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By analyzing the corresponding characteristic equations, the local stability of a disease-free equilibrium and an endemic equilibrium is established. We calculate the certain threshold known as the control reproduction number\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                      <mml:mrow>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi>\u211b<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi>v<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    . If\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                      <mml:mrow>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi>\u211b<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi>v<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                      <mml:mrow>\n                        <mml:mo>&lt;<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                      <mml:mrow>\n                        <mml:mo>1<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    , we obtain sufficient conditions for the global asymptotic stability of the disease-free equilibrium; the diseases will be eliminated from the community. By comparison of arguments, it is proved that if\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                      <mml:mrow>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi>\u211b<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi>v<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\">\n                      <mml:mrow>\n                        <mml:mo>&gt;<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\">\n                      <mml:mrow>\n                        <mml:mo>1<\/mml:mo>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    , the disease persists and the unique endemic equilibrium is globally asymptotically stable, which is obtained by the second compound matrix techniques and autonomous convergence theorems. We perform sensitivity analysis of\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\">\n                      <mml:mrow>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi>\u211b<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi>v<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    on the parameters in order to determine their relative importance to disease transmission and show that an imperfect vaccine is always beneficial in reducing disease spread within the community.\n                  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