Study will be conducted with 2 stages.

The first stage will explore the suitable dose of duvelisib when combined with SG001 injection in patients with advanced solid tumors who had failed with prior systemic therapy.

The second stage will explore the safety and tolerability, preliminary anti-tumor efficacy and PK data of duvelisib monotherapy and combo regime with SG001 injection in patients with advanced solid tumors which will including but not limited with esophageal carcinoma, gastric carcinoma, colorectal cancer and head and neck squamous carcinoma.

Inclusion Criteria:

1. Age ≥18 years old;
2. Histologically or cytologically proven metastatic or locally advanced solid tumors, including but not limited to esophageal cancer, gastric cancer, colorectal cancer and head and neck squamous cancer;
3. Prior treatment with PD-1 inhibitor containing regimen and disease progression on imaging or cytohistopathology;
4. Eastern Cooperative Oncology Group performance status (ECOG) performance status of 0 to 1;
5. At least one measurable lesion per RECIST v1.1;
6. Adequate laboratory function including hepatic function, renal function, and blood cell examination;
7. Ability to provide archived tumour tissue samples or newly obtained puncture biopsies or excisional biopsies from tumour lesions that have not previously received radiotherapy;
8. Life expectancy ≥12 weeks; Fully understand the study and voluntarily sign the informed consent form.

Exclusion Criteria:

1. Have hypersensitivity experience with content of duvelisib capsule or tislelizumab;

2. Has received prior therapy with other PI3K inhibitors or BTK inhibitors;

3. Has received anti-tumour agent (including but not limited to chemotherapy, target therapy, anti-angiogenesis therapy, immune therapy, radiotherapy, and tumour embolism therapy etc.) within 28 days before the first dose administration;

4. Has administrated with systemic immune inhibitors within 28 days prior to the first, except: topical glucocorticoids by nasal spray, inhalation or other routes, or physiological doses of systemic glucocorticoids (not beyond 10 mg/d of prednisone or equivalent dose);

5. Has received a live virus vaccine within 28 days of firs dose or planned during the trial period. Seasonal influence vaccine without live virus vaccine is permitted;

6. Has prior allograft solid or blood stem cell transplant;

7. Has had major surgery within 28 days prior to the first dose or un-healed wound, ulceration, or bone fraction at screening;

8. Presence of toxicity not recovered to CTCAE v5.0 ≤ Grade 1 from previous anti-tumour therapy prior to first dose, except for alopecia or no clinically significant abnormalities in laboratory tests;

9. Has hydrothorax or ascites or hydropericardium with symptom or need drainage therapy. Just radiological minor hydrothorax or ascites or hydropericardium without symptom was not excluded;

10. Has an active autoimmune disease requiring systemic treatment or immunosuppressive agents within the past 2 years. Replacement therapy is permitted (e.g. thyroxine, insulin or physiological corticosteroids for adrenal or pituitary insufficiency);

11. Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Patients with previously treated brain metastases may participate provided they are stable, and meet the following criteria: neurological symptoms have recovered to CTCAE v5.0 ≤ Grade 1 at least 2 weeks prior to the first dose; no imaging evidence of new brain metastases or enlarged brain metastases within 4 weeks prior to the first dose; patients has not been treated with corticosteroids since at least 3 days prior to the first dose or is receiving a stable dose, or a tapered dose of ≥ 10 mg/day of prednisone (or equivalent);

12. Has a history of any of the following cardiovascular conditions:

    1. Class Ⅱ or above congestive heart failure per New York Heart Association, unstable angina pectoris, myocardial infarction within 6 month prior to first dose, arrythmia require treatment, LVEF<50% during screening;
    2. Primary myocardiopathy (e.g. dilated cardiomyopathy, hypertrophic cardiomyopathy, arrhythmogenic right ventricular cardiomyopathy, restrictive cardiomyopathy, undefined cardiomyopathy);
    3. History of clinically significant corrected for heart rate (QTc) interval prolongation, or screening QTc > 470 msec (female) or > 450 msec (male);
    4. Coronary heart disease requiring drug therapy during screening;
    5. Has cerebrovascular incidence (including transient cerebral ischemic attack);
    6. Poorly controlled hypertension with oral drugs, systolic blood pressure (SBP) ≥ 150 mm Hg and/or diastolic blood pressure (DBP) ≥ 100 mm Hg;
    7. Other cardio-cerebral vascular disease with investigator's judgement;

13. Interstitial lung disease proved by CT or MRI during screening, or history of lung disease requiring oral or intravenous steroid hormone therapy within 6 months prior to first dose, or a previous history of severe lung function damage;

14. Active lung tuberculosis or received anti-tuberculosis therapy within 1 years prior to first dose;

15. Has a clinically significant gastrointestinal abnormality unable to oral take duvelisib capsule, such as major intestine resection, intra-intestinal plant, Crohn disease, ulcerative colitis, continuous diarrhoea, or gastro-intestinal perforation; gastrointestinal perforation and/or fistula, peptic ulcer, fully intestinal obstruction, or incomplete intestinal obstruction requiring full-extra intestinal nutrition within 6 months prior to first dose;

16. Has an active infection requiring systemic intravenous antibiotics contented therapy;

17. Has known active Hepatitis B (Hepatitis B surface antigen positive with HBV-DNA ≥10^4copies/mL) or Hepatitis C virus (HCV antibody positive with HCV RNA ≥10^3copies/mL). HIV/AIDS positive or other severe infectious disease;

18. Patients cannot receive prohibition therapy for Pneumocystis Kanosh's, herpes simplex virus (HSV), herpes zoster (VZV) during trial process base on investigator's judgement;

19. Has a history of pulmonary embolism within 6 months prior to first dose, or deep vein thrombosis or any other serious venous thromboembolic event within 3 months prior to first dose;

20. Has a known history of additional malignancy. Carcinoma in situ, non-melanoma cutaneous carcinoma which resolved at least 2 years prior to first dose and will not requiring additional treatment during the study period;

21. Cannot or unwilling agree to use an adequate method of contraception, starting with the first dose of study therapy through 12 weeks after the last dose of duvelisib or 150 days after the lase dose of tislelizumab whichever occurs first. Or pregnant or breasting women;

22. Other conditions that, in the opinion of the investigator, make participation in the study unsuitable, including but not limited with psychiatric or substance abuse disorders.

When the research is completed, we will publish the results in the form of conference reports and papers.