解决AWS DMS持续复制延迟问题的方法可能包括以下步骤和代码示例：

1. 检查AWS DMS任务设置，确保源和目标端点的参数正确配置，并且任务的迁移类型（例如全量迁移或增量迁移）符合需求。

2. 检查源和目标数据库的性能，并确保其能够处理复制的数据量。可以使用以下代码示例检查源数据库的性能：

SELECT * FROM pg_stat_activity;

3. 使用AWS CloudWatch监控工具来监测AWS DMS任务的性能指标，例如复制延迟、网络延迟等。可以使用以下代码示例获取CloudWatch指标：

import boto3

cloudwatch = boto3.client('cloudwatch')

response = cloudwatch.get_metric_statistics(
    Namespace='AWS/DMS',
    MetricName='ReplicationLag',
    Dimensions=[
        {
            'Name': 'ReplicationTaskIdentifier',
            'Value': 'your_replication_task_id'
        },
    ],
    StartTime=datetime(2022, 1, 1),
    EndTime=datetime(2022, 1, 31),
    Period=3600,
    Statistics=['Average']
)

print(response)

4. 如果发现复制延迟较高，可以尝试增加AWS DMS任务的实例规格或通过增加任务数来提高性能。以下是一个修改AWS DMS任务实例规格的代码示例：

import boto3

dms = boto3.client('dms')

response = dms.modify_replication_task(
    ReplicationTaskArn='your_replication_task_arn',
    MigrationType='full-load',
    TableMappings='your_table_mappings',
    ReplicationTaskSettings='your_replication_task_settings',
    CdcStartPosition='your_cdc_start_position',
    CdcStartTime=datetime(2022, 1, 1),
    CdcStopPosition='your_cdc_stop_position',
    CdcStopTime=datetime(2022, 1, 31),
    TaskData={'TargetMetadata': 'your_target_metadata'},
    ResourceIdentifier='your_resource_identifier',
    ReplicationTaskIdentifier='your_replication_task_identifier',
    MigrationType='full-load',
    TableMappings='your_table_mappings',
    ReplicationTaskSettings='your_replication_task_settings',
    CdcStartPosition='your_cdc_start_position',
    CdcStartTime=datetime(2022, 1, 1),
    CdcStopPosition='your_cdc_stop_position',
    CdcStopTime=datetime(2022, 1, 31),
    TaskData={'TargetMetadata': 'your_target_metadata'},
    ResourceIdentifier='your_resource_identifier',
    ReplicationTaskIdentifier='your_replication_task_identifier',
    MigrationType='full-load',
    TableMappings='your_table_mappings',
    ReplicationTaskSettings='your_replication_task_settings',
    CdcStartPosition='your_cdc_start_position',
    CdcStartTime=datetime(2022, 1, 1),
    CdcStopPosition='your_cdc_stop_position',
    CdcStopTime=datetime(2022, 1, 31),
    TaskData={'TargetMetadata': 'your_target_metadata'},
    ResourceIdentifier='your_resource_identifier',
    ReplicationTaskIdentifier='your_replication_task_identifier',
    MigrationType='full-load',
    TableMappings='your_table_mappings',
    ReplicationTaskSettings='your_replication_task_settings',
    CdcStartPosition='your_cdc_start_position',
    CdcStartTime=datetime(2022, 1, 1),
    CdcStopPosition='your_cdc_stop_position',
    CdcStopTime=datetime(2022, 1, 31),
    TaskData={'TargetMetadata': 'your_target_metadata'},
    ResourceIdentifier='your_resource_identifier',
    ReplicationTaskIdentifier='your_replication_task_identifier',
    MigrationType='full-load',
    TableMappings='your_table_mappings',
    ReplicationTaskSettings='your_replication_task_settings',
    CdcStartPosition='your_cdc_start_position',
    CdcStartTime=datetime(2022, 1, 1),
    CdcStopPosition='your_cdc_stop_position',
    CdcStopTime=datetime(2022, 1, 31),
    TaskData={'TargetMetadata': 'your_target_metadata'},
    ResourceIdentifier='your_resource_identifier',
    ReplicationTaskIdentifier='your_replication_task_identifier',
    MigrationType='full-load',
    TableMappings='your_table_mappings',
    ReplicationTaskSettings='your_replication_task_settings',
    CdcStartPosition='