Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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A MATHEMATICAL MODEL OF TRANSMISSION OF PLASMODIUM VIVAX MALARIA WITH A CONSTANT TIME DELAY FROM INFECTION TO INFECTIOUS

  • Kammanee, Athassawat (Applied Analysis Research Unit Department of Mathematics and Statistics Faculty of Science Prince of Songkla University) ;
  • Tansuiy, Orawan (Department of Mathematics and Statistics Faculty of Science Prince of Songkla University)
  • Received : 2018.04.20
  • Accepted : 2019.02.20
  • Published : 2019.04.30
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Abstract

This research is focused on a continuous epidemic model of transmission of Plasmodium vivax malaria with a time delay. The model is represented as a system of ordinary differential equations with delay. There are two equilibria, which are the disease-free state and the endemic equilibrium, depending on the basic reproduction number, $R_0$, which is calculated and decreases with the time delay. Moreover, the disease-free equilibrium is locally asymptotically stable if $R_0<1$. If $R_0>1$, a unique endemic steady state exists and is locally stable. Furthermore, Hopf bifurcation is applied to determine the conditions for periodic solutions.

Keywords

DBSHCJ_2019_v34n2_685_f0001.png 이미지

FIGURE 1. Flows in a compartmental model for the transmission of Plasmodium vivax malaria

DBSHCJ_2019_v34n2_685_f0002.png 이미지

FIGURE 2. Computer simulations of the model equations (8)-(10) demonstrating the case R0 > 1 and τ < τ0 with a stable endemic state. The plots show time traces of the solution in (a) (Ih, Dh)-plane (b) (Ih, Iv)-plane, (c) (Dh, Iv)-plane and (d) plotting Ih(−), Dh(−−), Iv(..) versus t.

DBSHCJ_2019_v34n2_685_f0003.png 이미지

FIGURE 3. Computer simulations of the model equations (8)-(10) demonstrating a case with R0 > 1 and τ = τ0 (i.e., the bi-furcation point) with a limit cycle expected theoretically. The plots show time traces in (a) (Ih, Dh)-plane (b) (Ih, Iv)-plane,(c)(Dh, Iv)-plane and (D) plotting Ih(−), Dh(−−), Iv(..) versus t.

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