Mathematical evaluation of the role of cross immunity and nonlinear incidence rate on the transmission dynamics of two dengue serotypes

Dengue fever is a common disease which can cause shock, internal bleeding, and death in patients if a second infection is involved. In this paper, a multi-serotype dengue model with nonlinear incidence rate is formulated to study the transmission of two dengue serotypes. The dynamical behaviors of the proposed model depend on the threshold value R0n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{{0}}^{{n}}$\end{document} known as the reproductive number which depends on the associated reproductive numbers with serotype-1 and serotype-2. The value of R0n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{{0}}^{{n}}$\end{document} is used to reflect whether the disease dies out or becomes endemic. It is found that the proposed model has a globally stable disease-free equilibrium if R0n≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{{0}}^{{n}}\leq 1$\end{document}, which indicates that if public health measures that make (and keep) the threshold to a value less than unity are carried out, the strategy in disease control is effective in the sense that the number of infected human and mosquito populations in the community will be brought to zero irrespective of the initial sizes of sub-populations. When R0n>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R_{{0}}^{{n}}>1$\end{document}, the endemic equilibria called the co-existence primary and secondary infection equilibria are locally asymptotically stable. The effects of cross immunity and nonlinear incidence rate are explored using data from Thailand to determine the effective strategy in controlling and preventing dengue transmission and reinfection.