to \text{Mildly infected}\to \text{Recovery},\text{or}$$

$$\text{Susceptible}\to \text{Mildly infected}\to \text{Severely infected}\to \text{Mildly infected}\to \text{Recovery}.$$

For fatality, all groups are subject to a common natural mortality rate $d$. Only severely infected individuals $I_S(t)$ face additional mortality due to disease progression. Both quarantined $I_S^Q(t)$ and non‐quarantined $I_S^U(t)$ severely infected populations are subject to infection‐induced fatality rates ${\overline{d}}_Q$ and ${\overline{d}}_U$, respectively. The fatality pathway is:

$$\text{Susceptible}\to \text{Mildly infected}\to \text{Severely infected}\to \text{Fatality}.$$

2.1.4. Dynamics of the Susceptible Population

New infections occur through effective contact between susceptible individuals and infectious individuals. Following the SIR framework, the number of new infections per unit time is ${\beta }_{M}S(t)I_M^U(t)+{\beta }_{S}S(t)I_S^U(t)$, where ${\beta }_M$ and ${\beta }_S$ are the effective contact rates for non‐quarantined mildly and severely infected individuals. Quarantined individuals $I_M^Q(t)$ and $I_S^Q(t)$ do not transmit the virus and are excluded from the infection term. We assume that newborns are susceptible (Sun et al. 2020; Gatto et al. 2020).

Therefore, the dynamics of the susceptible population are given by:

$$\dot{S}(t)=-{\beta }_{M}S(t)I_M^U(t)-{\beta }_{S}S(t)I_S^U(t)-dS(t)+gN(t)+A_S,$$

where $A_S>0$ is the influx of susceptible individuals immigrating into the region. The parameters $d$ and $g$ are the natural death and birth rates, respectively, with $d>g$.

2.1.5. Dynamics of the Infected Population

Susceptible individuals who become infected initially enter the non‐quarantined mildly infected group $I_M^U(t)$. From there, they may either progress to non‐quarantined severe infection $I_S^U(t)$ at rate $w_U$ or be quarantined to $I_M^Q(t)$ at rate $q_M$. Those in $I_S^U(t)$ may improve back to $I_M^U(t)$ at rate $b_U$ or be quarantined to $I_S^Q(t)$ at rate $q_S$. Quarantined mildly infected individuals $I_M^Q(t)$ may deteriorate to $I_S^Q(t)$ at rate $w_Q$, and quarantined severely infected individuals $I_S^Q(t)$ may improve to $I_M^Q(t)$ at rate $b_Q$. Once quarantined, individuals cannot return to a non‐quarantined state; they either recover or die. Recovery rates for mildly infected individuals are $v_U$ for non‐quarantined and $v_Q$ for quarantined. In addition, quarantined individuals receive timely medical intervention, resulting in lower mortality rates and higher recovery rates than non‐quarantined individuals.

Therefore, the complete nonlinear dynamical system is:

$$\left\{\begin{array}{cc}& {\dot{I}}_M^U(t)={\beta }_{M}S(t)I_M^U(t)+{\beta }_{S}S(t)I_S^U(t)-(v_U+q_M+d+w_U)I_M^U(t)+b_U{I}_S^U(t)\hfill \\ & {\dot{I}}_S^U(t)=w_U{I}_M^U(t)-(b_U+q_S+d+{\overline{d}}_U)I_S^U(t)\hfill \\ & {\dot{I}}_M^Q(t)=q_M{I}_M^U(t)+b_Q{I}_S^Q(t)-(v_Q+d+w_Q)I_M^Q(t)\hfill \\ & {\dot{I}}_S^Q(t)=q_S{I}_S^U(t)+w_Q{I}_M^Q(t)-(b_Q+d+{\overline{d}}_Q)I_S^Q(t)\hfill \\ & \dot{S}(t)=-{\beta }_{M}S(t)I_M^U(t)-{\beta }_{S}S(t)I_S^U(t)-dS(t)+gN(t)+A_S\hfill \\ & \dot{R}(t)=v_U{I}_M^U(t)+v_Q{I}_M^Q(t)-dR(t).\hfill \end{array}\right.$$ (1)

The epidemic model, which incorporates multiple infected population types, is illustrated in Figure 1. Parameter definitions are summarized in Table 1.

FIGURE 1.

The epidemic model with multiple types of infected people.

TABLE 1.

Model parameters.

Parameter	Definition
${\beta }_M$, ${\beta }_S$	Effective contact rates for $I_M^U$ and $I_S^U$, respectively
$q_M$, $q_S$	Quarantine rates for $I_M^U$ and $I_S^U$, respectively
$v_U$, $v_Q$	Recovery rates for $I_M^U$ and $I_M^Q$, respectively
$w_U$, $w_Q$	Deterioration rates for $I_M^U$ and $I_M^Q$, respectively
$b_U$, $b_Q$	Improvement rates for $I_S^U$ and $I_S^Q$, respectively
${\overline{d}}_U$, ${\overline{d}}_Q$	Additional fatality rates for $I_S^U$ and $I_S^Q$, respectively
$A_S$	Increment of susceptible individuals per unit of time
$d$	Natural death rate
$g$	Natural birth rate

2.2. The Basic Reproduction Number

The basic reproduction number $R_0$ is a key metric in epidemiology that measures the transmissibility of infectious diseases. It represents the expected number of secondary infections produced by a single infected individual in a fully susceptible population (Diekmann et al. 1990).

Before deriving $R_0$ for model (1), we introduce two equilibrium states: the Disease‐Free Equilibrium (DFE) and the Endemic Equilibrium (EE). From model (1), when the disease is absent, the total population $N(t)$ satisfies:

$$\dot{N}(t)=A_S+(g-d)N(t).$$ (2)

The total population converges to $N_0\triangleq \frac{A_S}{d-g}$. The set

$$\begin{array}{c}\hfill \mathbf{\Omega }=\left\{\left(S,I_M^U,I_S^U,I_M^Q,I_S^Q,R\right)\in {\mathbb{R}}_{+}^6:N⩽N_0\right\}\end{array}$$ (3)

is a positive invariant set of model (1), ensuring that all populations remain non‐negative and bounded above by $N_0$. The DFE in $\mathbf{\Omega }$ is:

$$\begin{array}{c}\hfill \left(S^0,I_M^{U,0},I_S^{U,0},I_M^{Q,0},I_S^{Q,0},R^0\right)=\left(N_0,0,0,0,0,0\right).\end{array}$$ (4)

This represents the state where the epidemic has been eradicated. It is straightforward to verify that the DFE for model (1) is unique.

In contrast, the EE is a steady state where the infected population persists. It represents a dynamic balance where the epidemic continues to spread. The EE in $\mathbf{\Omega }$ is:

$$\begin{array}{c}\hfill \left(S^{*},I_M^{U*},I_S^{U*},I_M^{Q*},I_S^{Q*},R^{*}\right),\end{array}$$ (5)

with $\left(I_M^{U*},I_S^{U*},I_M^{Q*},I_S^{Q*}\right)\gg (0,0,0,0)$. Typically, the existence of the EE depends on $R_0$. Below, we present $R_0$ for the model described in Equation (1):

Theorem 1

The basic reproduction number for model (1) is:

$$\begin{array}{c}\hfill R_0=\frac{A_S\left[{\beta }_M(d+{\overline{d}}_U+b_U+q_S)+{\beta }_S{w}_U\right]}{(d-g)\left[(v_U+d+w_U+q_M)(d+{\overline{d}}_U+b_U+q_S)-w_U{b}_U\right]}.\end{array}$$ (6)

In the next section, we analyze how $R_0$ influences the stability of the DFE and the existence of the EE.

2.3. Properties of Equilibrium

In this section, we study the existence and stability of the DFE and EE. The DFE represents a state with no infections. If the DFE is asymptotically stable, the disease will eventually vanish. The EE represents a state where infection persists at a steady level, indicating that the disease continues to spread. The basic reproduction number $R_0$ plays a critical role in determining the properties of these equilibria.

Proposition 1

For model (1) and $R_0$ given in (6), the following hold:

• 1.If $R_0<1$, the DFE is globally asymptotically stable.
• 2.If $R_0>1$, the DFE is unstable and the epidemic persists.
• 3.If $R_0>1$, there exists at least one EE.

Proposition 1 shows that when $R_0<1$, each infected individual produces fewer than one new infection on average, and the disease dies out. When $R_0>1$, the disease spreads, the DFE becomes unstable, and the EE exists. This forms the basis for public health interventions, as reducing $R_0$ below 1 through measures such as vaccination, quarantine, or reduced contact rates is essential for outbreak control.

From the expression for $R_0$ in Equation (6), certain key parameters can be adjusted to reduce $R_0$. Consider the quarantine rates $q_M$ and $q_S$. We derive the following theorem:

Proposition 2

$R_0$ decreases with respect to $q_M$ and $q_S$.

Proposition 2 implies that increasing quarantine rates can reduce $R_0$. We focus on quarantine rates for two reasons. First, raising quarantine rates as a method to reduce $R_0$ has been widely and successfully implemented in the global fight against COVID‐19. Second, in the early stages of a pandemic, effective medications or vaccines are typically limited. Under such circumstances, increasing quarantine rates becomes one of the few viable options to contain the spread.

A substantial body of literature supports these claims. Sachs et al. (2020) highlight that countries in the Asia‐Pacific region, such as South Korea, New Zealand, and Australia, were more successful in reducing infection rates due to well‐organized public health systems, robust quarantine protocols, and excellent hygiene practices. Smetters (2020) argue that personal protective measures, such as mask‐wearing and isolation, are highly effective against COVID‐19. The importance of isolation and social distancing, particularly in the absence of vaccines and treatments, has been emphasized by Kantner and Koprucki (2020) and Oke et al. (2023). Using data from Spain, Aleta and Moreno (2020) concluded that the optimal strategy involves early detection and isolation of symptomatic individuals. Valencia et al. (2020) reached similar conclusions using data from Puerto Rico. Anderson et al. (2020), in their analysis of China, Italy, and the United Kingdom, argued that in the short term, without vaccines or antiviral drugs, isolating infected individuals is essential for reducing $R_0$. Similar conclusions are drawn by Vicentini et al. (2020), Tam et al. (2020), Hellewell et al. (2020), and Hollingsworth et al. (2011) using data from China, Italy, the United States, and the United Kingdom.

Next, we develop an insurance scheme based on model (1) and explore its relationship with $R_0$. By analyzing the impact of the insurance mechanism on key parameters, we assess the effectiveness of insurance schemes in controlling epidemic spread.

3. Catastrophe Insurance Scheme

3.1. Catastrophe Insurance for Pandemic

Catastrophe insurance provides financial protection against large‐scale disasters, including natural disasters and pandemics, where traditional policies are insufficient to cover extensive losses. Government‐led programs, such as the US National Flood Insurance Program, have historically played a key role in disaster risk management, with modern strategies incorporating reinsurance and catastrophe bonds to enhance resilience. Premiums are calculated using models that assess hazards, vulnerabilities, and potential losses. For pandemics, catastrophe insurance addresses economic impacts such as business closures and unemployment. Epidemic risk models are essential for premium calculations, and epidemic catastrophe bonds are often used to fund recovery efforts. Examples include the World Bank's Pandemic Emergency Financing Facility and the UN's Central Emergency Response Fund, which are designed to rapidly mobilize funds for pandemic containment and recovery.

In this study, we focus on pandemic catastrophe insurance, a long‐term mechanism designed to mitigate the financial impact of pandemics. Its general framework is illustrated in Figure 2. Prior to a pandemic, the government typically funds this system through regular contributions from public finances, serving as the premium for purchasing catastrophe insurance. These funds are directed to the insurance sector or invested in capital markets. When a pandemic occurs and trigger conditions are met, the government receives compensation as stipulated in the insurance agreement. This compensation acts as an external funding source, allowing the government to implement mitigation strategies without imposing excessive financial strain on insurance companies. Upon receiving these funds, the government must allocate them efficiently, prioritizing the acquisition of essential medical resources and targeted treatment for infected populations. This strategic allocation is crucial for maximizing the effectiveness of pandemic control.

FIGURE 2.

Catastrophe insurance for pandemic.

In practice, many catastrophe insurance systems are government‐sponsored. China's catastrophe insurance system has evolved significantly in response to natural disaster risks such as earthquakes, floods, and typhoons. Efforts to establish a comprehensive disaster risk management framework began in the early 2000s, gaining momentum after the 2008 Wenchuan earthquake. The government initiated pilot programs in high‐risk regions, and by 2014, the China Insurance Regulatory Commission began promoting nationwide catastrophe insurance initiatives. In 2016, Shenzhen launched the first comprehensive catastrophe insurance pilot project. Recently, China has integrated financial instruments such as catastrophe bonds and reinsurance to enhance system resilience.

China's Catastrophic Medical Insurance program has proven effective in reducing catastrophic health expenditures, particularly in urban and rural areas (Zhao et al. 2021a). Key factors influencing such expenditures include healthcare needs and demographics (Li et al. 2012), while the New Cooperative Medical System has improved rural healthcare coverage (Yi et al. 2009). Despite these improvements, disparities persist, with rural households facing similar expenditure levels to uninsured households (Li et al. 2014), and poorer households experiencing higher burdens (Sun and Lyu 2020). To address these issues, Zhang et al. (2019) proposed a two‐layer critical illness insurance model.

Several studies focus on catastrophe insurance development in China. He (2016) emphasizes the government's role in disaster risk management, while Kong and Wang (2022) and Shi et al. (2008) explore challenges and strategies for climate catastrophe insurance. In rural areas, Tian and Yao (2015) and Tang et al. (2022) examine factors driving demand for earthquake insurance. Li et al. (2023) introduce the government–market–public partnership model for disaster management, and Wang and Tian (2016) and Liu et al. (2022) address catastrophe insurance funding and cost reduction. Finally, Wang et al. (2023) and Zhao et al. (2021b) analyze risk‐sharing mechanisms in catastrophe insurance and their impact on the reinsurance industry.

3.2. Premium and Claim Payments

From an actuarial perspective, susceptible individuals facing infection risk contribute premiums to an insurance fund. In the event of an epidemic, infected policyholders receive claim payments to cover medical expenses. Death benefits may also be provided in the event of disease‐related mortality. The premium is determined by the actuarial fairness principle, which requires that

$$\begin{array}{c}\hfill \mathbb{E}(\text{compensation})=\mathbb{E}(\text{premium}).\end{array}$$ (7)

The time horizon considered may be the full duration of the epidemic, defined as the first instant $T$ when no more than one infection remains in the population, or an alternative observation window such as the implementation period of a specific policy.

Given that this paper assumes insurance benefits are centrally managed by policymakers, we focus exclusively on insurance payouts to infected individuals. This assumption is reasonable. In China's COVID‐19 response, for example, all infected individuals were treated at designated public hospitals (Jin et al. 2022; An et al. 2022). Treatment costs were covered by social health insurance or commercial insurance purchased by patients. Any expenses beyond insurance coverage were supplemented by government funding (Yuan et al. 2023; Ma et al. 2022).² In this context, it is plausible for insurance payouts to be centrally managed and allocated by policymakers.

Inspired by Feng and Garrido (2011) and Lefèvre et al. (2017), we introduce the claim payments of this insurance scheme as follows.

Continuous benefits for infected individuals. Under this scheme, medical expenses for each infected policyholder are continuously reimbursed throughout the treatment period. Coverage ceases immediately upon death or recovery. Let $M_c(t)$ denote the total benefits paid to all infected individuals at time $t$:

$$M_c(t)={\rho }_m\left(I_M^U(t)+I_M^Q(t)\right)+{\rho }_s\left(I_S^U(t)+I_S^Q(t)\right),$$

where ${\rho }_m$ and ${\rho }_s$ are the compensation amounts per unit time for mild and severe cases, respectively. The premium is the actuarial present value of the continuous benefits:

$${\pi }_c={\int }_0^T{e}^{-\delta t}{M}_c(t)\mathrm{d}t,$$

with $\delta$ denoting the discount rate.

Lump‐sum benefits for newly infected individuals. Under this scheme, a lump‐sum compensation is paid immediately upon infection. All newly infected individuals are assumed to have mild symptoms. Let ${\rho }_l$ be the lump‐sum payment per newly infected individual. The total lump‐sum compensation at time $t$ is

$$M_l(t)={\rho }_l\left({\beta }_{M}S(t)I_M^U(t)+{\beta }_{S}S(t)I_S^U(t)\right).$$

The premium of the lump‐sum benefits is

$${\pi }_l={\int }_0^T{e}^{-\delta t}{M}_l(t)\mathrm{d}t.$$

Combined benefits. In this case, we consider a combination of the two schemes. The total benefits paid at time $t$ are

$$\begin{array}{c}\hfill M(t)=M_c(t)+M_l(t).\end{array}$$ (8)

The premium of the combined benefits is

$$\begin{array}{c}\hfill \pi ={\int }_0^T{e}^{-\delta t}M(t)\mathrm{d}t={\pi }_c+{\pi }_l.\end{array}$$ (9)

3.3. Utilization of Insurance Compensation

From an epidemiological perspective, $I_M^U(t)$ and $I_S^U(t)$ have distinct characteristics, leading to different efficiencies of the same medical resources in enhancing quarantine rates. For example, a makeshift hospital with limited staff can effectively isolate and treat a large number of mildly infected patients, as their condition does not require extensive medical intervention. In contrast, severely infected patients typically need specialized wards, advanced medical equipment, and multiple healthcare providers. A similar modeling framework is presented in Chen et al. (2021), which categorizes resources into durable and disposable medical supplies and investigates their optimal distribution across regions.

As shown in Proposition 2, $R_0$ decreases with the quarantine rates $q_M$ and $q_S$. In this section, we examine how to use the total claim payments $M(t)$ from the combined benefits model (8) to adjust these rates and minimize $R_0$. Specifically, we consider using $M(t)$ to purchase medical resources and allocate them between mild and severe cases. Let $\alpha (t)$ be the proportion of $M(t)$ allocated to $I_M^U(t)$, with $1-\alpha (t)$ allocated to $I_S^U(t)$. Define $m_1$ and $m_2$ as the marginal quarantine efficiencies (MQE) for $I_M^U(t)$ and $I_S^U(t)$, respectively, representing the number of infected individuals that can be quarantined per unit of compensation.

Let $q_M^{\ast }(t)$ and $q_S^{\ast }(t)$ denote the quarantine rates after utilizing $M(t)$. These are given by:

$$\left\{\begin{array}{cc}& q_M^{\ast }(t)=\frac{q_M{I}_M^U(t)+\alpha (t)M(t)m_1}{I_M^U(t)},\hfill \\ & q_S^{\ast }(t)=\frac{q_S{I}_S^U(t)+(1-\alpha (t))M(t)m_2}{I_S^U(t)}.\hfill \end{array}\right.$$

To ensure $q_M^{\ast }(t),q_S^{\ast }(t)\in [0,1]$, we require for any $\alpha (t)\in [0,1]$:

$$\begin{array}{cc}& q_M{I}_M^U(t)+\alpha (t)M(t)m_1\le I_M^U(t),\hfill \\ & q_S{I}_S^U(t)+(1-\alpha (t))M(t)m_2\le I_S^U(t).\hfill \end{array}$$

Define

$$k_1(t)=\frac{M(t)m_1}{I_M^U(t)},k_2(t)=\frac{M(t)m_2}{I_S^U(t)}.$$

The quarantine rates can then be expressed as:

$$\left\{\begin{array}{cc}& q_M^{\ast }(t)=k_1(t)\alpha (t)+q_M,\hfill \\ & q_S^{\ast }(t)=k_2(t)(1-\alpha (t))+q_S,\hfill \end{array}\right.$$ (10)

where $k_1(t)\in (0,1-q_M]$ and $k_2(t)\in (0,1-q_S]$.

Rather than incorporating $q_M^{\ast }(t)$ and $q_S^{\ast }(t)$ as dynamic coefficients into model (1) to derive a time‐varying $R_0$, we treat them as exogenous parameters. This allows us to substitute them directly into the original $R_0$ formulation, yielding the real‐time basic reproduction number $R_0(\alpha (t))$:

$$R_0(\alpha (t))=\frac{N_0{\beta }_M[d+{\overline{d}}_U+b_U+q_S+k_2(t)(1-\alpha (t))]+N_0{\beta }_S{w}_U}{(v_U+d+w_U+q_M+k_1(t)\alpha (t))[d+{\overline{d}}_U+b_U+q_S+k_2(t)(1-\alpha (t))]-w_U{b}_U}.$$ (11)

At each time $t$, we seek the optimal allocation ${\alpha }^{\ast }(t)$ that minimizes $R_0(\alpha (t))$:

$${\alpha }^{\ast }(t)=arg\underset{\alpha (t)\in [0,1]}{\min }{R}_0(\alpha (t)).$$ (12)

This approach is justified for two reasons. First, while the insurance compensation scheme interacts with the epidemic progression, it is not a core component of the epidemiological dynamics. Relative to the mechanisms driving the epidemic, the insurance scheme is exogenous, and $q_M^{\ast }(t)$ and $q_S^{\ast }(t)$ can be treated as short‐term adjustable parameters. Second, virus mutation and environmental complexity make long‐term epidemiological predictions difficult. Treating these quarantine rates as exogenous avoids the need for such predictions. Indeed, estimating current epidemiological parameters to calculate real‐time $R_0$ has become common practice during COVID‐19, as seen in studies such as Tang et al. (2020) and Eikenberry et al. (2020).

There are two primary reasons for adopting this approach. First, although the insurance compensation scheme and the epidemic's progression are interrelated in our model, the insurance plan is not a core feature of the epidemic's epidemiological dynamics. In other words, compared to the epidemiological mechanisms driving the epidemic, the insurance scheme acts as an exogenous factor. Thus, we can treat $q_M^{*}(t)$ and $q_S^{*}(t)$ as exogenous parameters that can be adjusted in the short term. Second, the mutation of the virus and the inherent complexity of real‐world environments make it difficult to accurately model the long‐term dynamics of certain epidemiological parameters. By considering $q_M^{*}(t)$ and $q_S^{*}(t)$ as exogenous parameters, we avoid the challenge of making long‐term predictions. Precisely due to the difficulty of such predictions, estimating current epidemiological parameters and calculating real‐time $R_0(t)$ has become a common practice in various countries and regions during the COVID‐19 outbreak. Similar frameworks can be observed in studies such as Tang et al. (2020) and Eikenberry et al. (2020). Hereafter, we omit $t$ in Equations (11) and (12), and problem (12) becomes:

$${\alpha }^{\ast }=arg\underset{\alpha \in [0,1]}{\min }{R}_0(\alpha ).$$ (13)

4. Theoretical Analysis of Optimal Compensation Usage Strategy

In this section, we derive the analytical solution for the optimal compensation usage strategy ${\alpha }^{\ast }$ and examine its relationship with key epidemiological parameters. Define the following notations:

$$A=v_U+d+q_M+w_U,B=d+{\overline{d}}_U+b_U+q_S,$$

$$\widehat{A}=N_0{\beta }_{M}B+N_0{\beta }_S{w}_U,\widehat{B}=AB-w_U{b}_U.$$

Define the auxiliary function $f(\alpha )$ as:

$$\begin{array}{cc}\hfill f(\alpha )=& N_0{\beta }_M{k}_1{k}_2^2{\alpha }^2-\left(2N_0{\beta }_M{k}_1{k}_2^2+2\widehat{A}{k}_1{k}_2\right)\alpha \hfill \\ & +N_0{\beta }_M{k}_1{k}_2^2+\left(\widehat{A}+N_0{\beta }_{M}B\right)k_1{k}_2\hfill \\ & +B\widehat{A}{k}_1-\left(\widehat{A}A-N_0{\beta }_M\widehat{B}\right)k_2.\hfill \end{array}$$ (14)

Based on this, we present the following theorem.

Theorem 2

The optimal compensation usage strategy ${\alpha }^{\ast }$ is determined as follows:

• 1.If $f(1)\ge 0$, then ${\alpha }^{\ast }=1$.
• 2.
  If $f(1)<0$ and $f(0)>0$, then there exists a unique ${\alpha }^{\ast }\in (0,1)$ given by:

  $$\begin{array}{cc}\hfill {\alpha }^{\ast }=& 1+\frac{B}{k_2}+\frac{{\beta }_S{w}_U}{{\beta }_M{k}_2}\hfill \\ & -\frac{\sqrt{w_U{k}_1[{\beta }_M{\beta }_S{k}_1{k}_2+({\beta }_{M}B+{\beta }_S{w}_U){\beta }_S{k}_1+({\beta }_M{b}_U+A{\beta }_S){\beta }_M{k}_2]}}{{\beta }_M{k}_1{k}_2}.\hfill \end{array}$$ (15)
• 3.If $f(0)\le 0$, then ${\alpha }^{\ast }=0$.

Theorem 2 provides the optimal allocation strategy. When ${\alpha }^{\ast }=1$, the entire compensation $M$ is allocated to isolate $I_M^U$. When ${\alpha }^{\ast }=0$, all of $M$ is allocated to isolate $I_S^U$. When ${\alpha }^{\ast }\in (0,1)$, the compensation is divided between the two groups. To compute ${\alpha }^{\ast }$, first obtain $M$ from Equation (8) based on the current epidemic state. Then compute $f(\alpha )$ using Equation (14). Finally, apply Theorem 2 to determine ${\alpha }^{\ast }$.

Next, we examine the monotonic relationship between ${\alpha }^{\ast }$ and key epidemiological parameters. We begin with the effective contact rates ${\beta }_M$ and ${\beta }_S$ of the two types of non‐quarantined infected individuals.

Proposition 3

Given a fixed value of ${\beta }_M$, ${\alpha }^{\ast }$ has the following properties:

• 1. ${\alpha }^{\ast }$ decreases with respect to ${\beta }_S$.
• 2.If ${\alpha }^{\ast }\in (0,1)$, then $\frac{{\partial }^2{\alpha }^{\ast }}{\partial {\beta }_S^2}\ge 0$.

Proposition 3 shows that as the effective contact rate of non‐quarantined infected individuals increases, more resource should be allocated on mildly infected individuals. Moreover, the second derivative being non‐negative indicates that the rate of decrease slows as ${\beta }_S$ rises. This implies that resources shift toward the more infectious group, but the marginal benefit of additional resources diminishes with higher infectiousness. These properties highlight the need to adjust compensation allocation based on variations in contact rates.

We now investigate the impact of $k_1$ and $k_2$. Recall that

$$k_1=\frac{M{m}_1}{I_M^U},k_2=\frac{M{m}_2}{I_S^U},$$

which represent the proportion of infections that can be isolated using compensation $M$ relative to the number of infections in each class. These ratios reflect the efficiency of compensation in isolating infected individuals. Define the following notations:

$$C={\beta }_M{\beta }_S{k}_1+{\beta }_M({\beta }_M{b}_U+A{\beta }_S),D={\beta }_S{k}_1({\beta }_{M}B+{\beta }_S{w}_U),E={\beta }_S^2{w}_U{k}_1.$$

Define the auxiliary variable $h$ as:

$$h=\min \left(1-q_S,\frac{2D\left[(D-E)+\sqrt{D(D-E)}\right]}{EC}\right).$$

Based on these notations, we have the following:

Proposition 4

${\alpha }^{\ast }$ has the following properties:

• 1. ${\alpha }^{\ast }$ increases with respect to $k_1$.
• 2.If $k_2\in (0,h)$, ${\alpha }^{\ast }$ decreases with respect to $k_2$.
• 3.If $k_2\in [h,1-q_S]$, ${\alpha }^{\ast }$ increases with respect to $k_2$.

Proposition 4 suggests that limited resources should be directed toward populations that can be quarantined more cost‐effectively. We now examine the minimized reproduction number ${\stackrel{\sim }{R}}_0=R_0({\alpha }^{\ast })$.

Proposition 5

${\stackrel{\sim }{R}}_0$ decreases with respect to both $k_1$ and $k_2$.

Proposition 5 shows that ${\stackrel{\sim }{R}}_0$ declines as the quarantine efficiency of compensation increases. In addition, to increase $k_1$ and $k_2$, one can improve the MQE $m_1$ and $m_2$ through better resource allocation, more effective medical supplies, and improved utilization of funds. Such improvements would further reduce ${\stackrel{\sim }{R}}_0$.

Finally, we examine the impact of the optimal compensation strategy on premiums. Let ${\pi }^0$ denote the premium without the strategy and ${\pi }^{\ast }$ the premium with the strategy. We then establish the following theorem.

Theorem 3

For any ${\alpha }^{\ast }\in [0,1]$, ${\pi }^{\ast }<{\pi }^0$.

Theorem 3 shows that implementing the proposed strategy ${\alpha }^{\ast }$ reduces the premium while achieving effective epidemic control, thereby lowering the overall societal costs of epidemic response. Moreover, the optimal compensation strategy is applicable beyond insurance contexts. The benefits $M$ can be interpreted as various types of continuously supplied medical resources, such as special medical funds, supplies, or personnel. Accordingly, $m_1$, $m_2$, $k_1$, and $k_2$ can be redefined as the marginal treatment rates and quarantine capacities associated with these resources.

5. Numerical Illustration

In this section, we illustrate the optimal compensation usage strategy ${\alpha }^{\ast }$ and the relationship between the minimized basic reproduction number ${\stackrel{\sim }{R}}_0$ and the parameters ${\beta }_M$, ${\beta }_S$, $k_1$, and $k_2$ through numerical simulation. The calibration of epidemiological parameters is presented in Table 2.

TABLE 2.

Parameter calibrations for numerical simulation.

Parameter	Value	Parameter	Value
${\overline{d}}_U$	1/7	${\overline{d}}_Q$	1/14
$v_U$	1/10	$v_Q$	1/7
$q_M$	1/5	$q_S$	1/5
$w_U$	1/5	$w_Q$	1/10
$b_Q$	1/7	$d$	1/100
$g$	1/200	$A_s$	1
$b_U$	1/14

The numerical results are shown in Figures 3 and 4. Figure 3 presents the optimal compensation usage strategy ${\alpha }^{\ast }$, and Figure 4 presents the minimized basic reproduction number ${\stackrel{\sim }{R}}_0$. In each figure, the horizontal axis represents $k_2$ and the vertical axis represents $k_1$. Different colors indicate different values of ${\alpha }^{\ast }$ and ${\stackrel{\sim }{R}}_0$, facilitating visual interpretation of the relationships among these variables. Both figures illustrate two scenarios based on distinct parameter pairs: Case 1 with $({\beta }_M,{\beta }_S)=(0.005,0.025)$, and Case 2 with $({\beta }_M,{\beta }_S)=(0.005,0.005)$.

FIGURE 3.

Optimal compensation usage strategy ${\alpha }^{*}$.

FIGURE 4.

Minimized basic reproduction number ${\stackrel{\sim }{R}}_0$.

Figure 3 shows that when $k_2$ is fixed, ${\alpha }^{\ast }$ increases with $k_1$. When $k_1$ is fixed, ${\alpha }^{\ast }$ decreases with $k_2$. These observations are consistent with Proposition 4 in Section 4. A comparison of the two cases in Figure 3 reveals that for any given pair $(k_1,k_2)$, the values of ${\alpha }^{\ast }$ in Case 2 are greater than or equal to those in Case 1. This suggests that, with ${\beta }_M$ is held constant, ${\alpha }^{\ast }$ does not increase with ${\beta }_S$, which aligns with Proposition 3. Figure 4 shows that when $k_2$ is fixed, ${\stackrel{\sim }{R}}_0$ decreases with $k_1$. When $k_1$ is fixed, ${\stackrel{\sim }{R}}_0$ decreases with $k_2$. These findings support Proposition 5 in Section 4. Overall, the numerical simulations reinforce the theoretical insights and highlight the critical relationships among the parameters.

Figure 3 also shows that when $k_1$ is small or $k_2$ is large, changes in the other parameter have minimal impact on ${\alpha }^{\ast }$. This observation corresponds to conclusions 1 and 3 in Theorem 2. The intuition is that if compensation is highly effective or nearly ineffective in isolating a specific type of infection, it will consistently be used or not used for that type. Figure 4 reveals a different pattern. When $k_1$ is fixed, ${\stackrel{\sim }{R}}_0$ is less sensitive to changes in $k_2$ than to changes in $k_1$ when $k_2$ is fixed. This indicates that increasing $k_1$ is more effective in reducing ${\stackrel{\sim }{R}}_0$.

When $k_1$ and $k_2$ are close in value, simultaneous changes in both parameters have little effect on ${\alpha }^{\ast }$. This reflects that ${\alpha }^{\ast }$ responds primarily to relative differences in quarantine efficiency. However, simultaneous adjustments in both parameters significantly affect ${\stackrel{\sim }{R}}_0$, which aligns with the intuition that higher quarantine efficiency leads to better epidemic control.

Figure 4 also shows that for any given ${\stackrel{\sim }{R}}_0$, a corresponding curve can be derived for the pair $(k_1,k_2)$. This is analogous to an indifference curve in expected utility theory and defines an equivalent basic reproduction number curve. Since different pairs $(k_1,k_2)$ correspond to different epidemic stages and resource conditions, the existence of such a curve suggests that regions with varying backgrounds can achieve the same level of epidemic control by adopting the optimal compensation strategy. This finding underscores the adaptability of the proposed approach across diverse contexts.

6. Discussion

6.1. Comparison With Existing Literature

Our research aligns with a substantial body of literature on epidemic control. In epidemiological models, the determination of an optimal compensation strategy is closely tied to managing epidemics under resource constraints, a topic explored extensively in prior studies. For example, Hansen and Day (2011) provided analytical solutions to minimize outbreak size with limited resources, while Dangerfield et al. (2019) emphasized concentrating treatment efforts on specific population subsets. Bolzoni et al. (2019) used simulations to minimize outbreak size and duration given limited resources, and Mu et al. (2019) highlighted the importance of hospital resources, government interventions, and threshold policies. Related findings can also be found in Dar et al. (2021), Dondorp et al. (2020), Kache et al. (2020), Nyiwul (2021), and Jiang and Zhou (2018).

While much of this literature focuses on long‐term objectives such as eradication or endemic equilibrium, the COVID‐19 pandemic has shifted attention toward real‐time epidemic control. This shift introduces new challenges, including virus mutations, public response dynamics, and resource uncertainty. Our study diverges from traditional control models by aiming to minimize $R_0$ through real‐time strategies. By deriving an explicit expression for $R_0$ at any time $t$, we enable decision‐makers to adjust strategies dynamically to evolving conditions. This real‐time, adaptive approach distinguishes our study from models that emphasize long‐term control.

Our research also contributes to the literature on insurance in epidemic interventions. The novelty lies in establishing a link between $R_0$ and insurance schemes, particularly in optimizing the use of insurance payouts. While most studies focus on the direct impacts of epidemics on insurance mechanisms, we show that an optimal payout strategy can reduce premiums. As the epidemic subsides due to effective interventions, the total societal cost of managing the epidemic also decreases. This dual benefit, reducing both insurance costs and societal expenditures, sets our study apart from existing literature.

6.2. Practical Implication of Theoretical Results

It is well recognized that a gap often exists between theoretical models and real‐world epidemic responses, particularly regarding data accuracy and resource availability. Several considerations arise when applying our theoretical insights to practice, which merit further discussion.

First, the calculation of optimal strategies relies on accurate estimation of epidemic parameters and population counts, especially the number of infected individuals. In practice, this task is often hindered by data inaccuracies. Starnini et al. (2021) examined the impact of data accuracy on evaluating mitigation policies using evidence from Italy and Spain. Barry (2022) noted that tracing applications were not widely adopted during COVID‐19, reflecting a tension between insurance solidarity, where interdependence is viewed as a source of rights, and epidemic solidarity, which imposes obligations. Gardner et al. (2014) investigated discrepancies between observed and expected epidemiology, attributing them to ongoing nonhuman sources, undetected human‐to‐human transmission, or a combination of both. Additionally, delays in detection and the presence of asymptomatic or latent individuals contribute to data inaccuracies (Adhikari et al. 2020; Hou et al. 2020; Wu et al. 2020; Hu et al. 2021).

Second, insufficient resources can lead to suboptimal outcomes. Experiences from various countries indicate that the initial phase of an outbreak is characterized by extremely rapid escalation (Remuzzi and Remuzzi 2020; Yue et al. 2020; Tang et al. 2020). This rapid increase often prevents the stockpiling and production of medical supplies needed to meet actual demand (Chen et al. 2021; Mu et al. 2019; Dangerfield et al. 2019; Jiang and Zhou 2018).

7. Conclusion

This study explores the optimal strategy for using catastrophe insurance compensation based on an epidemiological model. By distinguishing between mildly and severely infected individuals, as well as quarantined and non‐quarantined individuals, we derive the basic reproduction number $R_0$ and analyze the properties of both disease‐free and endemic equilibria. This approach offers insights into the persistence or eradication of an epidemic under different conditions. A key contribution of this work is the introduction of a pandemic catastrophe insurance plan that links insurance compensation usage to real‐time $R_0$. Our model examines how compensation can influence quarantine rates and epidemic dynamics, identifying the optimal allocation strategy ${\alpha }^{\ast }$ to minimize $R_0$. By dynamically adjusting ${\alpha }^{\ast }$, policymakers can allocate resources efficiently for epidemic control.

Several important conclusions emerge. First, resources should be prioritized for populations with higher infection rates to reduce epidemic spread efficiently. Second, resources should target populations where quarantine measures are most cost‐effective, maximizing transmission reduction relative to cost. Third, optimizing compensation‐driven quarantine measures improves epidemic control by reducing $R_0$ and limiting transmission, while also lowering the financial costs of managing the epidemic. Our findings show that this strategy not only enhances epidemic control but also reduces insurance premiums, as the need for compensation decreases when the epidemic is contained. This dual benefit, improved control and reduced financial burden, underscores the value of the proposed allocation strategy.

The broader implications of this research are significant for real‐time epidemic management. By providing a dynamic framework that adjusts compensation usage based on real‐time data, our model enables policymakers to respond quickly, target high‐risk populations, and maximize the efficiency of quarantine measures. This adaptability is crucial in rapidly evolving epidemics like COVID‐19, where timely interventions are essential. Moreover, the proposed strategy is adaptable across different epidemic contexts and regions. Regardless of specific conditions, the model provides a practical tool for managing epidemic risks and minimizing societal costs.

In conclusion, this paper presents a novel approach to epidemic control by linking catastrophe insurance compensation to $R_0$. By identifying an optimal allocation strategy, we show how limited compensation can be effectively deployed to curb epidemic spread while reducing both insurance premiums and overall societal costs. These findings provide valuable insights for policymakers and contribute to the broader literature on epidemic management and catastrophe insurance.

One limitation of our study is the treatment of government insurance as an external source without explicitly modeling participant costs. While this simplification allows us to focus on epidemic dynamics and compensation strategies, it overlooks the financial burden on individuals. Additionally, the model does not account for population heterogeneity, such as age or mobility, which may affect both epidemic spread and intervention effectiveness. Future research could address these aspects to provide a more comprehensive analysis of how demographic factors influence epidemic control and insurance mechanisms.

Appendix A. A.1. Proof of Theorem 1

A.1.

For simplicity, we define the following constants:

$$A=v_U+d+q_M+w_U,B=d+{\overline{d}}_U+b_U+q_S,C=v_Q+d+w_Q,D=b_Q+d+{\overline{d}}_Q.$$

We can then rewrite model (1) as:

$$\dot{x}=f(x)=\mathcal{F}(x)-\mathcal{V}(x),$$

where

$$\mathcal{F}(x)=\left[\begin{array}{c}{\beta }_{M}S{I}_M^U+{\beta }_{S}S{I}_S^U\\ 0\\ 0\\ 0\\ 0\\ 0\end{array}\right],\mathcal{V}(x)=\left[\begin{array}{c}A{I}_M^U-b_U{I}_S^U\\ -w_U{I}_M^U+B{I}_S^U\\ -q_M{I}_M^U+C{I}_M^Q-b_Q{I}_S^Q\\ -q_S{I}_S^U-w_Q{I}_M^Q+D{I}_S^Q\\ {\beta }_{M}S{I}_M^U+{\beta }_{S}S{I}_S^U+dS-gN-A_S\\ -v_U{I}_M^U-v_Q{I}_M^Q+dR\end{array}\right].$$

Denote $\mathrm{D}f(x^0)$ as the Jacobian matrix evaluated at the DFE, $x^0$. It is straightforward to prove that when $\mathcal{F}(x)=0$, all eigenvalues of $\mathrm{D}f(x^0)$ have negative real parts. According to Lemma 1 in Van Den Driessche and Watmough (2002), we obtain the matrices $\mathbb{F}$ and $\mathbb{V}$ as:

$$\mathbb{F}=\left[\begin{array}{cccc}{N}_0{\beta }_M& N_0{\beta }_S& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],\mathbb{V}=\left[\begin{array}{cccc}A& -b_U& 0& 0\\ -w_U& B& 0& 0\\ -q_M& 0& C& -b_Q\\ 0& -q_S& -w_Q& D\end{array}\right].$$

We partition $\mathbb{F}$ and $\mathbb{V}$ as:

$$\mathbb{F}=\left[\begin{array}{cc}{\mathbb{F}}_1& {\mathbb{F}}_2\\ {\mathbb{F}}_3& {\mathbb{F}}_4\end{array}\right],\mathbb{V}=\left[\begin{array}{cc}{\mathbb{V}}_1& {\mathbb{V}}_2\\ {\mathbb{V}}_3& {\mathbb{V}}_4\end{array}\right],$$

where

$${\mathbb{F}}_1=\left[\begin{array}{cc}{N}_0{\beta }_M& N_0{\beta }_S\\ 0& 0\end{array}\right],{\mathbb{V}}_1=\left[\begin{array}{cc}A& -b_U\\ -w_U& B\end{array}\right].$$

The inverse of $\mathbb{V}$ is:

$${\mathbb{V}}^{-1}=\left[\begin{array}{cc}{\mathbb{V}}_1^{-1}& 0\\ -{\mathbb{V}}_4^{-1}{\mathbb{V}}_3{\mathbb{V}}_1^{-1}& {\mathbb{V}}_4^{-1}\end{array}\right],$$

and the product $\mathbb{F}{\mathbb{V}}^{-1}$ is:

$$\mathbb{F}{\mathbb{V}}^{-1}=\left[\begin{array}{cc}{\mathbb{F}}_1{\mathbb{V}}_1^{-1}& 0\\ 0& 0\end{array}\right],$$

where

$${\mathbb{F}}_1{\mathbb{V}}_1^{-1}=\left[\begin{array}{cc}\frac{N_0{\beta }_{M}B+N_0{\beta }_S{w}_U}{AB-w_U{b}_U}& \frac{N_0{\beta }_M{b}_U+N_0{\beta }_{S}A}{AB-w_U{b}_U}\\ 0& 0\end{array}\right].$$

Thus, the basic reproduction number $R_0$ is given by:

$$R_0=\rho (\mathbb{F}{\mathbb{V}}^{-1})=\rho ({\mathbb{F}}_1{\mathbb{V}}_1^{-1})=\frac{N_0{\beta }_{M}B+N_0{\beta }_S{w}_U}{AB-w_U{b}_U}.$$

Finally, substituting the values of $A$, $B$, $C$, and $D$, we obtain:

$$R_0=\frac{N_0{\beta }_M(d+{\overline{d}}_U+b_U+q_S)+N_0{\beta }_S{w}_U}{(v_U+d+w_U+q_M)(d+{\overline{d}}_U+b_U+q_S)-w_U{b}_U}.$$

Thus, the theorem is proved.

A.2. Proof of Proposition 1

First, we show that if $R_0<1$, then the DFE is globally asymptotically stable. Below, we define a Lyapunov function $Q$ for model (1) and show that it decreases over time under the given conditions.

We define:

$$Q=\frac{{\beta }_{M}B+{\beta }_S{w}_U}{{\beta }_S(AB-w_U{b}_U)}{I}_M^U+\frac{{\beta }_M{b}_U+{\beta }_{S}A}{{\beta }_S(AB-w_U{b}_U)}{I}_S^U.$$

Next, we verify that $Q$ is indeed a Lyapunov function for the model within the positive invariant set $\mathbf{\Omega }$. First, observe that the coefficients of $I_M^U$ and $I_S^U$ in the expression for $Q$ are positive, since:

$$\frac{{\beta }_{M}B+{\beta }_S{w}_U}{{\beta }_S(AB-w_U{b}_U)}>0,\frac{{\beta }_M{b}_U+{\beta }_{S}A}{{\beta }_S(AB-w_U{b}_U)}>0.$$

Thus, $Q>0$ in $\mathbf{\Omega }$.

Now, we compute the derivative of $Q$:

$$\dot{Q}=\frac{{\beta }_{M}B+{\beta }_S{w}_U}{{\beta }_S(AB-w_U{b}_U)}{\dot{I}}_M^U+\frac{{\beta }_M{b}_U+{\beta }_{S}A}{{\beta }_S(AB-w_U{b}_U)}{\dot{I}}_S^U.$$

Substituting the equations for ${\dot{I}}_M^U$ and ${\dot{I}}_S^U$, we obtain:

$$\dot{Q}=\frac{({\beta }_M{I}_M^U+{\beta }_S{I}_S^U)[S({\beta }_{M}B+{\beta }_S{w}_U)-(AB-w_U{b}_U)]}{{\beta }_S(AB-w_U{b}_U)}.$$

If $R_0$ is given by:

$$R_0=\frac{N_0{\beta }_{M}B+N_0{\beta }_S{w}_U}{AB-w_U{b}_U},$$

then when $R_0<1$, we have:

$$S({\beta }_{M}B+{\beta }_S{w}_U)\le N_0{\beta }_{M}B+N_0{\beta }_S{w}_U<AB-w_U{b}_U.$$

This implies:

$$\dot{Q}<0,$$

showing that $Q$ decreases over time, confirming that $Q$ is a valid Lyapunov function. Therefore, model (1) is globally asymptotically stable in $\mathbf{\Omega }$ when $R_0<1$. This proves that the disease will eventually die out in the population when $R_0$ is less than 1.

Second, we prove that if $R_0>1$, the DFE is unstable. Considering the the Lyapunov function $Q$ we define above, we know that when $R_0>1$, we find that if both $I_M^U>0$ and $I_S^U>0$, and $S=N_0$, then $\dot{Q}>0$. This indicates that solutions in $\mathbf{\Omega }$ that are sufficiently close to the DFE will move away from it, implying that the DFE is unstable. Furthermore, as noted in Li et al. (1999) and Freedman et al. (1994), when the DFE is unstable, infectious diseases become persistent within the population.

Finally, we prove that if $R_0>1$, there exists at least one EE. For simplicity, we define the following quantities:

$$E=CD-w_Q{b}_Q,F=B{q}_{M}D+w_U{q}_S{b}_Q,G=B{q}_M{w}_Q+w_U{q}_{S}C,H=E(w_U+B)+F+G.$$

Next, we set all equations in model (1) equal to zero. Since $I^U=I_M^U+I_S^U$, from the second equation in the model, we can derive:

$$I_M^U=\frac{B}{w_U+B}{I}^U,I_S^U=\frac{w_U}{w_U+B}{I}^U.$$

Similarly, since $I^Q=I_M^Q+I_S^Q$, we can deduce from the third and fourth equations:

$$I_M^Q=\frac{F}{F+G}{I}^Q,I_S^Q=\frac{G}{F+G}{I}^Q.$$

Noting that $I=I^U+I^Q$, we can express $I^U$ and $I^Q$ in terms of $I$:

$$I^U=\frac{F(w_U+B)}{H}I,I^Q=\frac{F+G}{H}I.$$

Thus, we have the following relationships:

$$I_M^U=\frac{BE}{H}I,I_S^U=\frac{w_{U}E}{H}I,I_M^Q=\frac{F}{H}I,I_S^Q=\frac{G}{H}I.$$

From the sixth equation in model (1), we know:

$$R=\frac{v_{U}BE+v_{Q}F}{dH}I.$$

From the first and fifth equations in the model, we can establish:

$$S=\frac{As}{d-g}+\frac{1}{(g-d)dH}[d(ABE-b_U{w}_{U}E)-g(dH+v_{U}BE+v_{Q}F)]I.$$

Using the expression for $I_M^U$ and the first equation in the model, we derive:

$$\frac{S}{N_0}=\frac{AB-b_U{w}_U}{({\beta }_{M}B+{\beta }_S{w}_U)N_0}=\frac{1}{R_0},$$

where $N_0=\frac{As}{d-g}$. Consequently, we obtain:

$$I=\left(\frac{1}{R_0}-1\right)\frac{N_0(g-d)dH}{d(ABE-b_U{w}_{U}E)-g(dH+v_{U}BE+v_{Q}F)}.$$

Denote $({*}_1)\triangleq d(ABE-b_U{w}_{U}E)-g(dH+v_{U}BE+v_{Q}F)$, and we find:

$$({*}_1)>({*}_2)\triangleq g(ABE-b_U{w}_{U}E-dH-v_{U}BE-v_{Q}F).$$

Substituting $A,F,G$, and $H$ into the equations, we define:

$$({*}_3)\triangleq B{q}_M[E-d(D+w_Q)],({*}_4)\triangleq w_U{q}_S[E-d(b_Q+C)],({*}_5)\triangleq w_U{\overline{d}}_{U}E-v_Q(B{q}_{M}D+w_U{q}_S{b}_Q).$$

Thus, we can express $({*}_2)$ as:

$$({*}_2)=({*}_3)+({*}_4)+({*}_5).$$

Substituting for $E$ and $D$, we find:

$$({*}_3)=B{q}_M{v}_{Q}D+B{q}_M{w}_Q{\overline{d}}_Q,$$

$$({*}_4)=w_U{q}_S{\overline{d}}_{Q}C+w_U{q}_S{b}_Q{v}_Q,$$

$$({*}_5)=v_{Q}D{w}_U{\overline{d}}_U-v_{Q}DB{q}_M+dD{w}_U{\overline{d}}_U+w_Q{w}_U{\overline{d}}_U(d+d^{(1)})-v_Q{b}_Q{w}_U{q}_S.$$

Consequently, we have:

$$({*}_2)=B{q}_M{w}_Q{\overline{d}}_Q+w_U{q}_S{\overline{d}}_{Q}C+dD{w}_U{\overline{d}}_U+w_Q{w}_U{\overline{d}}_U(d+d^{(1)})+v_{Q}D{w}_U{\overline{d}}_U>0.$$

Since $d>g$, we conclude that if $R_0>1$, then $I>0$, indicating the existence of the EE. Thus, the proposition is proved.

A.3. Proof of Proposition 2

To show that $R_0$ decreases with respect to $q_M$ and $q_S$, we examine the partial derivatives.

For $q_M$, recall that

$$R_0=\frac{A_S\left[{\beta }_M(d+{\overline{d}}_U+b_U+q_S)+{\beta }_S{w}_U\right]}{(d-g)\left[(v_U+d+w_U+q_M)(d+{\overline{d}}_U+b_U+q_S)-w_U{b}_U\right]}.$$

Treat $R_0$ as $R_0=C/D(q_M)$, where $C$ is independent of $q_M$:

$$C=\frac{A_S\left[{\beta }_M(d+{\overline{d}}_U+b_U+q_S)+{\beta }_S{w}_U\right]}{d-g},$$

$$D(q_M)=(v_U+d+w_U+q_M)(d+{\overline{d}}_U+b_U+q_S)-w_U{b}_U.$$

Then

$$\frac{\partial R_0}{\partial q_M}=-C·\frac{D^{\prime }(q_M)}{{[D(q_M)]}^2}.$$

Since

$$D^{\prime }(q_M)=d+{\overline{d}}_U+b_U+q_S>0,$$

we obtain

$$\frac{\partial R_0}{\partial q_M}=-\frac{A_S\left[{\beta }_M(d+{\overline{d}}_U+b_U+q_S)+{\beta }_S{w}_U\right](d+{\overline{d}}_U+b_U+q_S)}{(d-g){\left[(v_U+d+w_U+q_M)(d+{\overline{d}}_U+b_U+q_S)-w_U{b}_U\right]}^2}<0.$$

Thus, $R_0$ decreases with respect to $q_M$.

For $q_S$, write $R_0=\frac{A_S·N(q_S)}{(d-g)D(q_S)}$, where

$$N(q_S)={\beta }_M(d+{\overline{d}}_U+b_U+q_S)+{\beta }_S{w}_U,$$

$$D(q_S)=(v_U+d+w_U+q_M)(d+{\overline{d}}_U+b_U+q_S)-w_U{b}_U.$$

Then

$$\frac{\partial R_0}{\partial q_S}=\frac{A_S}{d-g}·\frac{N^{\prime }(q_S)D(q_S)-N(q_S)D^{\prime }(q_S)}{{[D(q_S)]}^2}.$$

Since $N^{\prime }(q_S)={\beta }_M$ and $D^{\prime }(q_S)=v_U+d+w_U+q_M$, we have

$$\frac{\partial R_0}{\partial q_S}=\frac{A_S}{d-g}·\frac{{\beta }_{M}D(q_S)-N(q_S)(v_U+d+w_U+q_M)}{{[D(q_S)]}^2}.$$

Substituting $D(q_S)$ and simplifying, we have:

$$\frac{\partial R_0}{\partial q_S}=-\frac{A_S{w}_U\left({\beta }_M{b}_U+{\beta }_S(v_U+d+w_U+q_M)\right)}{(d-g){\left[(v_U+d+w_U+q_M)(d+{\overline{d}}_U+b_U+q_S)-w_U{b}_U\right]}^2}<0.$$

Thus, $R_0$ decreases with respect to $q_S$. This completes the proof.

A.4. Proof of Theorem 2

Given the notations, we express $R_0(\alpha )$ as follows:

$$R_0(\alpha )=\frac{-N_0{\beta }_M{k}_2\alpha +N_0{\beta }_M{k}_2+\widehat{A}}{-k_1{k}_2{\alpha }^2+(k_1{k}_2+B{k}_1-A{k}_2)\alpha +A{k}_2+\widehat{B}}.$$

Next, we define:

$$g(\alpha )=-k_1{k}_2{\alpha }^2+(k_1{k}_2+B{k}_1-A{k}_2)\alpha +A{k}_2+\widehat{B}.$$

Thus, the derivative of $R_0(\alpha )$ with respect to $\alpha$ becomes:

$$\begin{array}{cc}\hfill \frac{\mathrm{d}{R}_0(\alpha )}{\mathrm{d}\alpha }& =\frac{-1}{g{(\alpha )}^2}\left[N_0{\beta }_M{k}_1{k}_2^2{\alpha }^2-\left(2N_0{\beta }_M{k}_1{k}_2^2+2\widehat{A}{k}_1{k}_2\right)\alpha +N_0{\beta }_M{k}_1{k}_2^2+(\widehat{A}+N_0{\beta }_{M}B)k_1{k}_2\right.\hfill \\ & \left.+B\widehat{A}{k}_1-(\widehat{A}A-N_0{\beta }_M\widehat{B})k_2\right],\hfill \end{array}$$

which simplifies to:

$$\frac{\mathrm{d}{R}_0(\alpha )}{\mathrm{d}\alpha }=-\frac{f(\alpha )}{g{(\alpha )}^2},$$

where

$$\begin{array}{cc}\hfill f(\alpha )& =N_0{\beta }_M{k}_1{k}_2^2{\alpha }^2-(2N_0{\beta }_M{k}_1{k}_2^2+2\widehat{A}{k}_1{k}_2)\alpha +N_0{\beta }_M{k}_1{k}_2^2+(\widehat{A}+N_0{\beta }_{M}B)k_1{k}_2\hfill \\ & +B\widehat{A}{k}_1-(\widehat{A}A-N_0{\beta }_M\widehat{B})k_2.\hfill \end{array}$$

Thus, the sign of $\frac{\mathrm{d}{R}_0(\alpha )}{\mathrm{d}\alpha }$ is determined by $f(\alpha )$. Since $f(\alpha )$ is a quadratic function of $\alpha$ and the coefficient of ${\alpha }^2$, $a=N_0{\beta }_M{k}_1{k}_2^2>0$, we now examine the properties of $f(\alpha )$.

The symmetry axis of $f(\alpha )$ is given by:

$$-\frac{b}{2a}=\frac{2N_0{\beta }_M{k}_1{k}_2^2+2\widehat{A}{k}_1{k}_2}{2N_0{\beta }_M{k}_1{k}_2^2}=1+\frac{\widehat{A}}{N_0{\beta }_M{k}_2}>1.$$

Next, we calculate the discriminant $\mathrm{\Delta }$ of $f(\alpha )$:

$$\mathrm{\Delta }=b^2-4ac={(2N_0{\beta }_M{k}_1{k}_2^2+2\widehat{A}{k}_1{k}_2)}^2-4N_0{\beta }_M{k}_1{k}_2^2[N_0{\beta }_M{k}_1{k}_2^2+(\widehat{A}+N_0{\beta }_{M}B)k_1{k}_2+B\widehat{A}{k}_1].$$

Simplifying $\mathrm{\Delta }$, we get:

$$\mathrm{\Delta }=4N_0{\beta }_M{N}_0{\beta }_S{w}_U{k}_1^2{k}_2^3+4\widehat{A}{N}_0{\beta }_S{w}_U{k}_1^2{k}_2^2+4N_0{\beta }_M(N_0{\beta }_M{w}_U{b}_U+N_{0}A{\beta }_S{w}_U)k_1{k}_2^3>0.$$

Thus, $f(\alpha )$ has a positive discriminant and is decreasing in $[0,1]$ with at most one root. Therefore, we analyze the following cases:

1. If $f(1)\ge 0$: In this case, $f(\alpha )\ge 0$ and $\frac{\mathrm{d}{R}_0(\alpha )}{\mathrm{d}\alpha }\le 0$ for all $\alpha \in [0,1]$, implying that ${\alpha }^{*}=1$.

2. If $f(0)\le 0$: Here, $f(\alpha )\le 0$ and $\frac{\mathrm{d}{R}_0(\alpha )}{\mathrm{d}\alpha }\ge 0$ for all $\alpha \in [0,1]$, implying that ${\alpha }^{*}=0$.

3. If $f(1)<0$ and $f(0)>0$: In this case, there exists a unique $\widehat{\alpha }\in (0,1)$ such that $f(\widehat{\alpha })=0$. For $\alpha \in [0,\widehat{\alpha }]$, we have $f(\alpha )\ge 0$ and $\frac{\mathrm{d}{R}_0(\alpha )}{\mathrm{d}\alpha }\le 0$; for $\alpha \in [\widehat{\alpha },1]$, we have $f(\alpha )\le 0$ and $\frac{\mathrm{d}{R}_0(\alpha )}{\mathrm{d}\alpha }\ge 0$. Thus, the optimal solution is ${\alpha }^{*}=\widehat{\alpha }$, where:

$$\widehat{\alpha }=1+\frac{B}{k_2}+\frac{{\beta }_S{w}_U}{{\beta }_M{k}_2}-\frac{\sqrt{w_U{k}_1[{\beta }_M{\beta }_S{k}_1{k}_2+({\beta }_{M}B+{\beta }_S{w}_U){\beta }_S{k}_1+({\beta }_M{b}_U+A{\beta }_S){\beta }_M{k}_2]}}{{\beta }_M{k}_1{k}_2}.$$

Thus, the theorem is proved.

A.5. Proof of Proposition 3

Denote $\stackrel{\sim }{\alpha }$ as the root of $f(\alpha )$ on the left‐hand side. From Theorem 2, we know that $\stackrel{\sim }{\alpha }$ exists and that ${\alpha }^{*}=\min \{1,\stackrel{\sim }{\alpha }\}{\mathbb{I}}_{\{\stackrel{\sim }{\alpha }>0\}}$. Therefore, it is sufficient to discuss the case where ${\alpha }^{*}\in (0,1)$.

We start by computing the derivative of ${\alpha }^{*}$ with respect to ${\beta }_S$:

$$\frac{\partial {\alpha }^{*}}{\partial {\beta }_S}=\frac{w_U}{{\beta }_M{k}_2}-\frac{\sqrt{w_U{k}_1}}{2{\beta }_M{k}_1{k}_2}\frac{2w_U{k}_1{\beta }_S+(B{k}_1+k_1{k}_2+A{k}_2){\beta }_M}{\sqrt{(B{k}_1+k_1{k}_2+A{k}_2){\beta }_M{\beta }_S+{\beta }_M^2{b}_U{k}_2+{\beta }_S^2{w}_U{k}_1}}.$$

Let $P=B{k}_1+k_1{k}_2+A{k}_2$ and $Q({\beta }_S)=2w_U{k}_1{\beta }_S+P{\beta }_M$. Additionally, define $g({\beta }_S)=2\sqrt{w_U{k}_{1}Q({\beta }_S)}-Q^{\prime }({\beta }_S)$, where $Q^{\prime }({\beta }_S)$ is the derivative of $Q({\beta }_S)$. Then, we can rewrite the derivative as:

$$\frac{\partial {\alpha }^{*}}{\partial {\beta }_S}=\frac{\sqrt{w_U}}{2{\beta }_M{k}_2\sqrt{k_{1}Q({\beta }_S)}}g({\beta }_S).$$

Next, we compute the second derivative of ${\alpha }^{*}$ with respect to ${\beta }_S$:

$$\frac{{\partial }^2{\alpha }^{*}}{\partial {\beta }_S^2}=\frac{1}{2{\beta }_M{k}_2}{\left(\frac{w_U}{k_1}\right)}^{\frac{1}{2}}\left[{\left(Q({\beta }_S)\right)}^{-\frac{1}{2}}\frac{w_U{k}_1}{Q({\beta }_S)}\left(Q^{\prime }({\beta }_S)-2\sqrt{w_U{k}_{1}Q({\beta }_S)}\right)-\frac{g({\beta }_S)}{2}{\left(Q({\beta }_S)\right)}^{-\frac{3}{2}}\right].$$

Since $P=B{k}_1+k_1{k}_2+A{k}_2⩾2\sqrt{(B{k}_1+k_1{k}_2)A{k}_2}⩾2\sqrt{BA{k}_1{k}_2}⩾2\sqrt{w_U{b}_U{k}_1{k}_2}$, we have $P^2⩾4w_U{b}_U{k}_1{k}_2$.

Given that $Q^{\prime }({\beta }_S)=P{\beta }_M{\beta }_S+{\beta }_M^2{b}_U{k}_2+{\beta }_S^2{w}_U{k}_1$, we conclude:

$$Q^{\prime 2}({\beta }_S)⩾4w_U{k}_{1}Q({\beta }_S)\Rightarrow Q^{\prime }({\beta }_S)⩾2\sqrt{w_U{k}_{1}Q({\beta }_S)}.$$

Thus, $g({\beta }_S)⩽0$, implying $\frac{\partial {\alpha }^{*}}{\partial {\beta }_S}⩽0$ and $\frac{{\partial }^2{\alpha }^{*}}{\partial {\beta }_S^2}⩾0$. Thus, the proposition is proved.

A.6. Proof of Proposition 4

We can rewrite ${\alpha }^{*}$ as follows:

$${\alpha }^{*}=1+\frac{B}{k_2}+\frac{{\beta }_S{w}_U}{{\beta }_M{k}_2}-\frac{\sqrt{w_U\left[{\beta }_M{\beta }_S{k}_2+({\beta }_{M}B+{\beta }_S{w}_U){\beta }_S+({\beta }_M{b}_U+A{\beta }_S){\beta }_M\frac{k_2}{k_1}\right]}}{{\beta }_M{k}_2}.$$

We first compute the derivative of ${\alpha }^{*}$ with respect to $k_1$:

$$\frac{\partial {\alpha }^{*}}{\partial k_1}=\frac{\sqrt{w_U}({\beta }_M{b}_U+A{\beta }_S){\beta }_M{k}_2}{2{\beta }_M{k}_1^2{k}_2\sqrt{{\beta }_M{\beta }_S{k}_2+({\beta }_{M}B+{\beta }_S{w}_U){\beta }_S+({\beta }_M{b}_U+A{\beta }_S){\beta }_M\frac{k_2}{k_1}}}⩾0.$$

Therefore, conclusion 1 is proved.

Next, we rewrite ${\alpha }^{*}$ as:

$${\alpha }^{*}=1+\frac{B}{k_2}+\frac{{\beta }_S{w}_U}{{\beta }_M{k}_2}-\frac{\sqrt{w_U{k}_1\left[{\beta }_M{\beta }_S{k}_1\frac{1}{k_2}+({\beta }_{M}B+{\beta }_S{w}_U){\beta }_S{k}_1\frac{1}{k_2^2}+({\beta }_M{b}_U+A{\beta }_S){\beta }_M\frac{1}{k_2}\right]}}{{\beta }_M{k}_1}.$$

Now, we compute the derivative of ${\alpha }^{*}$ with respect to $k_2$:

$$\frac{\partial {\alpha }^{*}}{\partial k_2}=-\frac{{\beta }_{M}B+{\beta }_S{w}_U}{{\beta }_M{k}_2^2}+\frac{\sqrt{w_U{k}_1}\left\{\frac{1}{k_2^2}\left({\beta }_M{\beta }_S{k}_1+({\beta }_M{b}_U+A{\beta }_S){\beta }_M\right)+\frac{2}{k_2^3}({\beta }_{M}B+{\beta }_S{w}_U){\beta }_S{k}_1\right\}}{2{\beta }_M{k}_1\sqrt{{\beta }_M{\beta }_S{k}_1\frac{1}{k_2}+({\beta }_{M}B+{\beta }_S{w}_U){\beta }_S{k}_1\frac{1}{k_2^2}+({\beta }_M{b}_U+A{\beta }_S){\beta }_M\frac{1}{k_2}}}.$$

This simplifies to:

$$\frac{\partial {\alpha }^{*}}{\partial k_2}=-\frac{D}{{\beta }_M{\beta }_S{k}_1{k}_2^2}+\frac{\sqrt{w_U{k}_1}\left(\frac{1}{k_2^2}C+\frac{2}{k_2^3}D\right)}{2{\beta }_M{k}_1\sqrt{\frac{1}{k_2}C+\frac{1}{k_2^2}D}}.$$

Further simplifying:

$$\frac{\partial {\alpha }^{*}}{\partial k_2}=\frac{-4D^2(C{k}_2+D)+E{(C{k}_2+2D)}^2}{2{\beta }_M{\beta }_S{k}_1{k}_2^3\sqrt{C+\frac{1}{k_2}D}\left[2D\sqrt{C+\frac{1}{k_2}D}+{\beta }_S\sqrt{w_U{k}_1{k}_2}(C+\frac{1}{k_2}2D)\right]}.$$

Now, denote:

$$f(k_2)=-4D^2(C{k}_2+D)+E{(C{k}_2+2D)}^2=E{C}^2{k}_2^2-4CD(D-E)k_2-4D^2(D-E),$$

where $f(k_2)$ and $\frac{\partial {\alpha }^{*}}{\partial k_2}$ have the same sign.

Considering $f(k_2)$ as a quadratic function of $k_2$, its discriminant is:

$$\mathrm{\Delta }=b^2-4ac=16C^2{D}^3(D-E)>0,$$

and the axis of symmetry is:

$$-\frac{b}{2a}=\frac{2CD(D-E)}{E{C}^2}>0.$$

Thus, the positive root of $f(k_2)$ is:

$$\frac{2D\left[(D-E)+\sqrt{D(D-E)}\right]}{EC}.$$

Given that $k_2\in (0,1-q_S]$, we conclude that: If $k_2\in (0,h)$, then $f(k_2)<0$, implying $\frac{\partial {\alpha }^{*}}{\partial k_2}<0$; If $k_2\in [h,1-q_S]$, then $\frac{\partial {\alpha }^{*}}{\partial k_2}⩾0$. Therefore, conclusions 2 and 3 are proved.

A.7. Proof of Proposition 5

Considering the partial derivatives of ${\stackrel{\sim }{R}}_0$ with respect to $q_S^{*}$ and $q_M^{*}$, we obtain:

$$\frac{\partial {\stackrel{\sim }{R}}_0}{\partial q_M^{*}}=\frac{-(d+{\overline{d}}_U+b_U+q_S^{*})}{{\left[(v_U+d+w_U+q_M^{*})(d+{\overline{d}}_U+b_U+q_S^{*})-w_U{b}_U\right]}^2}<0,$$

$$\frac{\partial {\stackrel{\sim }{R}}_0}{\partial q_S^{*}}=-\frac{N_0{\beta }_M{w}_U{b}_U+N_0{\beta }_S{w}_U(v_U+d+w_U+q_M^{*})}{{\left[(v_U+d+w_U+q_M^{*})(d+{\overline{d}}_U+b_U+q_S^{*})-w_U{b}_U\right]}^2}<0.$$

Since $\frac{\partial q_M^{*}}{\partial k_1}<0$ and $\frac{\partial q_S^{*}}{\partial k_2}<0$, we conclude that $\frac{\partial {\stackrel{\sim }{R}}_0}{\partial k_1}<0$ and $\frac{\partial {\stackrel{\sim }{R}}_0}{\partial k_2}<0$. Thus, the proposition is proved.

A.8. Proof of Theorem 3

Here, we use the superscripts $o$ and $n$ to distinguish the variables in model (1) when the optimal strategy is employed versus when it is not employed. By the comparison theorem for ordinary differential equations, it is straightforward to verify that for every $t$, $S^o(t)\ge S^n(t)$, $I_M^{U,o}(t)\le I_M^{U,n}(t)$, and $I_S^{U,o}(t)\le I_S^{U,n}(t)$. Consequently, for the endpoint of the epidemic, we have $T^o\le T^n$. Furthermore, for any $t$, we find that $M^o(t)\le M^n(t)$, which implies that ${\pi }^{\ast }\le {\pi }^0$.