To reduce the numbers of infected and hospitalized individuals while controlling intervention costs, we define the objective functional as

where T is the terminal time; A 1 , A 2 > 0 represent the social costs associated with infected and hospitalized individuals, respectively; and B 1 , B 2 , B 3 > 0 represent the implementation costs of the three control measures. These coefficients are selected to balance epidemic reduction and intervention costs in the numerical experiments.

Theorem 1: The optimal control problem defined by system (2) and objective functional (3) admits an optimal control

such that

Proof: To prove the existence of the optimal control, we verify the conditions of the Fleming-Rishel optimal control existence theorem.

First, the admissible control set U is nonempty, closed, and convex.

Second, for any u i ∈ U , i = 1 , 2 , 3 , since 1 + α I 2 > 0 , the right-hand side of system (2) is continuous with respect to the state variables in the feasible region. Moreover, the system satisfies the local Lipschitz condition with respect to the state variables. Therefore, the state system admits a unique solution for every admissible control.

Third, since the uncontrolled system (1) is bounded, the controlled system (2) is also bounded. In addition, the terms related to the control variables in system (2) can be rewritten as

and

Hence, the state system is linear with respect to the control variables u 1 , u 2 , u 3 .

Fourth, the integrand of the objective functional is convex with respect to the control variables. Define

Since the quadratic terms are positive and B 1 , B 2 , B 3 > 0 , L is convex with respect to u 1 ( t ) , u 2 ( t ) , u 3 ( t ) .

Finally, the integrand satisfies the required growth condition:

Therefore, all the assumptions of the Fleming-Rishel optimal control existence theorem are satisfied. Hence, there exists an optimal control

such that

The optimal control problem can be transformed into an equivalent Hamiltonian system by applying Pontryagin’s Maximum Principle [8]. For this purpose, the Hamiltonian function is defined as

where λ i , i = 1 , 2 , 3 , 4 are the adjoint variables associated with the state variables S , I , H , R , respectively.

Theorem 2: Let u * = ( u 1 * , u 2 * , u 3 * ) be the optimal control. Then there exist adjoint variables λ i , i = 1 , 2 , 3 , 4 satisfying the following adjoint system:

with the transversality conditions λ i ( T ) = 0 , i = 1 , 2 , 3 , 4 . The optimal controls are given by

Proof: According to Pontryagin’s Maximum Principle, there exist adjoint variables λ i , i = 1 , 2 , 3 , 4 such that the Hamiltonian satisfies d λ i d t = − ∂ H ∂ x i , x i = S , I , H , R , with the transversality conditions λ i ( T ) = 0 , i = 1 , 2 , 3 , 4 .

Taking the partial derivatives of the Hamiltonian with respect to the state variables S , I , H , R , respectively, yields the adjoint system (5).

Furthermore, the optimality conditions satisfy d H d u i = 0 , i = 1 , 2 , 3 .

Solving these equations gives the corresponding interior optimal controls:

Considering the control constraints

the optimal controls are obtained by projecting the interior solutions onto the admissible control set, which leads to Equations (6)-(8).

The parameter values used in the numerical simulations are selected based on biological plausibility and previous epidemic modeling studies. Since this work focuses on evaluating the qualitative effects of different control strategies, the parameters are not obtained through calibration with a specific outbreak dataset. The parameter values used in the numerical simulations are selected as follows (Table 1):

Table 1. Parameter values used in the numerical simulations.

For these parameter values, the basic reproduction number is calculated as R 0 = 4.183 > 1 .

The control cost parameters are selected as A 1 = 1 , A 2 = 1.4 , B 1 = 120 , B 2 = 8 , B 3 = 80 .

Considering the limitations of medical technology and implementation costs, the upper bounds of the three control variables are set as u 1 , max = 0.75 , u 2 , max = 0.95 , u 3 , max = 0.35 .

To investigate the effects of the main parameters on disease transmission dynamics, the parameters β , α , k , and r are varied individually while keeping the other parameters unchanged under the uncontrolled scenario. Figure 1 and Figure 2 illustrate the effects of these parameters on the number of infected individuals I ( t ) and hospitalized individuals H ( t ) , respectively.

As shown in Figure 1, the number of infected individuals increases significantly with the increase of the transmission rate β , indicating that reducing the effective transmission rate is a key measure for controlling the scale of disease transmission. With the increase of the nonlinear inhibition coefficient α , the number of infected individuals decreases due to the enhanced inhibitory effect of the nonlinear incidence term. This result reflects the potential role of preventive awareness and social intervention measures in reducing disease transmission.

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Figure 1. Effects of β , α , k , and r on I ( t ) .

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Figure 2. Effects of β , α , k , and r on H ( t ) .

Comparing Figure 1 and Figure 2, it can be observed that the medical capacity k and the hospitalization transfer rate r jointly affect the distribution relationship between infected individuals and hospitalized individuals. A larger k alleviates the limitation of hospital capacity, allowing more infected individuals to receive hospitalization treatment. Consequently, the number of infected individuals I ( t ) decreases, while the number of hospitalized individuals H ( t ) increases. A larger r accelerates the transition from infected individuals to hospitalized individuals, which is reflected by a reduction in the number of infected individuals and an increase in the number of hospitalized individuals. These results indicate that medical capacity and hospitalization transfer rate do not directly eliminate infections, but rather modify the dynamic transition process between infected and hospitalized compartments.

To evaluate the effects of the three control measures, the following five control strategies are considered:

1) No control, namely u 1 = u 2 = u 3 = 0 ;

2) Only the transmission reduction control u 1 ;

3) Only the medical-resource saturation mitigation control u 2 ;

4) Only the treatment efficiency improvement control u 3 ;

5) The combined control strategy involving u 1 , u 2 , u 3 .

As shown in Figure 3 and Figure 4, different control strategies have distinct effects on the dynamics of infected and hospitalized individuals. When only the transmission reduction control u 1 is implemented, the number of infected individuals decreases significantly, and the number of hospitalized individuals also decreases accordingly. This indicates that reducing the effective transmission rate plays an important role in controlling disease spread.

When only the treatment efficiency improvement control u 3 is applied, the number of hospitalized individuals is significantly reduced, while its effect on reducing the number of infected individuals is relatively limited. This is because u 3 mainly improves the recovery efficiency of hospitalized individuals rather than directly affecting the infection process.

When only the medical-resource saturation mitigation control u 2 is adopted, the number of infected individuals is close to that in the uncontrolled scenario, and the number of hospitalized individuals does not decrease significantly, or may even be slightly higher than that without control. This is because the primary role of u 2 is not to directly reduce infections or accelerate recovery, but to alleviate the restriction caused by limited medical capacity on the hospitalization process.

The combined control strategy simultaneously reduces transmission risk, alleviates medical-resource saturation, and improves treatment efficiency. Therefore, it achieves the most significant reduction in both I ( t ) and H ( t ) , demonstrating the superiority of the integrated intervention strategy.

Figure 3. Dynamics of I ( t ) under different control strategies.

Figure 4. Dynamics of H ( t ) under different control strategies.

Figure 5 presents the optimal trajectories of the three control variables under the combined control strategy. It can be observed that the transmission reduction control u 1 remains almost at its upper bound throughout the entire simulation period. This indicates that, under the selected weighting parameters, reducing the effective transmission rate is the dominant intervention strategy for controlling disease transmission.

The medical-resource saturation mitigation control u 2 initially remains close to its maximum value during the early stage of the epidemic, indicating that medical-resource allocation is particularly important when the disease burden is high. Subsequently, u 2 decreases rapidly and remains at a moderate level during the later period, suggesting that the pressure on healthcare resources gradually declines as the number of infected individuals decreases.

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Figure 5. Optimal trajectories of u 1 , u 2 , u 3 under the combined control strategy.

The treatment efficiency improvement control u 3 maintains a positive value throughout the simulation period, although its magnitude is lower than those of u 1 and u 2 . This indicates that improving treatment efficiency serves as a continuous auxiliary measure, which works together with transmission reduction and medical-resource management to alleviate hospitalization pressure.

These results demonstrate that, under the given parameter settings and cost weights, reducing the effective transmission rate remains the most cost-effective strategy for controlling disease spread, while medical-resource allocation and treatment enhancement play important complementary roles.

Since the social cost terms in the objective functional are positively associated with I ( t ) and H ( t ) , the cumulative disease burden can be used to evaluate the long-term effects of different control strategies. Therefore, we define B I = ∫ 0 T I ( t ) d t and B H = ∫ 0 T H ( t ) d t , which represent the cumulative infection burden and cumulative hospitalization burden, respectively.

Figure 6. Cumulative infection burden under different control strategies.

Figure 7. Cumulative hospitalization burden under different control strategies.

Figure 6 and Figure 7 illustrate the comparison of these two indicators under different control strategies. It can be observed that the combined control strategy achieves the lowest cumulative infection burden and cumulative hospitalization burden.

Specifically, under the no-control scenario, the cumulative infection burden and cumulative hospitalization burden are B I = 9080.7874 and B H = 1856.3709 , respectively. Under the combined control strategy, these values decrease to 512.8115 and 233.6586, respectively.

When only the transmission reduction control u 1 is implemented, the cumulative infection burden and cumulative hospitalization burden decrease to 1116.5956 and 566.7082, respectively, indicating that reducing the effective transmission rate plays a significant role in controlling disease spread.

When only the treatment efficiency improvement control u 3 is applied, the cumulative hospitalization burden decreases to 1347.2779, while the cumulative infection burden remains relatively high at 8109.1681. This demonstrates that improving treatment efficiency mainly alleviates hospitalization pressure but has a limited effect on reducing infection transmission.

When only the medical-resource saturation mitigation control u 2 is adopted, the cumulative infection burden is 9080.3431, which is close to the uncontrolled scenario, while the cumulative hospitalization burden is 1856.6231, slightly higher than that without control. This result indicates that medical-resource saturation mitigation mainly improves the capacity for hospitalization rather than directly reducing the number of infections. Therefore, it should be combined with transmission reduction and treatment enhancement strategies to achieve better epidemic control.

Since u 2 is introduced to alleviate medical-resource saturation, the effective saturation pressure ( 1 − u 2 ) H ( t ) k is further investigated.

Figure 8. Medical-resource saturation pressure under three different control strategies.

As shown in Figure 8, without control, the medical-resource saturation pressure H k remains at a relatively high level throughout the simulation period.

When only the medical-resource saturation mitigation control u 2 is applied, this indicator does not decrease significantly, indicating that increasing medical-resource allocation alone cannot fundamentally reduce hospitalization pressure.

In contrast, the combined control strategy significantly reduces the effective saturation pressure. This result demonstrates that medical-resource mitigation measures need to be implemented together with transmission reduction and treatment efficiency improvement strategies to achieve better epidemic control effects.

The medical capacity parameter k represents a key indicator of limited healthcare resources in the proposed model. In system (1), the effective transition rate from

infected individuals to hospitalized individuals is given by r I ( 1 − H k ) , where H k represents the proportion of hospitalized individuals relative to the available

medical capacity. When H approaches k , the ability of infected individuals to enter the hospitalization stage is significantly restricted due to healthcare resource saturation.

Therefore, in this section, the effects of different medical capacities on disease transmission dynamics are investigated while keeping all other parameters unchanged. The following four scenarios are considered:

where k 0 = 16.8749 . To highlight the role of medical capacity, the effects of different values of on the numbers of infected individuals, hospitalized individuals, and medical-resource saturation pressure are simulated under the combined optimal control strategy.

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Figure 9. Effects of different medical capacities on I ( t ) , H ( t ) and H k .

As shown in Figure 9, the number of infected individuals decreases significantly with increasing medical capacity k . This is because an increase in healthcare capacity weakens the restriction imposed by limited medical resources on the hospitalization transition process, allowing infected individuals to enter the hospitalization stage more efficiently and reducing the number of individuals who continue transmitting the disease in the community.

In Figure 9(b), the hospitalization curves under different values of k exhibit similar peak values during the early stage of the epidemic. However, smaller medical capacities lead to higher numbers of hospitalized individuals in the later stage, indicating that insufficient healthcare capacity prolongs hospitalization pressure.

More importantly, the medical-resource saturation pressure H k decreases significantly as k increases, demonstrating that medical capacity expansion can effectively alleviate healthcare system saturation.

Furthermore, the peak number of infected individuals I max , the peak number of hospitalized individuals H max , the cumulative infection burden B I = ∫ 0 T I ( t ) d t , the cumulative hospitalization burden B H = ∫ 0 T H ( t ) d t , and the maximum medical-resource saturation pressure max ( H k ) are calculated. The results are presented in Table 2.

Table 2. Effects of different medical capacities on epidemic indicators.

As shown in Table 2, when the medical capacity k increases from 8.4375 to 33.7498, the peak number of infected individuals decreases from 47.4327 to 26.9087. Meanwhile, the cumulative infection burden decreases from 796.1782 to 439.4709, and the cumulative hospitalization burden decreases from 315.9099 to 207.5966.

At the same time, the maximum medical-resource saturation pressure decreases significantly from 0.7467 to 0.1885. Although the peak number of hospitalized individuals H max slightly increases with the increase of k , this phenomenon occurs because a larger medical capacity weakens the restriction caused by limited healthcare resources, allowing more infected individuals to enter the

hospitalization stage during the early period of the epidemic. From the perspectives of the cumulative hospitalization burden and the saturation pressure H k , a larger medical capacity can effectively alleviate long-term hospitalization pressure.

Therefore, the expansion of medical capacity not only reduces the saturation pressure of the healthcare system but also decreases the peak number of infected individuals and the overall disease burden by improving the transition efficiency from infected individuals to hospitalized individuals. This result highlights the importance of incorporating medical capacity constraints into epidemic models. In practical epidemic control, increasing healthcare capacity should not only be regarded as improving hospital admission capability, but also as an intervention measure that can influence disease dynamics by modifying the transition process between community infection and hospitalization.

In summary, the numerical results demonstrate that although individual control measures can reduce the number of infected or hospitalized individuals to some extent, their effects remain limited. The combined control strategy, together with sufficient medical capacity, can simultaneously reduce infection burden, hospitalization burden, and medical-resource saturation pressure, providing a more effective and comprehensive approach for epidemic control.