# American Institute of Mathematical Sciences

## Persistence of mosquito vector and dengue: Impact of seasonal and diurnal temperature variations

 1 Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182, USA, Computational Science Research Center, San Diego State University, San Diego, CA 92182, USA, Viral Information Institute, San Diego State University, San Diego, CA 92182, USA 2 Department of Natural Science in the Center for General Education, Chang Gung University, Guishan, Taoyuan 333, Taiwan, Community Medicine Research Center, Chang Gung Memorial Hospital, Keelung Branch, Keelung 204, Taiwan

Received  August 2020 Revised  January 2021 Published  February 2021

Dengue, a mosquito-borne disease, poses a tremendous burden to human health with about 390 million annual dengue infections worldwide. The environmental temperature plays a major role in the mosquito life-cycle as well as the mosquito-human-mosquito dengue transmission cycle. While previous studies have provided useful insights into the understanding of dengue diseases, there is little emphasis put on the role of environmental temperature variation, especially diurnal variation, in the mosquito vector and dengue dynamics. In this study, we develop a mathematical model to investigate the impact of seasonal and diurnal temperature variations on the persistence of mosquito vector and dengue. Importantly, using a threshold dynamical system approach to our model, we formulate the mosquito reproduction number and the infection invasion threshold, which completely determine the global threshold dynamics of mosquito population and dengue transmission, respectively. Our model predicts that both seasonal and diurnal variations of the environmental temperature can be determinant factors for the persistence of mosquito vector and dengue. In general, our numerical estimates of the mosquito reproduction number and the infection invasion threshold show that places with higher diurnal or seasonal temperature variations have a tendency to suffer less from the burden of mosquito population and dengue epidemics. Our results provide novel insights into the theoretical understanding of the role of diurnal temperature, which can be beneficial for the control of mosquito vector and dengue spread.

Citation: Naveen K. Vaidya, Feng-Bin Wang. Persistence of mosquito vector and dengue: Impact of seasonal and diurnal temperature variations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021048
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A schematic diagram of the dengue transmission model. $A, M_s, M_e, M_i$: aquatic, susceptible, exposed, and infected mosquitos. $H_s, H_e, H_i, H_r$: susceptible, exposed, infected, and recovered humans. Solid arrows represent birth, maturation, infection, transfer, death, while dashed arrows indicate the effects of time dependent periodic environmental temperature, $T(t)$
] for $\delta (T)$ (oviposition rate), $\mu_a(T)$ (aquatic phase mortality rate), $\theta (T)$ (mosquito emergence rate from acuatic phase), and $\mu_m(T)$ (mosquito mortality rate)">Figure 2.2.  Best-fit curves provided by the experimental data [44] for $\delta (T)$ (oviposition rate), $\mu_a(T)$ (aquatic phase mortality rate), $\theta (T)$ (mosquito emergence rate from acuatic phase), and $\mu_m(T)$ (mosquito mortality rate)
] for the transmission probability from human to mosquito, $\beta_m (T)$">Figure 2.3.  Best-fit curve provided by the data generated from the previous estimates [17] for the transmission probability from human to mosquito, $\beta_m (T)$
Temperature profile of $T_m(t)$ [Left], $\Psi_d(t)$ [Middle], and $T(t)$ [Right]. Parameters used are $T_0 = 25 ^oC, \epsilon_m = 5 ^oC, \tau_m$ = 365 day, $\phi_m = 0$, $\epsilon_d = 5 ^oC$, $\tau_d = 1$ day, and $\phi_d = 0$
Mosquito reproduction number ($\mathcal{R}^M$) [Left] and infection invasion threshold ($\mathcal{R}^0$) [right] for different values of the mean temperature ($T_0$) with amplitudes of seasonal temperature and diurnal temperature fixed at $\epsilon_m = 5$ $^oC$ and $\epsilon_d = 5$ $^oC$, respectively. For comparison purposes, $\mathcal{R}^M$ and $\mathcal{R}^0$ for the constant temperature (i.e., $\epsilon_m = \epsilon_d = 0$ $^oC$) are also plotted
Mosquito reproduction number ($\mathcal{R}^M$) [left column] and infection invasion threshold ($\mathcal{R}^0$) [right column] for different values of the amplitudes of seasonal temperature ($\epsilon_m$) with the amplitude of diurnal temperature fixed at $\epsilon_d = 5$ $^oC$ and the mean temperature fixed at $T_0 = 16$ $^oC$ [top row], $T_0 = 28$ $^oC$ [middle row], and $T_0 = 38$ $^oC$ [bottom row]
Mosquito reproduction number ($\mathcal{R}^M$) [left column] and infection invasion threshold ($\mathcal{R}^0$) [right column] for different values of the amplitudes of diurnal temperature ($\epsilon_d$) with the amplitude of seasonal temperature fixed at $\epsilon_m = 5$ $^oC$ and the mean temperature fixed at $T_0 = 16$ $^oC$ [top row], $T_0 = 28$ $^oC$ [middle row], and $T_0 = 38$ $^oC$ [bottom row]
Model parameters
 Parameter Description Value Reference $k$ Fraction of female larvae from eggs 0.5 (0-1) [18,26] $b$ Per capita biting rate 0.1 [6,26] $\mu_h$ Natural death rate of humans 4.22$\times 10^{-5}$ d$^{-1}$ Calculated, [16] $1/\gamma_h$ Intrinsic period 10 days [6,16,18,26] $\alpha_h$ Human recovery rate 0.1 d$^{-1}$ [18,26] $\delta_m$ In $\delta(t)$ 9.531 Data fitting $\delta_h$ In $\delta(t)$ 22.55 Data fitting $N_{\delta}$ In $\delta(t)$ 7.084 Data fitting $a_{\delta}$ In $\delta(t)$ 0 Data fitting $\epsilon_{\delta}$ In $\delta(t)$ $10^{-6}$ Data fitting $a_{0\mu_a}$ In $\mu_a(t)$ 2.914 Data fitting $a_{1\mu_a}$ In $\mu_a(t)$ -0.4986 Data fitting $a_{2\mu_a}$ In $\mu_a(t)$ 0.03099 Data fitting $a_{3\mu_a}$ In $\mu_a(t)$ -0.0008236 Data fitting $a_{4\mu_a}$ In $\mu_a(t)$ 7.975$\times 10^{-6}$ Data fitting $a_{0\theta}$ In $\theta(t)$ 8.044$\times 10^{-5}$ Data fitting $a_{1\theta}$ In $\theta(t)$ 11.386 Data fitting $a_{2\theta}$ In $\theta(t)$ 40.1461 Data fitting $\epsilon_{\theta}$ In $\theta(t)$ $10^{-6}$ Data fitting $a_{0\mu_m}$ In $\mu_m(t)$ 0.1901 Data fitting $a_{1\mu_m}$ In $\mu_m(t)$ -0.0134 Data fitting $a_{2\mu_m}$ In $\mu_m(t)$ 2.739$\times 10^{-4}$ Data fitting $a_{0\gamma_m}$ In $\gamma_m(t)$ 5$\times 10^{4/3}$ Data fitting $a_{1\gamma_m}$ In $\gamma_m(t)$ 0.0768 Data fitting $\beta_{mh}$ In $\beta_m(t)$ 18.9871 Data fitting $N_{\beta_m}$ In $\beta_m(t)$ 7 Data fitting $\epsilon_{\beta m}$ In $\beta_m(t)$ $10^{-6}$ Data fitting $a_{\beta m}$ In $\beta_m(t)$ 0 Data fitting $a_{0\beta_h}$ In $\beta_h(t)$ 1.044$\times 10^{-3}$ Data fitting $a_{1\beta_h}$ In $\beta_h(t)$ 12.286 Data fitting $a_{2\beta_h}$ In $\beta_h(t)$ 32.461 Data fitting $\epsilon_{\beta h}$ In $\beta_h(t)$ $10^{-6}$ Data fitting
 Parameter Description Value Reference $k$ Fraction of female larvae from eggs 0.5 (0-1) [18,26] $b$ Per capita biting rate 0.1 [6,26] $\mu_h$ Natural death rate of humans 4.22$\times 10^{-5}$ d$^{-1}$ Calculated, [16] $1/\gamma_h$ Intrinsic period 10 days [6,16,18,26] $\alpha_h$ Human recovery rate 0.1 d$^{-1}$ [18,26] $\delta_m$ In $\delta(t)$ 9.531 Data fitting $\delta_h$ In $\delta(t)$ 22.55 Data fitting $N_{\delta}$ In $\delta(t)$ 7.084 Data fitting $a_{\delta}$ In $\delta(t)$ 0 Data fitting $\epsilon_{\delta}$ In $\delta(t)$ $10^{-6}$ Data fitting $a_{0\mu_a}$ In $\mu_a(t)$ 2.914 Data fitting $a_{1\mu_a}$ In $\mu_a(t)$ -0.4986 Data fitting $a_{2\mu_a}$ In $\mu_a(t)$ 0.03099 Data fitting $a_{3\mu_a}$ In $\mu_a(t)$ -0.0008236 Data fitting $a_{4\mu_a}$ In $\mu_a(t)$ 7.975$\times 10^{-6}$ Data fitting $a_{0\theta}$ In $\theta(t)$ 8.044$\times 10^{-5}$ Data fitting $a_{1\theta}$ In $\theta(t)$ 11.386 Data fitting $a_{2\theta}$ In $\theta(t)$ 40.1461 Data fitting $\epsilon_{\theta}$ In $\theta(t)$ $10^{-6}$ Data fitting $a_{0\mu_m}$ In $\mu_m(t)$ 0.1901 Data fitting $a_{1\mu_m}$ In $\mu_m(t)$ -0.0134 Data fitting $a_{2\mu_m}$ In $\mu_m(t)$ 2.739$\times 10^{-4}$ Data fitting $a_{0\gamma_m}$ In $\gamma_m(t)$ 5$\times 10^{4/3}$ Data fitting $a_{1\gamma_m}$ In $\gamma_m(t)$ 0.0768 Data fitting $\beta_{mh}$ In $\beta_m(t)$ 18.9871 Data fitting $N_{\beta_m}$ In $\beta_m(t)$ 7 Data fitting $\epsilon_{\beta m}$ In $\beta_m(t)$ $10^{-6}$ Data fitting $a_{\beta m}$ In $\beta_m(t)$ 0 Data fitting $a_{0\beta_h}$ In $\beta_h(t)$ 1.044$\times 10^{-3}$ Data fitting $a_{1\beta_h}$ In $\beta_h(t)$ 12.286 Data fitting $a_{2\beta_h}$ In $\beta_h(t)$ 32.461 Data fitting $\epsilon_{\beta h}$ In $\beta_h(t)$ $10^{-6}$ Data fitting
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