科技部專題研究計畫申請書

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科技部專題研究計畫申請書

一、基本資料：

申請條碼：105WFD2650161

105WFD2650161

計畫類別 ( 單選 ) 一般研究計畫

研究型別個別型

計畫歸屬自然司

申請機構/系所（單位）長庚大學通識教育中心

本計畫主持人姓名王埄彬職　稱副教授身分證號碼 P12244****

本計畫名稱

中文反應擴散方程組模擬季節性及潛伏期對登革熱傳播的影響

英文 Reaction-diffusion equations modeling influence of seasonality and latency on the spread of dengue fever

整合型總計畫名稱

整合型總計畫主持人身分證號碼

全程執行期限自民國 105 年 08 月 01 日起至民國 108 年 07 月 31 日

研究學門學　門　代　碼學　門　名　稱

M02090003 生物系統建模與模擬

研究性質 ■純基礎研究　□導向性基礎研究　□應用研究　□技術發展　

【請考量己身負荷，申請適量計畫】

本年度申請主持科技部各類研究計畫(含預核案)共 2 件。(共同主持之計畫不予計入) 本計畫是否為國際合作研究：■否；　□是

本計畫是否申請海洋研究船：■否；　□是，請務必填寫表CM15。

1.本計畫是否有進行下列實驗/研究：（勾選下列任一項，須附相關實驗/研究同意文件）

□人體試驗/人體檢體　□人類胚胎/人類胚胎幹細胞　□基因重組實驗　□基因轉殖田間試驗　□第二級以上感染性生物材料　□動物實驗(須同時加附動物實驗倫理3R說明)　

2.本計畫是否為人文司行為科學研究計畫：□是(請檢附已送研究倫理審查之證明文件)；　■否計畫連絡人姓名：王埄彬電話：(公) (03)2118800 # 5684 (宅/手機) 0937-

684131

通訊地址桃園市桃園區龍鳳里8鄰龍鳳二街25之2巷9號

傳真號碼 (03)2118700 E-MAIL fbwang@mail.cgu.edu.tw

計畫主持人簽章：　　　　　　日期：　　　　　　

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二、研究計畫中英文摘要：請就本計畫要點作一概述，並依本計畫性質自訂關鍵詞。

（一）計畫中文摘要。（五百字以內）

關鍵字: 登革熱傳染,擁擠效應,時間延滯,基本再生數,週期解,傳播速率, 週期行波解。

摘要:

登革熱在亞熱帶及熱帶地區常造成嚴重問題。他是透過埃及斑蚊叮咬傳給人類,並且有四種相關病毒已被確定。因為其高傳染力與高死亡率, 控制其傳播是公共衛生與健康的重要課題。此計畫致力於對登革熱傳染數學模型之改進與分析。為了解時空奇異性, 擁擠效應對登革熱傳染之影響,我們先探討一個帶時間週期的反應擴散方程組。其次, 我們用另一個帶時間週期及時間延滯的反應擴散方程組去模擬季節性與疾病潛伏期在登革熱傳播中扮演之重要性。最後,我們想利用生物入侵的概念去研究登革熱傳染。因此,我們對一個一維度之週期反應擴散方程組探討其傳播速率及週期行波解。

申請人本年度申請主持科技部各類研究計畫共 2 件,其中包含優秀年輕學者研究計畫:1 件,一般研究計畫:1 件,預核案:0 件。本件在本年度所申請之計畫中優先順序為第二。

共 2 頁第 1 頁表 CM02

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（二）計畫英文摘要。（五百字以內）

Keywords:. Dengue transmission, Crowding effect, Time delays, Basic reproduction number, Periodic solutions, Spreading speeds, Periodic traveling waves

Abstract:

Dengue disease is a serious problem in the subtropical and tropical regions of the world, including Taiwan. It is transmitted to humans by the bite of Aedes mosquitoes, and four serotypes have been identified. The control of the spread of the disease is an important issue in the public health due to the high infection rate of dengue fever and high death rate of its dengue hemorrhagic fever. This project is devoted to the study of Dengue transmission. Firstly, we study a periodic reaction-diffusion system to investigate how the spatiotemporally heterogeneity and crowding effect influence the dynamics of the spread of the disease. Secondly, we investigate a time periodic

reaction–diffusion epidemic system modeling the incorporation of seasonality, disease latency and mobility of the individuals. Thirdly, we shall study a one-dimensional model which is temporally heterogeneous, and investigate the periodic travelling waves by wing and wind.

請概述執行本計畫可能產生對社會、經濟、學術發展等面向的預期影響性 (一百五十字內)。

今年登革熱在台灣南部大流行,造成許多人死亡及嚴重的經濟損失。在已有的數學模型基礎上,我們想更進一步了解季節性在登革熱傳播中扮演之角色。期望對台灣公共衛生及登革熱防治可以有一些貢獻。

共 2 頁第 2 頁表 CM02

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三、研究計畫內容（以中文或英文撰寫）：

(一) 研究計畫之背景。請詳述本研究計畫所要探討或解決的問題、重要性、預期影響性及國內外有關本計畫之研究情況、重要參考文獻之評述等。如為連續性計畫應說明上年度研究進度。

(二) 研究方法、進行步驟及執行進度。請分年列述：1.本計畫採用之研究方法與原因。2.預計可能遭遇之困難及解決途徑。3.重要儀器之配合使用情形。4.如為須赴國外或大陸地區研究，請詳述其必要性以及預期效益等。

(三) 預期完成之工作項目及成果。請分年列述：1.預期完成之工作項目。2.對於參與之工作人員，預期可獲之訓練。3.預期完成之研究成果（如期刊論文、研討會論文、專書、技術報告、專利或技

術移轉等質與量之預期成果）。4.學術研究、國家發展及其他應用方面預期之貢獻。

(四) 整合型研究計畫說明。如為整合型研究計畫請就以上各點分別說明與其他子計畫之相關性。

（一）研究計畫之背景、目的:

Dengue fever disease is a serious problem in the subtropical and tropical regions of the world, including Taiwan. It is transmitted to humans by the bite of Aedes mosquitoes, and four serotypes have been identified. Infection by any single type of virus usually gives lifelong immunity to that type, but only short-term immunity to the other types. The mosquitoes never recover from the infection and their infective period ends with their death. The control of the spread of the disease is an important issue in the public health due to the high infection rate of dengue fever and high death rate of its hemorrhagic fever.

Biological invasion is also an important topic in the spread of dengue epidemics. The main reason for local population dispersal is that winged female A. aegypti can search for human blood freely, and wind currents may also result in an advection movement of mosquitoes. Thus, a diffusion process is usually used to model the random search movements of winged A. aegypti, and a constant advection is added to simulate the result of wind transportation. On the other hand, large numbers of larvae are frequently carried by unattended water containers, commonly found inside transportation trucks. Those factors also affect the spread of disease.

For those facts, we shall study the following topics related to Dengue fever disease models:

(1) To incorporate the factors of seasonality and random movement of population, we will study a spatiotemporal dengue fever disease model.

(2) To incorporate the factors of seasonality, the incubation period of mosquitoes and random movement of population, we will study a time-periodic and time-delayed reaction-diffusion system.

(3)In order to study biological invasion of dengue epidemics in the time varying environments, we will study a one-dimensional model which is temporally heterogeneous, and investigate the periodic travelling waves by wing and wind.

詳細研究計畫及欲探討之數學模型請參看下頁起共 11 頁

表 CM03 共 13 頁第 1 頁

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The Proposal of My Future Research

FENG-BIN WANG

1 Motivations

Dengue fever is a serious disease in the subtropical and tropical regions of the world.

It might cause the dengue shock syndrome and potential to death. Dengue disease is transmitted to humans by the bite of Aedes mosquitoes. Four serotypes (I ∼ IV) have been identiﬁed. Infection by any single type of virus usually gives lifelong immunity to that type, but only short-term immunity to the other serotypes ([21]).

The mosquitoes never recover from the infection and their infective period ends with their death ([3]). The control of the spread of the disease is an important issue in the public health due to the high infection rate of dengue fever and high death rate of its hemorrhagic fever. I ﬁrst brieﬂy outline my recent works related to disease models:

• In [24], we studied a periodic version of the system which describes a popu- lation where the individuals alternate between mobile and non-mobile states, and only the mobile reproduce. Such behaviour is typical for invertebrates living in small ponds in arid climates, which dry up and reappear subject to rainfall. For the model in [24], we established the existence of the spreading speed and showed that it coincides with the minimal wave speed for monotone periodic traveling waves by appealing to the theory of asymptotic speeds of spread and traveling waves for monotone periodic semiﬂow. Finally, we considered the case where the spatial domain is bounded. A threshold result on the global attractivity of either zero or a positive periodic solution were established.

• In [19], we propose a mathematical model to describe the avian inﬂuenza dynamics in wild birds with bird mobility and heterogeneous environment incorporated. In addition to establishing the basic properties of solutions

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to the model, we also prove the threshold dynamics which can be expressed either by the basic reproductive number or by the principal eigenvalue of the linearization at the disease free equilibrium. When the environment factor in the model becomes a constant (homogeneous environment), we are able to find explicit formulas for the basic reproductive number and the principal eigenvalue. We also perform numerical simulation to explore the impact of the heterogeneous environment on the disease dynamics. Our analytical and numerical results reveal that the avian influenza dynamics in wild birds is highly affected by both bird mobility and environmental heterogeneity.

• In [4], we derive and analyze an infectious disease model containing a fixed latency and non-local infection caused by the mobility of the latent individuals in a continuous bounded domain. The model is given by a spatially non-local reaction-diffusion system carrying a discrete delay associated with the zero-flux condition on the boundary. By appealing to the theory of monotone dynamical systems and uniform persistence, we show that the model has the global threshold dynamics which can be described either by the principal eigenvalue of a linear non-local scalar reaction diffusion equation or equiva- lently by the basic reproduction number R0 for the model.

• In an effort to understand the dynamics of the spread of the dengue fever, Esteva and Vargas [1] proposed a SIR v.s. SI epidemiological model without crowding effect and spatial heterogeneity. They found a threshold parameter R₀, if R₀ < 1, then the disease will die out; if R₀ > 1, then the disease will always exist. In [7], we modify the autonomous system provided in [1] to obtain a reaction-diffusion system to investigate how the spatial heterogeneity and crowding effect influence the dynamics of the spread of the disease. We first define the basic reproduction number in an abstract way and then employ the comparison theorem and the theory of uniform persistence to study the global dynamics of the modified system.

• In [11], we intended to understand the influences of the spatial heterogene- ity, crowding effect and non-local infection caused by the movements of the latent mosquitoes on the dynamics of dengue transmission and we modified the systems provided in [1, 7] to obtain a nonlocal and time-delayed reaction- diffusion system with the Neumann condition on the boundary. Then we showed that the global threshold dynamics of the model system can be de- termined by R₀, the basic reproduction number for the model system.

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Next, I will propose my detailed models of this project in the following sections, respectively.

2 Previous models

In order to understand the dynamics of the spread of the disease, Esteva and Var- gas [1] proposed a SIR v.s. SI epidemiological model. Basically, they studied the mechanisms that allows the invasion and persistence of a serotype of dengue in a region. Their mathematical model for the dynamics of dengue disease contains only one type of virus and ignore the disease-related death rate. In [7], the authors modiﬁed the model proposed in [1] to incorporate the crowding eﬀect in spatially heterogeneous environments. Assume that Ω is a spatial habitat with the smooth boundary ∂Ω and n is the outward normal to ∂Ω. Let S_H, I_H, and R_H denote the number of the susceptible, infectious and immune class in the human popula- tion; SV, IV denote the number of the susceptible, infectious class in the mosquito population. Thus, N_H := S_H + I_H + R_H and N_V := S_V + I_V represent the pop- ulation sizes of humans and mosquitoes, respectively. The constants µ_b, µ_d, and γ_H stand for the birth, death and recover rate of human species; A and µ_V denote the recruitment and the per capita mortality rate of mosquitoes, respectively. The biting rate b of mosquitoes is the average number of bites per mosquito per day.

Mosquitoes bite not only human but also pets. Thus, we assume m is the number of alternative hosts available as blood sources. Let β_H be the transmission prob- ability from infectious mosquitoes to susceptible humans; βV be the transmission probability from infectious humans to susceptible mosquitoes. Then the dynamics of dengue fever was described by the following system of diﬀerential equations [7]:











∂SH

∂t = d_H∆S_H + µ_bN_H − c(x)SHN_H − ^β_N^H_H^(x)b(x)_+m(x)S_HI_V − µdS_H,

∂IH

∂t = d_H∆I_H +_N^β^H^(x)b(x)

H+m(x)S_HI_V − c(x)IHN_H − (µd+ γ_H)I_H,

∂RH

∂t = d_H∆R_H + γ_HI_H − c(x)RHN_H − µdR_H, x∈ Ω, t > 0,

∂SV

∂t = d_V∆S_V + A(x)−_N^β^V_H^(x)b(x)_+m(x)S_VI_H − µVS_V,

∂IV

∂t = dV∆IV +_N^β^V^(x)b(x)

H+m(x)SVIH − µVIV,

∂S_H

∂n = ^∂I_∂n^H = ^∂R_∂n^H = ^∂S_∂n^V = ^∂I_∂n^V = 0, x∈ ∂Ω, t > 0.

(2.1)

Here, we consider a closed environment in the sense that the ﬂuxes for each of these subpopulations are zero, and hence, we have proposed the Neumann boundary con-

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ditions to the equations (2.1) on the boundary. Furthermore, the crowding eﬀect terms (see, e.g., [6]) in the susceptible class, the infectious class and the immune

class in the human population are respectively given by c(x)S_HN_H, c(x)I_HN_H and c(x)R_HN_H. Assume that the spatial dependent functions A(x), b(x), c(x), m(x), β_H(x), β_V(x)

are positive; ∆ is the usual Laplacian operator; d_H > 0, d_V > 0 denote the diﬀusion coeﬃcients for humans and mosquitoes, respectively.

In order to include the latency of mosquitoes, the authors in [11] proposed and analyzed the following system:











∂u1

∂t = d_H∆u₁ + µ_bK(x)−_K(x)+m(x)^β^H^(x)b(x) u₁u₄− D1(x)u₁,

∂u2

∂t = dH∆u2 +_K(x)+m(x)^β^H^(x)b(x) u1u4− D2(x)u2,

∂u3

∂t = d_V∆u₃+ A(x)− _K(x)+m(x)^β^V^(x)b(x) u₂u₃− µVu₃,

∂u4

∂t = d_V∆u₄(x, t)− µVu₄(x, t) +e^−µ^V^τ∫

ΩΓ(d_Vτ, x, y)_K(y)+m(y)^β^V^(y)b(y) u₂(y, t− τ)u3(y, t− τ)dy,

(2.2)

in (x, t) ∈ Ω × (0, ∞) with the boundary conditions

∂u₁

∂ν = ∂u₂

∂ν = ∂u₃

∂ν = ∂u₄

∂ν = 0, x∈ ∂Ω, t > 0, (2.3) where

D₁(x) := c(x)K(x) + µ_d and D₂(x) := c(x)K(x) + µ_d+ γ_H, (2.4) µb > µd and τ is the average incubation period. In (2.2) (u1, u2, u3, u4) represents (S_H, I_H, S_V, I_V).

3 A spatiotemporal model

It was known that seasonality also play an important role in the spread of dengue fever. Then we incorporate this factor into the model in (2.1), and we investigate

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the following system:











∂SH

∂t =∇ · [dH(x, t)∇SH] + µ_b(x, t)N_H − c(x, t)SHN_H

−^β^H(x,t)b(x,t)

NH+m(x,t)SHIV − µd(x, t)SH,

∂IH

∂t =∇ · [dH(x, t)∇IH] + ^β^H(x,t)b(x,t)

NH+m(x,t) S_HI_V − c(x, t)IHN_H − (µd(x, t) + γ_H(x, t))I_H,

∂RH

∂t =∇ · [dH(x, t)∇RH] + γ_H(x, t)I_H − c(x, t)RHN_H − µd(x, t)R_H, x∈ Ω, t > 0,

∂SV

∂t =∇ · [dV(x, t)∇SV] + A(x, t)− ^β^V(x,t)b(x,t)

NH+m(x,t)S_VI_H − µV(x, t)S_V,

∂IV

∂t =∇ · [dV(x, t)∇IV] + ^β^V(x,t)b(x,t)

NH+m(x,t)S_VI_H − (µV(x, t) + σ(x, t))I_V, [d_H(x, t)∇P (x, t)] · n = 0, x ∈ ∂Ω, t > 0, P = SH, I_H, R_H,

[d_V(x, t)∇Q(x, t)] · n = 0, x ∈ ∂Ω, t > 0, Q = SV, I_V,

(3.1) where σ(x, t) is the disease-related death rate. We assume that there is an ω > 0 such that

ϑ(x, t + ω) = ϑ(x, t) > 0, for all x∈ Ω, t > 0,

and ϑ ≡ dH, d_V, µ_b, c, b, m, µ_d, γ_H, β_H, A, β_V, µ_V, σ. We point out that the main differences between systems (2.1) and (3.1) are as follows: The system (2.1) is temporally homogeneous, while system (3.1) is temporally heterogeneous. The mathematical arguments in (2.1) do NOT work for (3.1) !

Difficulties and Goals:

• For the basic reproduction number, the theory developed in [22] can be only applied to a temporally heterogeneous system, while the theory in [23] is only suitable to spatially heterogeneous system. Thus, we need to combine those results in [22, 23] to deﬁne the basic reproduction number for the spatiotemporal system (3.1).

• By using the theory of uniform persistence, we will show that the basic re- production number plays a threshold index in the extinction/persistence of the disease.

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4 A time-periodic and time-delayed reaction-diffusion system

In [11], the authors included spatial heterogeneity, crowding effect and non-local infection caused by the movements of the latent mosquitoes on the dynamics of dengue transmission. Then they modified the system (2.1) to obtain a nonlocal and time-delayed reaction-diffusion system with the Neumann condition on the boundary. It is reported that the population dynamics of infectious disease is affected by the time varying environments. Therefore, it is natural to incorporate tempo- ral heterogeneity into the disease model in [11], which leads to non-autonomous evolution systems.

Note that mosquitoes infected by the dengue disease in one location can move freely in the habitat when this individual becomes infectious. That is, mosquitoes may not stay at the same location in space during the incubation period and the mobility of the individuals in the latent period will result in a delay term with spatial averaging on Ω. We point out that those observations were also discussed in the previous papers [4, 12, 13, 14, 15, 23]. To formulate this process with the latency properly, we introduce the notion of infection age, and we adopt the standard model on describing age structured population with spatial diﬀusion (see e.g. [16]). Then we can use the ideas in [4, 11, 27] to obtain a modiﬁed system of (3.1):











∂u1

∂t =∇ · [dH(x, t)∇u1] + µ_bK(x, t)− ^β^H(x,t)b(x,t)

K(x,t)+m(x,t)u₁u₄− D1(x, t)u₁,

∂u2

∂t =∇ · [dH(x, t)∇u2] + ^β^H(x,t)b(x,t)

K(x,t)+m(x,t)u₁u₄− D2(x, t)u₂,

∂u3

∂t =∇ · [dV(x, t)∇u3] + A(x, t)− ^β^V(x,t)b(x,t)

K(x,t)+m(x,t)u2u3− D3(x, t)u3,

∂u4

∂t =∇ · [dI(x, t)∇u4]− D4(x, t)u₄ +∫

ΩΓ(t, t− τ, x, y)[ ^β^V^(y,t−τ)b(y,t−τ)

K(y,t−τ)+m(y,t−τ)u₂(y, t− τ)u3(y, t− τ)]dy, (4.1) in (x, t) ∈ Ω × (0, ∞) with the boundary conditions







[d_H(x, t)∇ui(x, t)]· n = 0, x ∈ ∂Ω, t > 0, i = 1, 2, [d_V(x, t)∇u3(x, t)]· n = 0, x ∈ ∂Ω, t > 0,

[d_I(x, t)∇u4(x, t)]· n = 0, x ∈ ∂Ω, t > 0,

(4.2)

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where

{D₁(x, t) := c(x, t)K(x, t) + µ_d, D₂(x, t) := c(x, t)K(x, t) + µ_d+ γ_H(x, t), D3(x, t) := µV(x, t) and D4(x, t) := µI(x, t).

(4.3) where, τ is the average incubation period of mosquitoes, (u₁, u₂, u₃, u₄) := (S_H, I_H, S_V, I_V) and Γ(t, s, x, y) with t > s ≥ 0 and x, y ∈ Ω is the fundamental solution associated with the partial diﬀerential operator the operator ∂_t− ∇ · [dL(·, t)∇] − µL(·, t) as- sociated with the zero ﬂux boundary condition (see e.g. [2]). In order to describe the function K(x, t), we consider

{∂NH

∂t =∇ · [dH(x, t)∇NH] + (µ_b(x, t)− µd(x, t))N_H − c(x, t)N_H², x∈ Ω, t > 0, [d_H(x, t)∇NH(x, t)]· n = 0, x ∈ ∂Ω, t > 0.

(4.4) Then the reaction-diﬀusion equation (4.4) is a logistic equation and it is known that (4.4) admits a unique positive τ−periodic solution K(x, t) such that (see, e.g.

[25, Theorem 3.1.5 and the proof of Theorem 3.1.6] ):

tlim→∞(N_H(x, t)− K(x, t)) = 0, uniformly in x ∈ ¯Ω, (4.5) for all solutions with nonnegative and nonzero initial datas provided that some suit- able conditions on µ_b(x, t)− µd(x, t), see [5]. In contrast to (2.2)-(2.3), system (4.1)-(4.2) is temporally heterogeneous. Thus, the mathematical argu- ments in (2.2)-(2.3) do NOT work for (4.1)-(4.2) !

Difficulties and Goals:

• To study the basic reproduction number, the theory developed in [17] provides a powerful tool for autonomous reaction-diffusion systems with or without time delay. However, it is difficult to apply abstract results of [17] to periodic and time-delayed population models with or without spatial diffusion. Our system (4.1)-(4.2) is a periodic and time-delayed reaction-diffusion system, so we need to use the new results in [26] to define the basic reproduction number R₀ of system (4.1)-(4.2).

• By appealing to the ideas in [27], the comparison arguments and persistence theory, we will show that the disease-free periodic solution is globally attrac- tive if R₀ < 1, while there is an endemic periodic solution and the disease is uniformly persistent if R₀ > 1.

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5 Dispersal dynamics: Periodic travelling waves by wing and wind

Biological invasion is also an important topic in the spread of dengue epidemics.

The main reason for local population dispersal is that winged female A. aegypti can search for human blood freely, and wind currents may also result in an advection movement of mosquitoes. Thus, a diﬀusion process is usually used to model the random search movements of winged A. aegypti, and a constant advection is added to simulate the result of wind transportation. On the other hand, large numbers of larvae are frequently carried by unattended water containers, commonly found inside transportation trucks. For those observations, the authors in [18] proposed a one-dimensional model which is temporally homogeneous, and studied the travelling waves by wing and wind. It was observed that seasonality also play an important role in the spread of dengue fever. Then we incorporate this factor into the model in [18], that is, we shall investigate the following system:

{_∂

∂tu₁(x, t) = D(t)_∂x^∂²2u₁(x, t)− ν(t)_∂x^∂ u₁(x, t) + γ(t)u₂(x, t)(1− ^u_k¹₁^(x,t)_(t) )− d1(t)u₁(x, t),

∂

∂tu₂(x, t) = α(t)(1− ^u_k²₂^(x,t)_(t) )u₁(x, t)− (d2(t) + γ(t))u₂(x, t), x∈ R, t > 0.

(5.1) Here, u₁(x, t) represents the spatial density of the winged A. aegypti (mature fe- male mosquitoes) at position x and time t; u₂(x, t) represents the aquatic form of mosquitoes (eggs, larvae and pupae) at location x and time t; γ(t) is the speciﬁc rate of maturation of the aquatic form into winged female mosquitoes, saturated by a carrying capacity k₁(t). The term α(t)(1− ^u_k²₂^(x,t)_(t) )u₁(x, t) describes the rate of production of the aquatic form, which ia produced only by female mosquitoes.

That is, we assume that the rate of production of the aquatic form is proportional to the density of female mosquitoes and it is also saturated by a carrying capacity k₂(t). The random ﬂying movement of female mosquitoes is represented by a dif- fusion process with coeﬃcient D(t), and ν(t) represents the wind advection. d₁(t) and d₂(t) represent the mortality rates of the mosquitoes and the aquatic forms, respectively. Next, we assume that there is an ω > 0 such that

ϑ(t + ω) = ϑ(t) > 0, for all t > 0, ϑ≡ D, ν, γ, k1, d₁, α, k₂, d₂.

The arguments in [18] are suitable to temporally homogeneous models, and hence, we can NOT use those arguments in [18] to study system (5.1).

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Difficulties and Goals:

• It is easy to see that system (5.1) is cooperative and its solution maps are monotone on an invariant set. Thus, we can use the general theory developed in [9, 10] to study the spreading speeds for periodic system (5.1).

• The solution maps of system (5.1) are not compact with respect to the com- pact open topology due to the lack of the diffusion term in the second equation. Thus, the theory in [9, 10] may not be applied to the study of the existence of time-periodic traveling waves for system (5.1). Instead, we will use the theory recently developed in [8] for monotone semiflows with weak compactness and the ideas in [20] to overcome this difficulty.

References

[1] L. Esteva and C. Vargas, Analysis of a dengue disease transmission model, Mathematical Biosciences, 150 (1998), pp. 131–151.

[2] A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Inc., Englewoo Cliﬀs, N.J., 1964.

[3] D.J. Gubler, Dengue. In: Monath T.P. (ed.), The arbovirus: Epidemiology and Ecology, Vol II (1986), CRC Press, Florida, USA, pp. 213-261.

[4] Zhiming Guo, Feng-Bin Wang and Xinfu Zou, Threshold dynamics of an infective disease model with a fixed latent period and non-local infections, J. of Math. Biol., 65, (2012), pp. 1387–1410.

[5] P. Hess, Periodic-parabolic Boundary Value Problem and Positivity, Pitman Res. Notes Math., 247, Longman Scientiﬁc and Technical, 1991.

[6] T.-W. Hwang and Y. Kuang, Host extinction dynamics in a simple parasite-host interaction model, Mathematical Biosciences and Engineering, vol. 2, no. 4 (2005), pp. 743–751.

[7] T. W. Hwang and F. B. Wang, Dynamics of a dengue fever transmission model with crowding effect in human population and spatial variation, Discrete and Continuous Dynamical System Series-B, vol. 18 (2013), pp. 147–161.

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[8] J. Fang and X.-Q. Zhao, Traveling Waves for Monotone Semiflows with Weak Compactness, SIAM J. Math. Anal., 46 (2014), pp. 3678–3704.

[9] X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution system, J. Diﬀ. Equations 231 (2006), pp. 57–77.

[10] X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math. 60 (2007), pp. 1-40

[11] Huei-li Lin and Feng-Bin Wang, On a reaction-diffusion system modeling the dengue transmission with nonlocal infections and crowding effects, Applied Mathematics and Computation, Vol. 248 (2014), pp. 184-194.

[12] J. Li and X. Zou, An epidemic model with non-local infections on a patchy environment, J. Math. Biol. 60(2010), pp. 645-686.

[13] J. Li and X. Zou, Generalization of the Kermack-McKendrick SIR model to a patchy environment for a disease with latency, Math. Model. Nat., Phe- nom.4(2) (2009), pp. 92-118.

[14] J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bulletin of Mathematical Biol- ogy, 71 (2009), pp. 2048-2079.

[15] Y. Lou and X.-Q. Zhao, A reaction-diffusion malaria model with incubation period in the vector population, J. of Math. Biol., 62 (2011), pp. 543-568.

[16] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, edited by J.A.J. Metz and O. Diekmann, Springer-Verlag, New York, 1986.

[17] H. R. Thieme, Spectral bound and reproduction number for infinite- dimensional population structure and time heterogeneity, SIAM, J. Appl.

Math. 70 (2009), pp. 188–211.

[18] L. T. Takahashi, N. A. Maidana, W. C. Ferreira Jr., P. Pulino and H. M. Yang, Mathematical models for the Aedes aegypti dispersal dynamics:

travelling waves by wing and wind, Bulletin of Mathematical Biology 67 (2005), pp. 509–528.

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[19] N. K. Vaidya, Feng-Bin Wang and Xinfu Zou, Avian influenza dy- namics in wild birds with bird mobility and spatial heterogeneous environment, Discrete and Continuous Dynamical System Series-B, Volume 17, Number 8, November 2012, pp 2829-2848.

[20] X. Yu and X.-Q. Zhao, A periodic reaction-advection-diffusion model for a stream population, J. Diﬀerential Equations, 258 (2015), pp. 3037–3062.

[21] World Health Organization, Dengue haemorrhagic fever: Diagnosis, treatment and control, Ginebra, 1986.

[22] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epi- demic models in periodic environments, J. Dyn. Diﬀer. Equ. 20 (2008), pp. 699–

717.

[23] W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM. J Appl. Math., 71 (2011), pp. 147-168.

[24] F. B. Wang, A Periodic Reaction-Diffusion Model with A Quiescent Stage , Discrete and Continuous Dynamical System Series-B , Vol. 17, No. 1 (2012) , pp. 283-295.

[25] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, New York, 2003.

[26] X.-Q. Zhao, Basic Reproduction Ratios for Periodic Compartmental Mod- els with Time Delay, J. Dynamics and Diﬀerential Equations, 2015, DOI 10.1007/s10884-015-9425-2.

[27] L. Zhang, Z.-C. Wang and X.-Q. Zhao, Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period, J. Diﬀerential Equations, 258 (2015), pp. 3011–3036.

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（二）研究方法、進行步驟及執行進度。

預期遭遇之困難及解決途徑

(1) For the basic reproduction number, the theory developed in ref.[28] can be only applied to a temporally heterogeneous system, while the theory in ref.[29] is only suitable to spatially heterogeneous system.Thus, we need to combine those results in ref.[28,29] to define the basic reproduction number for the

spatiotemporal system (3.1). Then we will show that the basic reproduction number plays a threshold index in the extinction/persistence of the disease by using the theory of uniform persistence.

(2)Since our system (4.1)-(4.2) is a periodic and time-delayed reaction-diffusion system, we can NOT apply abstract results of ref.[21] to our case. We need to use the new results in ref.[32] to define the basic reproduction number $R_0$ of system (4.1)-(4.2).By appealing to the ideas in ref.[33], the comparison arguments and persistence theory, we will show that the disease-free periodic solution is globally

attractive if $R_0<1$, while there is an endemic periodic solution and the disease is uniformly persistent if $R_0>1$.

(3) Since solution maps of system (5.1) are monotone on an invariant set, we can use the general theory developed in ref.[9,10] to study the spreading speeds.However,the solution maps of system (5.1) are not compact with respect to the compact open topology due to the lack of the diffusion term in the second equation.Thus, the theory in ref.[9,10] may not be applied to the study of the existence of time-periodic traveling waves. Instead, we will use the theory recently developed in ref.[7] for monotone semiflows with weak compactness and the ideas in ref.[26] to overcome this difficulty.

（三）預期完成之工作項目、成果及績效。

這邊分三年度說明:

1.第一年(105 年 8 月 1 日至 106 年 7 月 31 日):

完成系統(3.1)之分析,希望可發表在 Journal of Mathematical Analysis and Applications。

2.第二年(106 年 8 月 1 日至 107 年 7 月 31 日):

完成系統(4.1)-(4.2)之分析,希望可發表在 Journal of Mathematical Biology。

3.第三年(107 年 8 月 1 日至 108 年 7 月 31 日):

完成系統(5.1)之分析,希望可發表在 Journal of Differential Equations 上。

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五、申請補助經費：

(一) 請將本計畫申請書之第七項(表CM07)、第八項(表CM08)、第九項(表CM09)、第十項(表CM10)、

第十一項(表CM11) 、第十二項(表CM12)所列費用個別加總後，分別填入「研究人力費」、「耗材、物品、圖書及雜項費用」、「國外學者來臺費用」、「研究設備費」、「國外差旅費-執行國際合作與移地研究」及「國外差旅費-出席國際學術會議」欄內。

(二) 管理費為申請機構配合執行本計畫所需之費用，其計算方式係依本部規定核給補助管理費之項目費用總和及各申請機構管理費補助比例計算後直接產生，計畫主持人不須填寫「管理費」欄

。

(三) 「貴重儀器中心使用額度」係將第十三項(表CM13)所列使用費用合計數填入。

(四) 請依各年度申請博士後研究之名額填入下表，如於申請時一併提出「補助延攬博士後研究(含大陸)員額/人才進用申請書」（表CIF2101、CIF2102），若計畫核定僅核定名額者應於提出合適人選後，另依據本部「補助延攬客座科技人才作業要點」規定向本部提出進用申請，經審查通過後，始得進用該名博士後研究。

(五) 申請機構或其他單位（含產業界）提供之配合項目，請檢附相關證明文件。

金額單位：新臺幣元　　　　　執行年次

　補助項目

業務費

　研究人力費

　耗材、物品、圖書及雜項　費用

　國外學者來臺費用

研究設備費

國外差旅費

　執行國際合作與移地研究　出席國際學術會議

管理費

合計

貴重儀器中心使用額度

博士後研究

國內、外

地區

大陸地區

第一年 (105年8月

～106年7月) 294,208 194,208 100,000 0 40,000 100,000 0 100,000 50,131 484,339 0 共 0 名共 0 名

第二年 (106年8月

～107年7月) 294,208 194,208 100,000 0 30,000 100,000 0 100,000 48,631 472,839 0 共 0 名共 0 名

第三年 (107年8月

～108年7月) 294,208 194,208 100,000 0 35,000 100,000 0 100,000 49,381 478,589 0 共 0 名共 0 名

第四年

共名共名

第五年

共名共名申請機構或其他單位（含產業界）提供之配合項目（無配合補助項目者免填）

配合單位名稱配合補助項目配合補助金額配合年次證明文件

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六、主要研究人力：

(一) 請依照「主持人」、「共同主持人」、「協同研究人員」及「博士後研究」等類別之順序分別填寫。

類別姓名服務機構/系所職稱在本研究計畫內擔任之具

體工作性質、項目及範圍

＊每週平均投入工作時數比率(%)

主持人王埄彬長庚大學通識教

育中心

副教授本研究計劃主要執行者,建立

或改進模型,對模型作數學分析,

解釋數學分析結果的生物意義

70%

※ 註：每週平均投入工作時數比率係填寫每人每週平均投入本計畫工作時數佔其每週全部工作時間之比率，以百分比表示（例如：50%即表示該研究人員每週投入本計畫研究工作之時數佔其每週全部工時之百分五十）。

(二) 如申請博士後研究，請另填表CIF2101及CIF2102(若已有人選者，請務必填註人選姓名，並將其個人資料表(表C301～表C303)併同本計畫書送本部)。

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七、研究人力費：

(一) 凡執行計畫所需助理人員費用，均得依預估研究人力（專任助理、兼任助理及臨時工）需求填寫，並請述明該助理人員在本計畫內擔任之具體內容、性質、項目及範圍，以利審查。

(二) 請分年列述。

第 1 年金額單位：新臺幣元

類別金額請述明在本計畫內擔任之具體內容、性質、項目及範圍

兼任助理(碩士生- 僱傭關係)

194,208 蒐集相關文獻及協助撰寫 Matlab 程式

(月支費用 6000.00元 x 12.00月 + [勞保費 777元, 健保費 955元, 勞工退休金/離職儲金 360元] x 12.00月) x 2名

合計 194,208

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八、耗材、物品、圖書及雜項費用：

(一) 凡執行研究計畫所需之耗材、物品(非屬研究設備者)、圖書及雜項費用，均可填入本表內。

(二) 說明欄請就該項目之規格、用途等相關資料詳細填寫，以利審查。

(三) 若申請單位有配合款，請於備註欄註明。

(四) 請分年列述。

項目名稱說明單位數量單價金額備註

消耗性器材文具,紙張,印刷,投影片

以及磁片,光碟片,碳粉等電腦相關耗材

1 1 40,000 40,000

雜支郵電差旅費,電腦維修費

,電腦週邊設備,印刷,裝訂

1 1 30,000 30,000

論文發表費研討會註冊費,論文發表

費

1 1 30,000 30,000

合　　　　　　　　計 100,000

1 1 40,000 40,000

1 1 30,000 30,000

費

1 1 30,000 30,000

合　　　　　　　　計 100,000

1 1 40,000 40,000

1 1 30,000 30,000

費

1 1 30,000 30,000

合　　　　　　　　計 100,000

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十、研究設備費：

(一) 凡執行研究計畫所需單價在新台幣一萬元以上且使用年限在二年以上與研究計畫直接有關之各項設備屬之。各類研究設備金額請於金額欄內分別列出小計金額。

(二) 購置設備單價在新臺幣二十萬元以上者，須檢附估價單。

(三) 若申請機構及其他機構有提供配合款，請務必註明提供配合款之機構及金額。

(四) 儀器設備單價超過六十萬元(含)以上者，請詳述本項設備之規格與功能(諸如靈敏度、精確度

…等)，其他重要特性與重要附件，以及申購本設備對計畫執行之必要性。本項設備若獲補助

，主持人應負維護保養之責，並且在不妨礙個人研究計畫或研究群計畫之工作下，同意提供他人共同使用，以避免設備閒置。

(五) 計畫主持人執行本項研究計畫，如欲申請購置單價新臺幣五百萬元(含)以上之大型儀器，請填表CM10-1。該項設備若獲本部核定補助新臺幣五百萬元(含)以上，則單獨核給一個規劃計畫

，主持人須遵守本部大型儀器之管考規定。

(六) 請分年列述。

類別設備名稱

(中文/英文) 說明數量單價金額

經費來源本部補助

經費需求

提供配合款之機構名稱及金額儀器及資

訊設備

桌上型電腦桌機主流機種含顯示器

1 40,000 40,000 40,000

合　　　　　　計 ^40,000 ^40,000

類別設備名稱

經費需求

訊設備

雷射多功能事務機

具掃描、彩色影印

、雙面影印等功能

1 30,000 30,000 30,000

合　　　　　　計 30,000 30,000

類別設備名稱

經費需求

訊設備

數位電子書閱讀器/Digital Paper

支援觸控和手寫輸入，便於使用者閱讀資料

1 35,000 35,000 35,000

合　　　　　　計 ^35,000 ^35,000

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十二、國外差旅費-出席國際學術會議：

(一) 計畫主持人及參與研究計畫之相關人員參加國際學術會議得申請本項經費。

(二) 請詳述預定參加國際學術會議之性質、預估經費、天數及地點。

(三) 機票費、生活費及其他費用之標準，請依照行政院頒布之「國外出差旅費報支要點」規定填列 (網址

http://law.dgbas.gov.tw/LawContentDetails.aspx?id=FL017584&KeyWordHL&StyleType=1)。

(四) 請詳述計畫主持人近三年參加國外舉辦之國際學術會議論文之發表情形。（包括會議名稱、時間、地點、發表之論文題目、補助機構，及後續收錄於期刊或專書之名稱、卷號、頁數、出版日期）

(五) 請分年列述。

出席國際學術會議

博士生人數共 0 名金　額 100,000

費用說明

預定參加國際學術會議之性質:

若能獲補助,申請人預定參加與偏微分方程、生物數學、動態系統等相關領域之國際學術會議

預估經費: 註冊費+機票+生活費約10萬地點:歐洲或美國或加拿大

會議時程:大約一週

近三年論文發表情形

(1)The 8th International Congress on Industrial and Applied Mathematics(ICIAM 2015) , National Convention Center inside the Beijing Olympic Green, China, August 10-14. (Title:A PDE system modeling the dengue transmission with nonlocal infections and crowding effects, 2015.)

(2)The 10th AIMS Conference on Dynamical Systems,Differential Equations and Applications, The Instituto de Ciencias Matemáticas (ICMAT) and the Universidad Autónoma de Madrid (UAM), July 7-11, 2014 (Title: Dynamics of a host-pathogen system in spatial

habitats.)

(3)The Fourth Conference on Computational and Mathematical Population Dynamics (CMPD4), North University of China,Taiyuan, China, May 29 -June 2, 2013 (Title: Dynamics of harmful algae in flowing habitats.)

(4)The 21st Annual Workshop on Differential Equations,, National Central University, Taiwan, 18-19 January, 2013. (Title: Global Dynamics of Zooplankton and Harmful Algae in Flowing Habitats Where the Algal Toxin Contains Little Limiting Nutrient.)

(5)2012 NCTS (Taiwan)-POSTECH (Korea) Joint Workshop on Partial Differential Equations, National Center for Theoretical Sciences, Taiwan 8-9 November, 2012.

(Title: Dynamics of a Bio-reactor Model with a Hydraulic Storage Zone.)

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費用說明

預估經費: 註冊費+機票+生活費約10萬地點: 歐洲或美國或加拿大

habitats.)

費用說明

預估經費: 註冊費+機票+生活費約10萬地點: 歐洲或美國或加拿大

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habitats.)

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十四、近三年內執行之研究計畫

　　（請務必填寫近三年所有研究計畫，不限執行本部計畫）

計畫名稱

（本部補助者請註明編號）

計畫內擔

任之工作起迄年月補助或委託機構執行情形經費總額

河岸生態系統之數學建模與應用(103-2115-M-182-001-MY2)

主持人 2014/08/01～

2016/07/31

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反應擴散方程組模擬有害藻類的成長及登革熱的傳播(101- 2115-M-182-003-MY2)

主持人 2012/09/01～

2014/07/31

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105 年度自然處專題計畫主持人近五年研究成果

姓名：王埄彬職稱：副教授服務機關系所：長庚大學

一、近五年內(2011/1/1～2015/12/31)已出版之最具代表性研究成果至多六篇，擇其中五篇電子檔上傳。

(請依序填寫：姓名,著作名稱,發表年份,期刊,卷數,頁次,IF,▲：被引用次數，並以＊號註記該篇所有的通訊作者)

註: （請務必更新個人資料表C302-C303，未來審查時將以該表之內容為準）

(1) James P. Grover, Sze-Bi Hsu and Feng-Bin Wang*, Competition between microorganisms for a single limiting resource with cell quota structure and spatial variation, Journal of Mathematical Biology, Vol.

64 (2012), pp. 713–743. (5-Year IF: 2.344 , Time cited: 6 )

(2) Sze-Bi Hsu, Feng-Bin Wang* and Xiao-Qiang Zhao, Global Dynamics of Zooplankton and Harmful Algae in Flowing Habitats, Journal of Differential Equations, Vol. 255 (2013), pp. 265-297.

(5-Year IF: 1.846, Time cited: 6 )

(3) James P. Grover* and Feng-Bin Wang, Competition and allelopathy with resource storage: Two resources, Journal of Theoretical Biology, Vol. 351 (2014), pp. 9-24. (5-Year IF: 2.239 , Time cited: 3) (4) Sze-Bi Hsu, Julian Lopez-Gomez*, Linfeng Mei and Feng-Bin Wang, A pivotal eigenvalue problem in

river ecology, Journal of Differential Equations, Vol. 259, (2015), pp. 2280-2316.

(5-Year IF: 1.846 , Time cited: 0 )

(5) Feng-Bin Wang*, Sze-Bi Hsu and Xiao-Qiang Zhao, A Reaction-Diffusion-Advection Model of

Harmful Algae Growth with Toxin Degradation, Journal of Differential Equations, Vol. 259 ( 2015), pp.

3178-3201. (5-Year IF: 1.846 , Time cited: 1)

(6) Sze-Bi Hsu, Linfeng Mei* and Feng-Bin Wang, On a nonlocal reaction–diffusion–advection system modelling the growth of phytoplankton with cell quota structure, Journal of Differential Equations, Vol.

259 (2015), pp. 5353-5378. (5-Year IF: 1.846 , Time cited: 0)

二、近五年內獲獎情形及重要會議邀請演講至多五項。

INVITED TALKS:

(1)The 8^th International Congress on Industrial and Applied Mathematics(ICIAM 2015) , National Convention Center inside the Beijing Olympic Green, China, August 10-14. (Title:A PDE system modeling the dengue transmission with nonlocal infections and crowding effects, 2015.)

(2)The 10th AIMS Conference on Dynamical Systems,Differential Equations and

Applications, The Instituto de Ciencias Matemáticas (ICMAT) and the Universidad Autónoma de Madrid (UAM), July 7-11, 2014 (Title: Dynamics of a host-pathogen system in spatial habitats.)

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三、近五年內其他資料：擔任國際重要學術學會理監事、國際知名學術期刊編輯或評審委員等。

曾擔任下列學術期刊評審委員:

‧Oikos (2015)

‧Journal of Mathematical Analysis and Applications (2014)

‧Nonlinear Analysis Series B: Real World Applications (2014)

‧Journal of Theoretical Biology (2014)

‧Journal of Mathematical Biology (2012)

‧Journal Mathematical Modelling of Natural Phenomena (2012)

‧International Journal of Biomathematics (2012)

‧Journal of Mathematical Analysis and Applications (2012)

‧Communications on Pure and Applied Analysis (2012)

‧Applicable Analysis (2011)

‧Applied Mathematic and Computation (2011)

‧Applicable Analysis (2010)

‧Discrete and Continuous Dynamical System Series-B (2010)

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四、請簡述上述代表性研究成果之個人重要貢獻（含合作研究成果之具體個人關鍵貢獻）（至多一頁）。

Competition between species for resources is a fundamental ecological process. Any mathematical model that explicitly addresses both resource and population dynamics must specify how much resource is consumed in the production of one new individual. To understand the roles of nutrient consumption, storage, and population growth in spatially varying environments, there are at least three possible ways to this issue.

In reference [1], we combined conventional partial differential equations (PDEs) for population structure, the McKendrick-Von Foerster equations, with the physical transport equations governing spatial distributions of populations and nutrients. Then we investigated the competition between microorganisms for a single limiting resource with cell quota structure and spatial variation. We showed that the model with quota structure may reverse the outcome of competition for the classical model without quota structure under appropriate assumptions.

In reference [6], we focus on the study of the dynamics of a single species in a water column in

eutrophic ecosystem, that is, the species depends only on light for its growth. As in reference [1], we assume the amount of light absorbed by individuals is proportional to cell size, which varies for populations that reproduce by simple division into two equally-sized daughters, and species move by vertical turbulent diffusion and advection (sinking or buoyant). Most of phytoplankton species have

tendency to sink as they are heavier than water while some species will float as they have a lower density than water and it is called buoyant. The production of toxins that act against competing species, known as allelopathy, is an important factor that potentially influences competitive dynamics. In reference [3], we extended the previous theoretical model to the one with allelopathy and resource exploitation by considering competition for two resources which are stored within individuals. The emergence of multiple attractors at high resource supplies suggests that blooms of harmful algae producing allelopathic toxins could be difficult to predict under such rich conditions.

Blooms of the harmful algae have increased the intensity worldwide in coastal as well as inland waters.

The blooms have direct impacts for human health, and food webs in aquatic ecosystems. For example, golden algae is responsible for such harmful algal blooms worldwide that have caused large fish kills and millions of dollars in economic losses. In reference [2,5], we investigated two advection-dispersion-reaction models arising from the dynamics of harmful algae and their toxins in flowing-water habitats where a main channel is coupled to a hydraulic storage zone, representing an ensemble of fringing coves on the shoreline.

We establish a threshold type result on the extinction/persistence in terms of the basic reproduction ratio for algae. Due to the lack of diffusion terms in the equations for hydraulic storage zone, the solution maps are not compact. So we need to address the existence of the principal eigenvalue for the associated eigenvalue

problem. In reference [4], we removed the additional assumptions in reference [2,5], and then we proved the principal eigenvalue for the associated eigenvalue problem still exists.

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