---

%PDF-1.5 %���� 1 0 obj << /S /GoTo /D (section.1) >> endobj 4 0 obj (1. Introduction) endobj 5 0 obj << /S /GoTo /D (section.2) >> endobj 8 0 obj (2. The model) endobj 9 0 obj << /S /GoTo /D (subsection.2.1) >> endobj 12 0 obj (2.1. Model formulation) endobj 13 0 obj << /S /GoTo /D (subsection.2.2) >> endobj 16 0 obj (2.2. Basic properties) endobj 17 0 obj << /S /GoTo /D (section.3) >> endobj 20 0 obj (3. Basic reproduction number R0 and disease-free equilibrium) endobj 21 0 obj << /S /GoTo /D (subsection.3.1) >> endobj 24 0 obj (3.1. Computation of R0) endobj 25 0 obj << /S /GoTo /D (subsection.3.2) >> endobj 28 0 obj (3.2. Sensitivity analysis of R0) endobj 29 0 obj << /S /GoTo /D (subsection.3.3) >> endobj 32 0 obj (3.3. Global stability of the disease-free equilibrium ) endobj 33 0 obj << /S /GoTo /D (section.4) >> endobj 36 0 obj (4. Endemic equilibrium) endobj 37 0 obj << /S /GoTo /D (subsection.4.1) >> endobj 40 0 obj (4.1. Existence and uniqueness) endobj 41 0 obj << /S /GoTo /D (subsection.4.2) >> endobj 44 0 obj (4.2. Global stability) endobj 45 0 obj << /S /GoTo /D (section.5) >> endobj 48 0 obj (5. Numerical simulations: Case study of Hepatitis B \(HBV\)) endobj 49 0 obj << /S /GoTo /D (subsection.5.1) >> endobj 52 0 obj (5.1. Simulations of stability results) endobj 53 0 obj << /S /GoTo /D (subsection.5.2) >> endobj 56 0 obj (5.2. Impact of effectiveness of vaccination) endobj 57 0 obj << /S /GoTo /D (subsection.5.3) >> endobj 60 0 obj (5.3. Impact of asymptomatic carriers on the transmission dynamics) endobj 61 0 obj << /S /GoTo /D (section.6) >> endobj 64 0 obj (6. Conclusion-discussion) endobj 65 0 obj << /S /GoTo /D (section*.13) >> endobj 68 0 obj (Acknowledgments) endobj 69 0 obj << /S /GoTo /D (section*.14) >> endobj 72 0 obj (REFERENCES) endobj 73 0 obj << /S /GoTo /D [74 0 R /Fit] >> endobj 87 0 obj << /Length 3790 /Filter /FlateDecode >> stream xڵZKs�F��W�HV�0f�Ξ$���_k)�J99@$L"

������)g�js������|�����̟�zry����d{Y����3cc/N�,NC/�i=�4�0�Ef�����[�����,�A�/����� ��y�T/n��~�ħ�ͬe���'�OKk�8�������;�g|���/��3��[��n��ټ�^K�����`�V�%ҿ%�/��\�#��"+�z��{��~��fb�`쏼b�:@Cճ��T�6�b?��c!�Rd�oCG�Tda���k�LD^7><�z���_(� ���?��z���ڌ�.����K� i��D���&Rx�X��Q���S���c��^ߒҲ؂�2�I�0�-C�űp~�FAw�H��69���
�����b���x�?)m��)��f�V�f`�-�V�zΈ��^�l�?5K�rY� F#�~�IiO0��
v�S�̋5���o��9���'��í�󖄍���]1�Ǐ���#�`��sT�ͼ��\��8->oqL0![� CI2��鷂K�������+�0���h"��S7�M2�����8:;���)��,'/ [��֗���SA�H�=��AN
�v\ɝ[����Yॄ�i &$���7@��� ��jbuGϋ=���'��2z> �>�#vD����\�x*�/i:Ϲ�P���+����ެJ�^lK��_B�"�.���*ڮ���
��0+��2�y*���{��ˆE�CzSvBV:�Q6��a�.t�U��� �f0���ɼ�&���3�@��"�|��*i���Rq�.��ox \�4p�g�ЌP�w���[�珰������5i�����ҙ�k��D��,���-����Z���=K��E0<������oڒ�Q��6���9���}�[�3c�\�j�N��2CN�~:UF�����`+,� ��:)��q��w�K��+d,aE��Qt{�lPb�w�%���h~�-d�r��!B'䴓��2��1�a�Us
f�#쬤��KB?�T��<��}p$'�$�v>�)P

---