STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN ECOLOGY AND EPIDEMICS

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11-2020

STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN ECOLOGY AND EPIDEMICS

APPLICATIONS IN ECOLOGY AND EPIDEMICS

Hebatallah Jamil Alsakaji

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College of Science

STOCHASTIC DELAY DIFFERENTIAL EQUATIONS WITH APPLICATIONS IN ECOLOGY AND EPIDEMICS

Hebatallah Jamil Alsakaji

This dissertation is submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy

Under the Supervision of Prof. Fathalla Ali Rihan

November 2020

Declaration of Original Work

I, Hebatallah Jamil Alsakaji, the undersigned, a graduate student at the United Arab Emi- rates University (UAEU), and the author of this dissertation, entitled "Stochastic Delay Differential Equations with Applications in Ecology and Epidemics”, hereby, solemnly declare that this dissertation is my own original research work that has been done and prepared by me under the supervision of Prof. Fathalla Ali Rihan, in the College of Sci- ence at the UAEU. This work has not previously been presented or published, or formed the basis for the award of any academic degree, diploma or a similar title at this or any other university. Any materials borrowed from other sources (whether published or un- published) and relied upon or included in my dissertation have been properly cited and acknowledged in accordance with appropriate academic conventions. I further declare that there is no potential conflict of interest with respect to the research, data collection, authorship, presentation and/or publication of this dissertation.

Student’s Signature Date 27.Dec.2020

Copyright c 2020 Hebatallah Jamil Alsakaji All Rights Reserved

Advisory Committee

1) Advisor: Fathalla A. Rihan Title: Professor

Department of Mathematical Sciences College of Science

2) Member: Farrukh Mukhamedov Title: Professor

Department of Mathematical Sciences College of Science

3) Member: Qasem Al Mdallal Title: Professor

Department of Mathematical Sciences College of Science

Approval of the Doctorate Dissertation

This Doctorate Dissertation is approved by the following Examining Committee Members:

1) Advisor (Committee Chair): Fathalla A. Rihan Title: Professor

Department of Mathematical Sciences College of Science

Signature Date

2) Member: Mohamed I. Syam Title: Professor

Department of Mathematical Sciences College of Science

Signature Date

3) Member: Adel Hashish Title: Associate Professor Department of Physics College of Science

Signature Date

4) Member (External Examiner): Gennady A. Bocharov Title: Professor

Institute of Numerical Mathematics

Russian Academy of Sciences, Russian Federation

Signature Date

25 Dec. 2020

25.12.2020 25-12-2020

25-12-2020

M.Syam

Adel Hashish

This Doctorate Dissertation is accepted by:

Dean of the College of Science: Professor Maamar Ben Kraouda

Signature Date

Dean of the College of Graduate Studies: Professor Ali Al-Marzouqi

Signature Date

Copy of

December 29, 2020

December 30, 2020

Abstract

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynam- ics, epidemiology, immunology, physiology, and neural networks. The memory or time- delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the evolution of the system at a certain time instant depends on the past history/memory. Introduction of such time-delays in a differential model significantly improves the dynamics of the model and enriches the complexity of the system.

Moreover, natural phenomena counter an environmental noise and usually do not follow deterministic laws strictly but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time.

Accordingly, stochastic delay differential equations (SDDEs) models play a prominent role in many application areas including biology, epidemiology and population dynamics, mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do. The SDDEs can be regarded as a generalization of stochastic differential equations (SDEs) and DDEs.

This dissertation, consists of eight Chapters, is concerned with qualitative and quantitative features of deterministic and stochastic delay differential equations with ap- plications in ecology and epidemics. The local and global stabilities of the steady states and Hopf bifurcations with respect of interesting parameters of such models are investi- gated. The impact of incorporating time-delays and random noise in such class of dif- ferential equations for different types of predator-prey systems and infectious diseases is studied. Numerical simulations, using suitable and reliable numerical schemes, are provided to show the effectiveness of the obtained theoretical results.

Chapter 1 provides a brief overview about the topic and shows significance of the study. Chapter 2, is devoted to investigate the qualitative behaviours (through local and global stability of the steady states) of DDEs with predator-prey systems in case of hunt- ing cooperation on predators. Chapter 3 deals with the dynamics of DDEs, of multiple time-delays, of two-prey one-predator system, where the growth of both preys populations subject to Allee effects, with a direct competition between the two-prey species having a common predator. A Lyapunov functional is deducted to investigate the global stabil- ity of positive interior equilibrium. Chapter 4, studies the dynamics of stochastic DDEs for predator-prey system with hunting cooperation in predators. Existence and unique-

ness of global positive solution and stochastically ultimate boundedness are investigated.

Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Chapter 5 is devoted to investigate Stochastic DDEs of three-species predator- prey system with cooperation among prey species. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution to the model are established, by constructing a suitable Lyapunov functional. Chapter 6 deals with stochas- tic epidemic SIRC model with time-delay for spread of COVID-19 among population.

The basic reproduction numberR₀^s for the stochastic model which is smaller thanR0 of the corresponding deterministic model is deduced. Sufficient conditions that guarantee the existence of a unique ergodic stationary distribution, using the stochastic Lyapunov functional, and conditions for the extinction of the disease are obtained. In Chapter 7, some numerical schemes for SDDEs are discussed. Convergence and consistency of such schemes are investigated. Chapter 8 summaries the main finding and future directions of research.

The main findings, theoretically and numerically, show that time-delays and ran- dom noise have a significant impact in the dynamics of ecological and biological systems.

They also have an important role in ecological balance and environmental stability of liv- ing organisms. A small scale of white noise can promote the survival of population;

While large noises can lead to extinction of the population, this would not happen in the deterministic systems without noises. Also, white noise plays an important part in con- trolling the spread of the disease; When the white noise is relatively large, the infectious diseases will become extinct; Re-infection and periodic outbreaks can also occur due to the time-delay in the transmission terms.

Keywords: Allee effect, Bifurcation, Brownian motion, Epidemic models, Lyapunov functionals, Predator-prey model, Sensitivity, SIRC, Stability, Stationary distribution, Stochastic perturbations, Time-delays.

Title and Abstract (in Arabic)

اﻟ ﻤ ﻌ ﺎ د ﻻ ت ا ﻟﺘ ﻔ ﺎ ﺿ ﻠﯿ ﺔ ا ﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ ذ ا ت ا ﻟ ﺬ ا ﻛ ﺮ ة و ﺗ ﻄ ﺒﯿ ﻘ ﺎﺗ ﮭ ﺎ ﻓ ﻲ ﻋ ﻠ ﻮ م ا ﻟﺒ ﯿﺌ ﺔ و ا ﻻ و ﺑﺌ ﺔ

اﻟ ﻤ ﻠ ﺨ ﺺ

ﺗ ﺴ ﺘ ﺨ ﺪ م ا ﻟ ﻤ ﻌ ﺎ د ﻻ ت ا ﻟﺘ ﻔﺎ ﺿ ﻠﯿ ﺔ ذا ت ا ﻟﺬ ا ﻛ ﺮ ة ) Delay Differential Equations, DDEs (

ﻋ ﻠ ﻰ ﻧ ﻄ ﺎ ق و ا ﺳ ﻊ ﻓ ﻲ ا ﻟﻨ ﻤ ﺬ ﺟ ﺔ

اﻟ ﺮ ﯾﺎ ﺿ ﯿ ﺔ (Mathematical Modeling) و

اﻟ ﺘﻨ ﺒ ﺆ ا ت ﻓ ﻲ ﻣ ﺨ ﺘﻠ ﻒ ﻣ ﺠ ﺎ ﻻ ت ﻋ ﻠ ﻮ م اﻟ ﺤ ﯿﺎ ة

، ﻋ ﻠ ﻰ ﺳ ﺒﯿ ﻞ اﻟ ﻤ ﺜﺎ ل د ﯾﻨ ﺎ ﻣ ﯿ ﻜ ﯿﺎ ت

اﻟ ﺴ ﻜ ﺎ ن (Dynamical Systems)

، و ﻋ ﻠﻢ ا ﻷ و ﺑﺌ ﺔ (Epidemiology)

، و ﻋ ﻠﻢ ا ﻟ ﻤ ﻨﺎ ﻋ ﺔ (Immunology)

، و ﻋ ﻠﻢ و ظ ﺎﺋ ﻒ

ا ﻷ ﻋ ﻀ ﺎ ء (Physiology)

، و اﻟ ﺸ ﺒ ﻜ ﺎ ت ا ﻟ ﻌ ﺼ ﺒﯿ ﺔ (Neural Networks) .

ﺣ ﯿ ﺚ ﺗ ﺮ ﺗﺒ ﻂ ا ﻟﺬ ا ﻛ ﺮ ة (Memory)

، ﻓ ﻲ ا ﻟﻨ ﻤ ﺎ ذ ج

اﻟ ﺮ ﯾﺎ ﺿ ﯿ ﺔ ﻟ ﻠﻔ ﺘ ﺮ ا ت ا ﻟ ﺰ ﻣ ﻨﯿ ﺔ

، ﻟ ﺘ ﻤ ﺜﯿ ﻞ ﺑ ﻌ ﺾ ا ﻟ ﻌ ﻤ ﻠﯿ ﺎ ت ا ﻟ ﺨ ﻔﯿ ﺔ ﻣ ﺜ ﻞ ﻣ ﺮ ا ﺣ ﻞ د و ر ة اﻟ ﺤ ﯿﺎ ة

، و ﻛ ﺬﻟ ﻚ ا ﻟ ﻮ ﻗ ﺖ ا ﻟ ﻼ ز م ﺑﯿ ﻦ إ ﺻ ﺎﺑ ﺔ ا ﻟ ﺨ ﻠﯿ ﺔ

و إﻧ ﺘﺎ ج ﻓﯿ ﺮ و ﺳ ﺎ ت ﺟ ﺪ ﯾ ﺪ ة

، و اﻟ ﻤ ﺪ ة اﻟ ﺰ ﻣ ﻨﯿ ﺔ ﻟ ﻠﻔ ﺘ ﺮ ا ت ا ﻟ ﻤ ﻌ ﺪ ﯾ ﺔ

، و اﻟ ﻔﺘ ﺮ ا ت ا ﻟ ﻤ ﻨﺎ ﻋ ﯿ ﺔ . و اﻟ ﺠ ﺪ ﯾ ﺮ ﺑﺎ ﻟﺬ ﻛ ﺮ ا ﻧ ﮫ ﻓ ﻲ ا ﻟ ﻤ ﻌ ﺎ د ﻻ ت ا ﻟﺘ ﻔﺎ ﺿ ﻠﯿ ﺔ

اﻟ ﻌ ﺎ د ﯾ ﺔ

، ﯾ ﺘ ﻢ ﺗ ﻘﯿ ﯿ ﻢ ا ﻟ ﺤ ﺎﻟ ﺔ ا ﻟ ﻤ ﺘ ﻐ ﯿ ﺮ ا ت ) (State Variables و

ﻣ ﺸ ﺘﻘ ﺎﺗ ﮭ ﺎ ﻓ ﻲ ا ﻟ ﻮ ﻗ ﺖ ا ﻵ ﻧ ﻲ ) (Current Time و

ﻟ ﻜ ﻦ ﻓ ﻲ ا ﻟ ﻤ ﻌ ﺎ د ﻻ ت

اﻟ ﺘﻔ ﺎ ﺿ ﻠﯿ ﺔ ذ ا ت ا ﻟﺬ ا ﻛ ﺮ ة DDEs

، ﯾ ﻌ ﺘ ﻤ ﺪ ﺗ ﻄ ﻮ ر ا ﻟﻨ ﻈ ﺎ م ﻓ ﻲ ا ﻟ ﻮ ﻗ ﺖ ا ﻵ ﻧ ﻲ و ﻋ ﻠ ﻰ ا ﻟﺘ ﺎ ر ﯾ ﺦ / اﻟ ﺬا ﻛ ﺮ ة اﻟ ﻤ ﺎ ﺿ ﯿ ﺔ (Memory/History)

. إد ﺧ ﺎ ل ﻣ ﺜ ﻞ ھ ﺬ ه اﻟ ﺘﺄ ﺧ ﯿ ﺮ ا ت ا ﻟ ﺰ ﻣ ﻨﯿ ﺔ ﻓ ﻲ ﻧ ﻤ ﻮ ذ ج ر ﯾﺎ ﺿ ﻲ ﺗ ﻔﺎ ﺿ ﻠ ﻲ ﯾ ﺤ ﺴ ﻦ ﺑ ﺸ ﻜ ﻞ ﻛ ﺒﯿ ﺮ د ﯾﻨ ﺎ ﻣ ﯿ ﻜ ﯿﺎ ت ا ﻟﻨ ﻤ ﻮ ذ ج و ﻣ ﺮ و ﻧﺘ ﺔ ﻟ ﺘ ﻤ ﺜﯿ ﻞ ا ﻷ ﻧ ﻈ ﻤ ﺔ

اﻟ ﻤ ﻌ ﻘ ﺪ ة ﻓ ﻲ ﻋ ﻠ ﻮ م ا ﻟ ﺤ ﯿﺎ ة.

ﻣ ﻦ اﻟ ﻤ ﻌ ﻠ ﻮ م ا ن اﻟ ﻈ ﻮ ا ھ ﺮ اﻟ ﻄ ﺒﯿ ﻌ ﯿ ﺔ ﻗ ﺪ ﺗ ﻮ ا ﺟ ﮫ ﺑ ﻌ ﺾ ا ﻻ ﺿ ﻄ ﺮ اﺑ ﺎ ت اﻟ ﺒﯿ ﺌﯿ ﺔ اﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ (Environmental Stochastic

Perturbations/Noise)

، ﻣ ﺜ ﻞ ﺗ ﺄﺛ ﯿ ﺮ ﺗ ﻐ ﯿ ﺮ ا ت ا ﻟ ﻄ ﻘ ﺲ و د ر ﺟ ﺎ ت ا ﻟ ﺤ ﺮ ا ر ة

، ا ﻟ ﺮ ط ﻮ ﺑ ﺔ و إﻟ ﺦ إ ﻟ ﺦ

، و ﻋ ﺎ د ة ھ ﺬ ه اﻟ ﻈ ﻮ ا ھ ﺮ ﻻ ﺗ ﺘﺒ ﻊ

اﻟ ﻘ ﻮ اﻧ ﯿ ﻦ ا ﻟﻘ ﻄ ﻌ ﯿ ﺔ و ﻟ ﻜ ﻨ ﮭ ﺎ ﺗﺘ ﺄ ر ﺟ ﺢ ﺑ ﺸ ﻜ ﻞ ﻋ ﺸ ﻮ اﺋ ﻲ ﺣ ﻮ ل ﺑ ﻌ ﺾ ا ﻟﻘ ﯿ ﻢ ا ﻟ ﻤ ﺘ ﻮ ﺳ ﻄ ﺔ

، ﺣ ﯿ ﺚ ﺗ ﺘﺄ ر ﺟ ﺢ ا ﻟ ﻜ ﺜﺎ ﻓ ﺔ ا ﻟ ﺴ ﻜ ﺎﻧ ﯿ ﺔ و ﻻ ﺗ ﺼ ﻞ أ ﺑ ﺪًا إ ﻟ ﻰ

ﻗﯿ ﻤ ﺔ ﺛﺎ ﺑﺘ ﺔ ﻣ ﻊ ﺗﻘ ﺪ م ا ﻟ ﻮ ﻗ ﺖ . و ﺑﻨ ءًﺎ ﻋ ﻠ ﻰ ذﻟ ﻚ

، ﻧﻘ ﺘ ﺮ ح ﻓ ﻲ ھ ﺬ ه اﻟ ﺮ ﺳ ﺎﻟ ﺔ ﺑ ﻌ ﺾ ا ﻟﻨ ﻤ ﺎ ذ ج ا ﻟ ﺮ ﯾﺎ ﺿ ﯿ ﺔ ﺑﺎ ﺳ ﺘ ﺨ ﺪا م ا ﻟ ﻤ ﻌ ﺎ د ﻻ ت ا ﻟﺘ ﻔﺎ ﺿ ﻠﯿ ﺔ

اﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ ذا ت ا ﻟﺬ ا ﻛ ﺮ ة ) SDDEs (

و ذﻟ ﻚ ﻟ ﺘ ﻤ ﺜﯿ ﻞ و ﻧ ﻤ ﺬ ﺟ ﺔ ﺑ ﻌ ﺾ ا ﻟ ﻈ ﻮ ا ھ ﺮ ا ﻟ ﻄ ﺒﯿ ﻌ ﯿ ﺔ ﻓ ﻲ ﻋ ﻠ ﻮ م ا ﻟﺒ ﯿﺌ ﺔ و ا ﻟ ﺤ ﯿﺎ ة

، ﺣ ﯿ ﺚ أ ﻧ ﮭ ﺎ ﺗ ﻮ ﻓ ﺮ

د ر ﺟ ﺎ ت إ ﺿ ﺎﻓ ﯿ ﺔ ﻣ ﻦ ا ﻟ ﻮ اﻗ ﻌ ﯿ ﺔ ﻣ ﻘﺎ ر ﻧ ﺔً

ﺑ ﻨ ﻈ ﯿ ﺮ اﺗ ﮭ ﺎ ﻏ ﯿ ﺮ ا ﻟ ﻌ ﺸ ﻮ اﺋ ﻲ .(Deterministic) ﯾ

ﻤ ﻜ ﻦ ا ﻋ ﺘﺒ ﺎ ر ا ﻟ ﻤ ﻌ ﺎ د ﻻ ت ا ﻟﺘ ﻔﺎ ﺿ ﻠﯿ ﺔ ا ﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ

ذا ت ا ﻟﺬ ا ﻛ ﺮ ة SDDEs ﺑ

ﻤ ﺜﺎ ﺑ ﺔ ﺗ ﻌ ﻤ ﯿ ﻢ ا ﻟ ﻤ ﻌ ﺎ د ﻻ ت ا ﻟﺘ ﻔﺎ ﺿ ﻠﯿ ﺔ ا ﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ (Stochastic Differential Equations, SDDEs)

و ا ﻟ ﻤ ﻌ ﺎ د ﻻ ت ا ﻟﺘ ﻔﺎ ﺿ ﻠﯿ ﺔ ذا ت ا ﻟﺬ ا ﻛ ﺮ ة.

ﺣ ﯿ ﺚ ﺗﻘ ﺪ م ھ ﺬ ه ﻷ ط ﺮ و ﺣ ﺔ د ر ا ﺳ ﺔ ﻣ ﻮ ﺳ ﻌ ﺔ ﻋ ﻦ ا ﻟ ﺴ ﻤ ﺎ ت و ا ﻟ ﺨ ﺼ ﺎﺋ ﺺ ا ﻟﻨ ﻮ ﻋ ﯿ ﺔ و اﻟ ﻜ ﻤ ﯿ ﺔ

(Qualitative and Quantitative Features) ﻟ

ﻠ ﻤ ﻌ ﺎ د ﻻ ت ا ﻟﺘ ﻔﺎ ﺿ ﻠﯿ ﺔ ا ﻟ ﺤ ﺘ ﻤ ﯿ ﺔ و اﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ ذا ت ا ﻟﺬ ا ﻛ ﺮ ة و ﺗ ﻄ ﺒﯿ ﻘﺎ ﺗ ﮭ ﺎ اﻟ ﻤ ﮭ ﻤ ﺔ

و ا ﻟ ﻤ ﺘ ﻌ ﺪ د ة ﻓ ﻲ ﻋ ﻠ ﻮ م ا ﻟﺒ ﯿﺌ ﺔ و اﻧ ﺘ ﺸ ﺎ ر و ا ﻷ و ﺑﺌ ﺔ .

ﺗﺘ ﻜ ﻮ ن ھ ﺬ ه ا ﻷ ط ﺮ و ﺣ ﺔ ﻣ ﻦ ﺛ ﻤ ﺎﻧ ﯿ ﺔ ﻓ ﺼ ﻮ ل

، ﯾ ﻘ ﺪ م ا ﻟﻔ ﺼ ﻞ ا ﻷ و ل ﻣ ﻘ ﺪ ﻣ ﺔ ﻋ ﺎ ﻣ ﺔ ﻟ ﻠ ﺮ ﺳ ﺎﻟ ﺔ و أ ھ ﻤ ﯿ ﺔ ﻟ ﮭ ﺬ ه اﻟ ﺪ ر ا ﺳ ﺔ . و ﯾ ﺨ ﺼ ﺺ ا ﻟﻔ ﺼ ﻞ

اﻟ ﺜﺎ ﻧ ﻲ ﻟ ﺪ ر ا ﺳ ﺔ ا ﻟ ﺨ ﺼ ﺎﺋ ﺺ ا ﻟﻨ ﻮ ﻋ ﯿ ﺔ ﻷ ﻧ ﻈ ﻤ ﺔ ا ﻟﻔ ﺮ ﯾ ﺴ ﺔ و اﻟ ﻤ ﻔﺘ ﺮ س (Prey-Predator Systems) ﻓ

ﻲ ﺣ ﺎﻟ ﺔ ا ﻟ ﺼ ﯿ ﺪ ﻣ ﻊ و ﺟ ﻮ د ﺗ ﻌ ﺎ و ن

ﺑﯿ ﻦ ا ﻟ ﺤ ﯿ ﻮ اﻧ ﺎ ت ا ﻟ ﻤ ﻔﺘ ﺮ ﺳ ﺔ و ذﻟ ﻚ ﺑ ﺎ ﺳ ﺘ ﺨ ﺪا م ﻧ ﻤ ﺎ ذ ج DDEs . ﯾﺘ ﻌ ﺎ ﻣ ﻞ ا ﻟﻔ ﺼ ﻞ ا ﻟﺜ ﺎﻟ ﺚ ﻣ ﻊ د ﯾﻨ ﺎ ﻣ ﯿ ﻜ ﯿﺎ ت DDEs

، ﻣ ﻦ ا ﻟﻔ ﺘ ﺮ ا ت ا ﻟ ﺰ ﻣ ﻨﯿ ﺔ

اﻟ ﻤ ﺘ ﻌ ﺪ د ة

، ﻷ ﻧ ﻈ ﻤ ﺔ ا ﻟﻔ ﺮ ﯾ ﺴ ﺔ و اﻟ ﻤ ﻔﺘ ﺮ س

، ﺣ ﯿ ﺚ ﯾ ﻜ ﻮ ن ﻧ ﻤ ﻮ ﻛ ﻠﺘ ﺎ اﻟ ﻔ ﺮ ﯾ ﺴ ﺘﯿ ﻦ ﻋ ﺮ ﺿ ﺔ ﻟ ﺘﺄ ﺛﯿ ﺮ ا ت Allee

، و ھ ﻨﺎ ك ﻣ ﻨﺎ ﻓ ﺴ ﺔ ﻣ ﺒﺎ ﺷ ﺮ ة ﺑﯿ ﻦ

ا ﻷ ﻧ ﻮ ا ع ذا ت ا ﻟﻔ ﺮ ﯾ ﺴ ﺔ ا ﻟﺘ ﻲ ﻟ ﺪ ﯾ ﮭ ﺎ ﻣ ﻔﺘ ﺮ س ﻣ ﺸ ﺘ ﺮ ك . ﯾﻘ ﺪ م ا ﻟﻔ ﺼ ﻞ ا ﻟ ﺮ اﺑ ﻊ د ر ا ﺳ ﺔ ﻋ ﻦ د ﯾﻨ ﺎ ﻣ ﯿ ﻜ ﯿﺎ ت ا ﻟ ﻤ ﻌ ﺎ د ﻻ ت ا ﻟﺘ ﻔﺎ ﺿ ﻠﯿ ﺔ ا ﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ ذا ت

اﻟ ﺬا ﻛ ﺮ ة SDDEs

ﻟﻨ ﻈ ﺎ م ا ﻟ ﻤ ﻔﺘ ﺮ س ا ﻟﻔ ﺮ ﯾ ﺴ ﺔ . ﺣ ﯿ ﺚ ﺗ ﻢ ا ﻟﺘ ﺤ ﻘ ﻖ ﻣ ﻦ و ﺟ ﻮ د و ﺗﻔ ﺮ د ا ﻟ ﺤ ﻞ ا ﻹ ﯾ ﺠ ﺎﺑ ﻲ

، ﺑ ﺎ ﻹ ﺿ ﺎﻓ ﺔ إ ﻟ ﻰ ا ﻟﺘ ﻮ ﺻ ﻞ ﻟ ﺒ ﻌ ﺾ

اﻟ ﺸ ﺮ و ط ا ﻟ ﻜ ﺎﻓ ﯿ ﺔ ﻟ ﻮ ﺟ ﻮ د و اﻧ ﻘ ﺮ ا ض ا ﻟﻔ ﺮ ﯾ ﺴ ﺔ و اﻟ ﻤ ﻔﺘ ﺮ س . اﻟ ﻔ ﺼ ﻞ ا ﻟ ﺨ ﺎ ﻣ ﺲ ﻣ ﻜ ﺮ س ﻟ ﺪ ر ا ﺳ ﺔ أ ﻧ ﻈ ﻤ ﺔ ا ﻟﻔ ﺮ ﯾ ﺴ ﺔ و اﻟ ﻤ ﻔﺘ ﺮ س ذا ت ا ﻷ ﻧ ﻮ ا ع

اﻟ ﺜ ﻼ ﺛ ﺔ ﻣ ﻊ و ﺟ ﻮ د ﺗ ﻌ ﺎ و ن ﺑ ﯿ ﻦ أ ﻧ ﻮ ا ع ا ﻟﻔ ﺮ اﺋ ﺲ و ذﻟ ﻚ ﺑ ﺎ ﺳ ﺘ ﺨ ﺪا م ﻧ ﻤ ﺎ ذ ج DDEs . ﺣ ﯿ ﺚ ﺗ ﺒﯿ ﻦ أ ﻧ ﮫ ﯾ ﻤ ﻜ ﻦ ﻟ ﻠ ﻀ ﻮ ﺿ ﺎ ء ا ﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ أ ن ﺗ ﻜ ﺒ ﺢ

اﻧ ﻔ ﺠ ﺎ ر ا ﻷ ﻧ ﻮ ا ع

، ﻓ ﻲ ﺣ ﺎﻟ ﺔ ﻛ ﻮ ﻧ ﮭ ﺎ ﻏ ﯿ ﺮ ﻣ ﺤ ﺪ و د ة ﻓ ﻲ ا ﻟﻨ ﻈ ﺎ م ا ﻟ ﺤ ﺘ ﻤ ﻲ . اﻟ ﻔ ﺼ ﻞ ا ﻟ ﺴ ﺎ د س

، ﯾ ﺘﻨ ﺎ و ل د ر ا ﺳ ﺔ ﻧ ﻤ ﻮ ذ ج SIRC ا ﻟ ﻮ ﺑﺎ ﺋ ﻲ ا ﻟ ﻌ ﺸ ﻮ اﺋ ﻲ

ﻣ ﻊ ا ﻟﺘ ﺄ ﺧ ﯿ ﺮ ا ت ا ﻟ ﺰ ﻣ ﻨﯿ ﺔ و ذﻟ ﻚ ﻟ ـﺪ ر ا ﺳ ﺔ د ﯾﻨ ﺎ ﻣ ﯿ ﻜ ﯿ ﺔ ا ﻧﺘ ﺸ ﺎ ر ﻓ ﯿ ﺮ و س ﻛ ﻮ ر و ﻧﺎ - ٢ (COVID-19) دا

ﺧ ﻞ ا ﻟ ﻤ ﺠ ﺘ ﻤ ﻊ . ﺣ ﯿ ﺚ ﺗ ﻢ ا ﺳ ﺘﻨ ﺒﺎ ط

اﻟ ﺸ ﺮ و ط ا ﻟ ﻼ ز ﻣ ﺔ ﻟ ﻠ ﻮ ﺻ ﻮ ل ا ﻟ ﻲ ﺣ ﺎﻟ ﺔ ا ﻻ ﺳ ﺘﻘ ﺮ ا ر و إ ﻣ ﻜ ﺎﻧ ﯿ ﺔ ا ﻟ ﺴ ﯿ ﻄ ﺮ ة ﻋ ﻠ ﻰ ا ﻧﺘ ﺸ ﺎ ر ا ﻟ ﻤ ﺮ ض . ﯾﻘ ﺪ م ا ﻟﻔ ﺼ ﻞ ا ﻟ ﺴ ﺎﺑ ﻊ د ر ا ﺳ ﺔ ا ﻟ ﺤ ﻠ ﻮ ل

اﻟ ﻌ ﺪ د ﯾ ﺔ و اﻟ ﺘﻘ ﺮ ﯾﺒ ﯿ ﺔ ﻟﻠ ﻤ ﻌ ﺎ د ﻻ ت اﻟ ﺘﻔ ﺎ ﺿ ﻠﯿ ﺔ اﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ (SDDEs) و

اﻟ ﺸ ﺮ و ط اﻟ ﻼ ز ﻣ ﺔ ﻻ ﺳ ﺘﻘ ﺮ ا ر اﻟ ﺤ ﻠ ﻮ ل اﻟ ﺘﻘ ﺮ ﯾﺒ ﯿ ﺔ

، ﺑﯿ ﻨ ﻤ ﺎ ﯾﻠ ﺨ ﺺ

اﻟ ﻔ ﺼ ﻞ ا ﻟﺜ ﺎ ﻣ ﻦ ﻣ ﺎ ﺗ ﻮ ﺻ ﻠﻨ ﺎ إﻟ ﯿ ﮫ ﻣ ﻦ ا ھ ﻢ ا ﻟﻨ ﺘﺎ ﺋ ﺞ و ا ﻻ ﺗ ﺠ ﺎ ھ ﺎ ت ﻣ ﺴ ﺘﻘ ﺒﻠ ﯿ ﺔ ﻟ ﻠﺒ ﺤ ﺚ .

ﺗ ﻈ ﮭ ﺮ ا ﻟﻨ ﺘﺎ ﺋ ﺞ ا ﻟﺘ ﻲ ﺗ ﻮ ﺻ ﻠﻨ ﺎ إﻟ ﯿ ﮭ ﺎ،

ﻧ ﻈ ﯾًﺎﺮ و ﻋ ﺪ ﯾًﺎد

، أ ﻓ ﻀ ﻠﯿ ﺔ ا ﻟ ﻤ ﻌ ﺎ د ﻻ ت ا ﻟﺘ ﻔﺎ ﺿ ﻠﯿ ﺔ ا ﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ ذا ت ا ﻟﺬ ا ﻛ ﺮ ة (SDDEs) ﻋ

ﻠ ﻰ ﻏ ﯿ ﺮ ھ ﺎ

ﻣ ﻦ ا ﻟﻨ ﻤ ﺎ ذ ج ا ﻟ ﺮ ﯾﺎ ﺿ ﯿ ﺔ ا ﻷ ﺧ ﺮ ى

، ﺣ ﯿ ﺚ ا ن ﻟ ﮭ ﺎ ﺗﺄ ﺛﯿ ﺮ ﻛ ﺒﯿ ﺮ ﻓ ﻲ د ﯾﻨ ﺎ ﻣ ﯿ ﻜ ﯿﺎ ت ا ﻷ ﻧ ﻈ ﻤ ﺔ ا ﻟﺒ ﯿﺌ ﯿ ﺔ و اﻟ ﺒﯿ ﻮ ﻟ ﻮ ﺟ ﯿ ﺔ . ﻛ ﻤ ﺎ أ ن ﻟ ﮭ ﺎ د رًو ا ﻣ ﻤًﮭ ﺎ ﻓ ﻲ

اﻟ ﺘ ﻮ ا ز ن و ا ﻻ ﺳ ﺘﻘ ﺮ ا ر ا ﻟﺒ ﯿﺌ ﻲ ﻟ ﻠ ﻜ ﺎﺋ ﻨﺎ ت ا ﻟ ﺤ ﯿ ﺔ . ﺣ ﯿ ﺚ ﺛ ﺒ ﺖ ا ن و ﺟ ﻮ د ا ﻟ ﻀ ﺠ ﯿ ﺞ ا ﻟ ﻄ ﻔﯿ ﻒ (Small Noise

، ﯾ ﻌ ﺰ ز ﺑ ﻘﺎ ء ا ﻟ ﻜ ﺎﺋ ﻨﺎ ت ا ﻟ ﺤ ﯿ ﺔ

(Persisting)

؛ ا ﻣ ﺎ ﻓ ﻲ ﺣ ﯿ ﻦ و ﺟ ﻮ د ا ﻟ ﻀ ﺠ ﯿ ﺞ ا ﻟ ﻜ ﺒﯿ ﺮ ﯾ ﻤ ﻜ ﻦ أ ن ﯾ ﺆ د ي إ ﻟ ﻰ ا ﻧﻘ ﺮ ا ض (Extinction)

ﺑ ﻌ ﺾ ا ﻷ ﻧ ﻮ ا ع (some

species) .

ﻛ ﻤ ﺎ و ﺟ ﺪ ا ن ا ﻻ ﺿ ﻄ ﺮ اﺑ ﺎ ت ا ﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ ا ﻟﺒ ﯿﺌ ﯿ ﺔ ﻣ ﻔﯿ ﺪ ة و ﺟ ﺎﻧ ﺐ ﺣ ﺘ ﻤ ﻲ ﻣ ﺆ ﺛ ﺮ ﻋ ﻠ ﻰ د ﯾﻨ ﺎ ﻣ ﯿ ﻜ ﯿﺎ ت أ ي ﻧ ﻈ ﺎ م ﺑ ﯿﺌ ﻲ

، ﻛ ﻤ ﺎ اﻧ ﮫ

ﻟ ﮫ د و ر ﻓ ﻌ ﺎ ل ﻟ ﻘ ﻤ ﻊ ا ﻧﻔ ﺠ ﺎ ر ﺳ ﻜ ﺎﻧ ﻲ ﻣ ﺤ ﺘ ﻤ ﻞ . و ﻣ ﻦ ﺛ ﻢ

، ا ﺳ ﺘ ﺨ ﺪا م ا ﻟ ﻤ ﻌ ﺎ د ﻻ ت ا ﻟﺘ ﻔﺎ ﺿ ﻠﯿ ﺔ ا ﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ ذا ت ا ﻟﺬ ا ﻛ ﺮ ة ﻓ ﻲ ﻧ ﻤ ﺬ ﺟ ﺔ د ﯾﻨ ﺎ ﻣ ﯿ ﻜ ﯿﺎ ت

اﻟ ﺴ ﻜ ﺎ ن و د ﯾﻨ ﺎ ﻣ ﯿ ﻜ ﯿ ﺔ و اﻧ ﺘ ﺸ ﺎ ر ا ﻷ و ﺑﺌ ﺔ اﻓ ﻀ ﻞ ﻣ ﻦ ﻏ ﯿ ﺮ ھ ﺎ ﻣ ﻦ اﻟ ﻨ ﻤ ﺎ ذ ج اﻟ ﺮ ﯾﺎ ﺿ ﯿ ﺔ ا ﻷ ﺧ ﺮ ى اﻟ ﺘ ﻲ ﺗﻔ ﺘﻘ ﺪ اﻟ ﺬا ﻛ ﺮ ة و ا ﻻ ﺿ ﻄ ﺮ اﺑ ﺎ ت

اﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ .

ﻣ ﻔ ﺎ ھ ﯿ ﻢ اﻟ ﺒ ﺤ ﺚ اﻟ ﺮ ﺋﯿ ﺴ ﺔ : ﺗﺄ ﺛﯿ ﺮ Allee

، اﻟ ﺘ ﺸ ﻌ ﺐ

، اﻟ ﺤ ﺮ ﻛ ﺔ اﻟ ﺒ ﺮ ا و ﻧﯿ ﺔ

، ﻧ ﻤ ﺎ ذ ج و ﺑﺎ ﺋﯿ ﺔ

، ﻧ ﻤ ﺎ ذ ج اﻟ ﻔ ﺮ ﯾ ﺴ ﺔ و اﻟ ﻤ ﻔﺘ ﺮ س

، SIRC

،

ا ﻻ ﺳ ﺘﻘ ﺮ ا ر

، ا ﻟﺘ ﻮ ز ﯾ ﻊ ا ﻟﺜ ﺎﺑ ﺖ ا ﻻ ﺿ ﻄ ﺮ اﺑ ﺎ ت ا ﻟ ﻌ ﺸ ﻮ اﺋ ﯿ ﺔ

، ا ﻟﺘ ﺄ ﺧ ﯿ ﺮ ا ت ا ﻟ ﺰ ﻣ ﻨﯿ ﺔ .

Acknowledgements

Firstly, all praise is to Allah, the Almighty, for giving me the strength, knowledge, ability and opportunity to undertake and complete this study.

Now, I would like to express my sincere gratitude to my supervisor Professor Fathalla A. Rihan for his continuous support during my PhD study and related research, for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research, complete and writing this dissertation. I could not have imagined having a better advisor and mentor for my PhD study.

I would also like to thank all the faculty members of Department of Mathematical Sciences, College of Graduate Studies (CGS) at UAEU, and coordinator of PhD program (Professor Mohamed I. Syam) for their support and cooperation during my study. My thanks should go to my colleagues and friends, especially Dr C. Rajivganthi, for dis- cussions, suggestions and criticism. I also would like to thank Professor Gennady A.

Bocharov (Moscow) for his constructive suggestions and comments.

This work was generously supported by UAE University. So, I am grateful to the UAE University that granted me a fellowship to pursue my PhD study.

I take this opportunity to express my sincere love and warmest gratitude to my family: my great parents (Mr Jamil Alsakaji and Mrs Maryam Alsakaji), my husband (Eng. Imad Abuhag), my kids (Basel, Yara, and Maria). Special thanks go my brothers and my sisters. I would have never accomplished this work without their continuous support and encouragements.

Dedication

To my great parents, my beloved husband and my lovely kids

Table of Contents

Title . . . i

Declaration of Original Work . . . ii

Copyright . . . iii

Advisory Committee . . . iv

Approval of the Doctorate Dissertation . . . v

Abstract . . . vii

Title and Abstract (in Arabic) . . . ix

Acknowledgments . . . xi

Dedication . . . xii

Table of Contents . . . xiii

List of Tables . . . xvii

List of Figures . . . xviii

List of Abbreviations . . . xxi

Chapter 1: Delay Differential Equations with Population Problems . . . 1

1.1 Introduction . . . 1

1.2 Existence and Uniqueness of Solutions for DDEs . . . 3

1.3 Stability of Equilibria and Lyapunov Functions for DDEs . . . 8

1.3.1 Lyapunov theory approach . . . 13

1.4 Stochastic Delay Differential Equations . . . 16

1.5 Existence and Uniqueness of the Solutions for SDDEs . . . 20

1.6 Stability Criteria for SDDEs . . . 23

1.7 Main Objectives and Significance of the Research . . . 27

1.7.1 Research significance . . . 27

Chapter 2: Delay Differential Equations of Predator-Prey Interactions with Hunting Cooperation in Predators . . . 30

2.1 Introduction . . . 30

2.2 Model Formulation . . . 31

2.3 Existence of Equilibrium Points . . . 33

2.3.1 Boundedness of the solutions . . . 33

2.4 Local Stability and Hopf Bifurcation . . . 35

2.5 Global Stability of the Interior Equilibrium Point . . . 39

2.6 Numerical Simulations . . . 41

2.7 Concluding Remarks . . . 43

Chapter 3: Delay Differential Equations of Three-Species Predator-Prey In- teractions with Allee Effect . . . 44

3.1 Introduction . . . 44

3.1.1 Allee effect . . . 44

3.2 Distribution of the Model with Allee Effect . . . 47

3.2.1 Non-negativity and boundedness of the solution . . . 49

3.3 Local Stability and Hopf Bifurcation . . . 51

3.3.1 Existence of equilibrium points . . . 51

3.3.2 Existence of bistability . . . 54

3.3.3 Stability and bifurcation analysis of the equilibriaE4andE^∗ . . . . 54

3.4 Global Stability of Interior Steady StateE^∗ . . . 63

3.5 Sensitivity Analysis . . . 65

3.5.1 Sensitivity to severity of Allee effect . . . 67

3.5.2 Sensitivity to time-delays . . . 68

3.6 Numerical Simulations . . . 69

3.7 Concluding Remarks . . . 73

Chapter 4: Stochastic DDEs of Predator-Prey System with Hunting Coop- eration in Predators . . . 75

4.1 Introduction . . . 75

4.2 Preliminaries . . . 76

4.2.1 Stochastic processes . . . 77

4.2.2 It ˆoformula . . . 79

4.3 SDDEs for Predator-Prey System . . . 80

4.3.1 Existence and uniqueness of positive solution . . . 81

4.4 Stochastically Ultimate Boundedness . . . 84

4.5 Almost Surely Asymptotic Properties . . . 86

4.6 Persistence and Extinction of the Solution . . . 89

4.6.1 Persistence . . . 89

4.6.2 Extinction . . . 93

4.7 Numerical Simulations . . . 96

4.8 Concluding Remarks . . . 100

Chapter 5: Stochastic DDEs of Three-Species Predator-Prey Systems with

Cooperation among Prey Species . . . 102

5.1 Introduction . . . 102

5.2 Existence and Uniqueness of Positive Solution . . . 105

5.3 Existence of Ergodic Stationary Distribution . . . 108

5.4 Extinction . . . 115

5.5 Numerical Simulations . . . 122

5.6 Concluding Remarks . . . 126

Chapter 6: Stochastic SIRC Epidemic Model with Time-Delay for COVID-19 . . . 128

6.1 Introduction . . . 128

6.2 Stochastic SIRC Epidemic Model . . . 131

6.3 Existence and Uniqueness of the Positive Solution . . . 134

6.4 Existence of Ergodic Stationary Distribution . . . 137

6.5 Extinction . . . 144

6.6 Numerical Simulations and Discussions . . . 147

6.7 Conclusion . . . 151

Chapter 7: Numerical Schemes for Stochastic Delay Differential Equations . . . . 153

7.1 Introduction . . . 153

7.2 Preliminaries . . . 154

7.3 Numerical Scheme for an Autonomous SDDEs . . . 156

7.3.1 Local and global errors . . . 158

7.3.2 Convergence and consistency . . . 158

7.4 Numerical Schemes for Non-autonomous SDDEs . . . 159

7.4.1 Taylor approximation . . . 160

7.4.2 Implicit strong approximations . . . 161

7.5 Milstien Scheme for SDDEs . . . 162

7.5.1 Convergence and mean square stability of Milstein scheme . . . 163

7.6 Concluding Remarks . . . 165

Chapter 8: Summary and Concluding Remarks . . . 166

8.1 Introduction . . . 166

8.2 Concluding Remarks and Findings . . . 167

8.3 Software Used . . . 169

8.4 Future Directions . . . 169

References . . . 171

List of Publications . . . 181

Appendices . . . 182 Appendix A: Software for Solving DDEs . . . 182 Appendix B: Sample of Matlab Program for Stochastic DDEs Using

Milstein Scheme . . . 187

List of Tables

Table 2.1: One biological meaning for the parameters of Model (2.2) . . . 32

Table 3.1: One biological meaning for the parameters of Model (3.3) . . . 49

Table 5.1: One biological meaning for the parameters of Model (5.3) . . . 103

Table 6.1: Incubation period of several common infectious diseases . . . 131

Table 6.2: Description of the model parameters . . . 132

List of Figures

Figure 1.1: Stability region for scalar DDEy⁰(t) =µ₁y(t−τ) +µ₀y(t) . . . . 11 Figure 1.2: Numerical simulations for DDEy⁰(t) =µ₁y(t−τ) +µ₀y(t) . . . . 12 Figure 1.3: Number of real eigenvalues for the characteristic equation

of DDEy⁰(t) =µ1y(t−τ) . . . 13 Figure 1.4: Numerical simulations of deterministic Hutchinson DDE

(1.27) and its corresponding SDDE (1.31) . . . 20 Figure 1.5: The impact of environmental Brownian noise that suppresses

explosions in population dynamics . . . 21 Figure 2.1: Stable population distribution (left) for Model (2.2) when

τ₁=τ₂=0 . . . 41 Figure 2.2: Stable population distribution (left) for Model (2.2) when

τ1=0.156<τ₁^∗andτ2=0.09<τ₂^∗ . . . 41 Figure 2.3: Bistability of the interior equilibriumE^∗andE1for Model

(2.2) . . . 42 Figure 2.4: Bifurcation diagram with respect toα for System (2.2) . . . 42 Figure 3.1: The left banner shows the per-capita growth rate _N^1dN_dt vs

populationN(t) . . . 45 Figure 3.2: Mathematical scheme of the three-species one-predator two-

prey System (3.3) . . . 47 Figure 3.3: Real parts of the approximated and corrected roots of the

characteristic Equation (3.12) . . . 57 Figure 3.4: The eigenvalues of the characteristic Equation (3.12) atE4 . . . 58 Figure 3.5: The roots of the characteristic equation of System (3.3)

with negative real parts at the stable steady stateE^∗ . . . 69 Figure 3.6: Numerical simulations of System (3.3) around the steady

stateE^∗ . . . 70 Figure 3.7: The Hopf bifurcation diagrams ofτ₁andτ₂for System (3.3) . . . 70 Figure 3.8: Bistability of two interior equilibria for the delayed System (3.3) . 71 Figure 3.9: The sensitivity of the dynamics of the System (3.3) due to

small changes in the severity of Allee effectα1andα2 . . . 71 Figure 3.10: Sensitivity functions of model solution of System (3.3) with

respect to Allee parametersα1andα2. . . 72 Figure 3.11: Sensitivity functions of model solution of System (3.3) with

respect to time delayτ₁ . . . 72 Figure 4.1: Numerical simulations of System (4.3) which displays the

persistence of the system, whenT₀^s>1 . . . 96

Figure 4.2: Numerical simulations of the solutions for System (4.3) and the corresponding deterministic System (2.2), when

T₀^s>1 . . . 97 Figure 4.3: Numerical simulations of the solutions for System (4.3)

and the corresponding deterministic System (2.2), when

T₀^s<1 . . . 98 Figure 4.4: Numerical simulations of the solutions for System (4.3)

and the corresponding deterministic System (2.2), when

T₀^s<1 withτ₁=τ₂=0.1,σ₁=0.001 andσ₂=0.1 . . . 98 Figure 4.5: Numerical simulations of the solutions for System (4.3)

and the corresponding deterministic System (2.2) with time

delayτ₁>τ₁^∗=2.1 . . . 99 Figure 5.1: Numerical simulations of deterministic DDEs (5.2) (left)

and SDDEs (5.3) (right), when τ₁ = 1.25, τ₂ = 0.6 and

τ₃=0.5 . . . 123 Figure 5.2: Numerical simulations of the solutions for System (5.3)

(right) and the corresponding undisturbed System (5.2) (left), withτ₁=1.25,τ₂=0.6 andτ₃=0.5, one can clearly see

that the predator goes to extinct . . . 123 Figure 5.3: Numerical simulations of the solutions for System (5.3)

(right) and the corresponding undisturbed System (5.2) (left), withτ₁=1.25,τ₂=0.6 andτ₃=0.5, one can clearly see

that all the species go to extinct . . . 124 Figure 5.4: Numerical simulations of the solutions for System (5.3)

(right) and the corresponding undisturbed System (5.2) (left),

withτ₁=10,τ₂=0.1 andτ₃=0.1 . . . 124 Figure 5.5: The effect of white noise to prevent the explosion of the

population, whenβ =0.5 . . . 125 Figure 6.1: Scheme of SIRC Model (6.1), assuming that the total pop-

ulationN=1. . . 132 Figure 6.2: Numerical simulations of stochastic Model (6.2), whenR₀^s=

1.3>1 . . . 147 Figure 6.3: Time domain behaviors of solutions of SDDEs Model (6.2)

(right) and the corresponding deterministic Model (6.1) (left),

whenR₀^s =0.38<1 . . . 147 Figure 6.4: Time domain behaviors of SDDEs Model (6.2) (right) and

corresponding deterministic Model (6.1) (left). The Figure

shows a periodic outbreak due to the time-delayτ . . . 148

Figure 6.5: Simulations of stochastic Model (6.2) (right) and the cor- responding deterministic Model (6.1) (left), when R₀^s =

0.38<1 . . . 148 Figure 6.6: Time domain behaviors of SDDEs Model (6.2) (right) and

corresponding deterministic Model (6.1) (left), whereτ=

1, whenR0=1.78>1 . . . 149 Figure 6.7: Time response of solutions for Model (6.2) (right) and cor-

responding deterministic Model (6.1) (left), whenτ =0 . . . 149

List of Abbreviations

N(0,1) A normally distributed random variable with zero mean and unit variance a.s. Almost Surely

R0 Basic reproduction number

R₀^s Basic reproductive number for stochastic models DDEs Delay Differential Equations

k.k Euclidean norm: kxk= q

x²₁+· · ·+x²_nforx∈Rⁿ Rⁿ Euclidean space of n-dimentions

E(X) Expected value ofX

ODEs Ordinary Differential Equations P(.) Probability measure

ℜ Real part

Z Set of all integers C Set of complex numbers

C(A,B) Set of continuous functions mapping A to B R+ Set of nonnegative real numbers

N Set of positive integers

C Short form forC([−τ,0],Rⁿ), whereτ>0 is a constant SDDEs Stochastic Delay Differential Equations

T₀^d Threshold parameter for deterministic models T₀^s Threshold parameter for stochastic models W(t) Winner process

B σ−algebra of Borel sets inRⁿ

Chapter 1: Delay Differential Equations with Population Problems

1.1 Introduction

In this chapter, some preliminaries about deterministic Delay Differential Equa- tions (DDEs) and Stochastic Delay Differential Equations (SDDEs) are introduced. Sec- tion 2 briefly discusses the existence and uniqueness of the solutions of DDEs. Section 3 provides some concepts about the stability of DDEs. Sections 4 and 5 provide some background about the randomness, environmental noise and the existence of the solutions of SDDEs. Section 6 provides some main concepts about the stability of SDDEs. The last Section introduces the main objectives and significance of the study.

DDEs are a class of differential equations that have received a considerable at- tention and been shown to model many real life problems, traditionally formulated by systems of Ordinary Differential Equations (ODEs), more naturally and more accurately.

Such class of DDEs are widely used for analysis and predictions of systems with memory such as population dynamics, epidemiology, immunology, physiology and neural net- works [5, 20, 74, 107, 114]. In ODEs, the unknown function and its derivatives are evaluated at the same time instant. However, in a DDE the evolution of the system at a certain time instant, depends on the state of the system at an earlier time. The delay can be related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on; See [111, 112].

In ecological systems, the individuals of the prey and predator species usually pass through various life stages during their entire life span and the involved morphology differs from one stage to another. Construction of delay differential equation models is a well known modelling strategy to take care of the stage specific activities which are responsible for significant change in the dynamics of interacting populations. In various existing literature, the biological processes like incubation, gestation, maturation, reaction

time, etc., are taken care of by introducing relevant time-delay parameters to the models for predator-prey and other types of interacting populations. Incorporating time-lags (or time-delays) in biological models makes the systems much more realistic, as it can desta- bilize the equilibrium points and give rise to a stable limit cycle, causing oscillations to grow, and enriching the dynamics of the model. Time-delays have been considered and extremely studied by many authors in predator-prey models and biological systems; See [15, 16, 117].

Most of the studies in ecology utilize deterministic models, which of course sup- ported the researchers with useful results for protecting species. In reality, natural phe- nomena counter an environmental noise and usually do not follow strictly deterministic laws but oscillate randomly about some average values, so that the population density never attains a fixed value with the advancement of time [43, 118]. Ecological systems are often subject to environmental noise, which is important factor in ecosystems, to sup- press a potential population explosion [120].

A key question in population biology is understanding the conditions under which populations coexist or go extinct. Extinction is one of the most important terms in pop- ulation dynamics. A species is said to be extinct when the last existing member dies.

Therefore, extinction becomes a certainty when there are no surviving individuals that can reproduce and create a new generation. In ecology, extinction is often used infor- mally to refer to local extinction, in which a species ceases to exist in the chosen area of study, but may still exist elsewhere. There are a variety of causes that can contribute directly or indirectly to the extinction of species or group of species, such as lack of food and space or toxic pollution of the entire population habitat, competition for food to bet- ter adapted competitors, predation, etc. [84]. Some examples in modelling population dynamics can be referred to [64, 84, 92, 100, 148].

In fact, stochastic perturbation factors, such as precipitation, absolute humid- ity, and temperature, have a significant impact on the infection force of all types of virus diseases to humans. Taking this into consideration enables a lot of authors to in-

troduce randomness into deterministic model of biological systems to reveal the effect of environmental variability, whether it is a random noise in the system of differential equations or environmental fluctuations in parameters. Moreover, stochastic epidemic models give an extra degree of realism in comparison with their deterministic models [26, 79, 85, 134, 142, 143].

In the next Section, the existence and uniqueness of the solutions of DDEs are discussed.

1.2 Existence and Uniqueness of Solutions for DDEs

Consider the Initial Value Problem (IVP) for the system of DDEs, with multiple discrete time-delays, of the following form

dy(t)

dt =f(t,y(t),y(t−τ₁), . . . ,y(t−τ_m)), t≥t₀, y(t) =φ(t) t≤t₀,

(1.1)

where y(t)∈Rⁿ, f is a nonlinear smooth function, with respect to all of its arguments, depending on delays τ_i>0, i=1, . . . ,m.Time-delayτ_i could be a constant, or variable in time τ_i(t) (i =1, . . . ,n) , or even state-dependent τ_i = τ_i(t,y(t)). If the right hand side of (1.1) is a function ofy⁰(t), then it be called Neutral Delay Differential Equations (NDDEs). The function φ(t) is defined in [ν,t₀], where ν = min

1≤i≤n

n

mint≥t₀(t−τ_i)o

. For

simplicity, consider DDEs of the form dy(t)

dt =f(t,y(t),y(t−τ(t))), t≥t₀, y(t) =φ(t) t≤t₀.

(1.2)

In general, initial discontinuityy⁰(t₀)⁺=f(t₀,y(t₀),φ(t₀−τ))may differ from the value φ⁰(t₀)⁻; and its propagation from initial pointt₀ along the integration interval and gives rise to subsequent discontinuity points where the solution is smoothed out more and more.

On the other hand, it is well-known that every step by step numerical method for the initial

value problem attains its own accuracy order provided that the solution is sufficiently smooth at each step interval[t_n,t_n+1]; More details are discussed in Appendix A.

Now, some essential results for the DDEs (1.2) will be introduced (see [17, 39]).

Theorem 1.2.1. (Local existence) Consider the equation

dy(t)

dt =f(t,y(t),y(t−τ(t))) t₀≤t<t_b y(t₀) =y₀.

(1.3)

Assume that the functionf(t,u,v)is continuous with respect to t on A⊆[t₀,t_b)×Rⁿ×Rⁿ and Lipschitz continuous with respect to u and v. Moreover, assume that the delay function τ(t)≥0is continuous in[t₀,t_b),τ(t₀) =0and, for someξ>0, t−τ(t)>t₀in the interval (t₀,t₀+ξ]. Then the Equation (1.3) has a unique solution in [t₀,t₀+δ)for some δ >0 and this solution continuously depends on the initial data.

To show the global existence theorem, under the same assumptions of Theorem 1.2.1, the solution can be carried on until a maximal solution defined in[t₀,l), witht₀<

l≤t_b.

Theorem 1.2.2. (Global existence) Under the assumptions of Theorem 1.2.1, if the unique maximal solution of (1.3) is bounded, then it exists on the entire interval[t₀,t_b).

Therefore, the following lemma is illustrated to define a bound for the solution.

Lemma 1.2.3. Under the assumptions of Theorem 1.2.1, assume that the functionf(t,u,v) satisfies the condition

kf(t,u,v)k ≤M₁(t) +M₂(t)(kuk+kvk)

in[t₀,t_b)×Rⁿ×Rⁿ, where M₁(t)and M₂(t)are continuous positive functions on[t₀,t_b).

Hence, the solution of (1.3) exists and is unique on the entire interval[t₀,t_b).

Driver [37] proved the above results with multiple delays. Particularly, the global existence, uniqueness and continuous dependence on the initial data for the solution of the linear DDEs

y⁰(t) =A₀y(t) +

r i=1

∑

A_i(t)y(t−τi(t)), t≥t₀ y(t) =φ(t), t≤t₀,

for any continuous functions A_i(t), i=0, . . . ,r, and φ(t), and for any set of continuous delaysτ_i(t)≥0.

Most DDEs don’t have analytic solutions, so it is generally essential to resort to numerical methods (See Appendix A). For linear DDEs with constant delay, considering solutions of exponential form; see [35]. Now, consider a scalar linear DDE of the form

dy

dt =µ1y(t−τ) +µ0y(t), t≥0, y(t) =φ(t), t∈[−τ,0].

(1.4)

Let the solutiony(t) =Ce^λt, whereC is constant; then substituting it into Equation (1.4) gives

Cλe^λt=µ₀Ce^λt+µ₁Ce^λ(t−τ), (1.5)

which can be simplify to

(λ−µ₀)e^{λ τ}−µ₁=0, (1.6)

Equation (1.6) is the characteristic equation and the root λi of (1.6) gives a solution to Equation (1.4) in the form ofCe^λit. The following Theorem [18] illustrates the general solution of (1.6).

Theorem 1.2.4. Assume that φ(t) is C[t₀−τ,t₀], and let {λ_i} be a sequence zeros of

Equation (1.6) arranged in order of decreasing real parts (or of increasing imaginary parts or absolute values). Then

y(t) =

∞

∑

i=1

p_i(t)e^λit, t≥t₀, (quasi-polynomial), (1.7)

is the solution of Equation (1.4), where p_i(t) is a polynomial of degree less than the multiplicity of the root{λ_i}.

The above approach, is not a practical method for solving DDEs. However, it provides useful information about solutions to DDEs. The following example affords the basic theory for the simplest method for solving DDEs, which is the method of steps [17].

Example 1.2.1. Assume a special case of (1.4) when µ0=0. Therefore, Equation (1.4) becomes

dy

dt =µ1y(t−τ), t≥0, y(t) =φ(t), t∈[−τ,0].

(1.8)

Letµ₁=−1,τ =1 andφ(t) =1+t; In the interval [0,1], one obtains

y₁(t) =y(0)− Z t

0

sds=1−t² 2,

in the interval [1,2], one gets

y₂(t) =y(1)− Z _t

1

(1−(s−1)²

2 )ds= (t−1)³

6 −t+3 2,

similarly, in the interval [2,3], the solution is

y₃(t) =−1 3−

Z _t

2

(t−2)³

6 −t+5 2

ds

= 8

3−(t−2)⁴ 24 +t²

2 −5 2t.

Now, one can check for discontinuity in the solutions. Att =0, φ(0) =y₁(0) =1, but φ⁰(0) =16=y⁰₁(0) =0. Thus, there is a discontinuity aty⁰(0). Att=1,y₁(1) =y₂(1) = ¹₂, y⁰₁(1) =y⁰₂(1) =−1, buty⁰⁰₁(1) =−16=y⁰⁰₂(1) =0. Thus, there is a discontinuity aty⁰⁰(1).

Similarly, att =2, there is a discontinuity aty⁰⁰⁰(3).

Applying Laplace transformation [18] to (1.8) also provides some useful facts about the solutions but do not mostly acquire explicit solutions. The Laplace transform L(y)ofy(t)is denoted byY(s), where¯ Y(s) =¯

Z _∞

0

y(t)e^−stdt. By taking Laplace transform ofy⁰(t) =−y(t−1), one obtains

Z _∞

0

y⁰(t)e^−stdt = Z _∞

0

−y(t−1)e^−stdt sY(s)¯ −1=−

Z _∞

−1y(w)e^−s(w+1)dw

=−e^−s Z _∞

0

y(w)e^−swdw−e^−s Z 0

−1

y(w)e^−swdw

=−e^−sY(s)¯ −e^−s Z ₀

−1(w+1)e^−swdw (s+e^−s)Y(s) =¯ 1−e^−s[−1

s²−1 s +e^s

s²]⇒Y(s) =¯ 1+^e−s_s +^e−s−1

s²

s+e^−s . Thus,y(t)can be found by the following inverse transform

y(t) =L⁻¹

h1+^e−s_s +^e−s−1

s²

s+e^−s i

.

Next, some stability criteria of DDEs are discussed.

1.3 Stability of Equilibria and Lyapunov Functions for DDEs

Definition 1.3.1. [12] The solution y(t) of (1.2) is stable if, given ε >0, there exists

∆≡∆(ε)such that sup_{t∈[t0−τ,t0]}|u(t)−φ(t)| ≤∆andu(t)is also a solution of (1.2), then δy(t):=u(t)−y(t)is uniformly bounded fort≥t₀and sup_t≥t0|δy(t)| ≤ε.The solution y(t)is asymptotically stable if it is stable and|δy(t)| →0 ast →∞for all∆sufficiently small.

There are two standard approaches for stability theory [72, 73], the first is stability in variation (first approximation), and the second is Lyapunov theory. First, stability in variation approach are introduced, which is based on local linearization of the DDE [89].

Consider a system of DDEs with multiple constant delays

dy

dt =f(y(t),y(t−τ1), . . . ,y(t−τm)), (1.9) wherey(t)∈Rⁿ, f:R^n(m+1) →Rⁿ is a nonlinear smooth function depending on delays τ_i>0, i=1, . . . ,m. An equilibrium, y^∗(t)≡y^∗, of (1.9) is a solution of the nonlinear algebraic system f(y^∗,y^∗, . . . ,y^∗), which is solved by Newton iteration starting from an initial guessy^∗. The local asymptotic stability ofy^∗is found out through the linearization of (1.9) aroundy^∗, i.e. through the following variation equation

du

dt =A₀u(t) +

m

∑

i=1

A_iu(t−τ_i), (1.10)

such thatA_istands for the partial derivatives offwith respect to theith variable, i.e. A_i:=

∂f

∂yⁱ

(y^∗,y^∗,...,y^∗),i=0,1, . . . ,m.The variational Equation (1.10) leads to the characteristic equation,

det

λI−A₀−

m

∑

i=1

A_ie^{−λ τi}

=0, (1.11)

whereIis the identity matrix andλ ∈C. The local asymptotic stability of the equilibrium y^∗ is determined by the real parts of the characteristic rootsλ, therefore, the solution is stable ifℜ(λ)is negative for allλ and it is unstable withℜ(λ)is positive. The number of roots for Equation (1.11) could be countably infinite. However, the number of roots in any right half-planeℜ(λ)>η,η∈ℜ, is finite such thatℜ(λ_j)→ −∞as j→∞.

Generally, a bifurcation (or threshold point) occurs when a real characteristic root passes through zero and a Hopf bifurcation occurs when a pair of complex conjugate char- acteristic roots passes the imaginary axis; While a transcritical bifurcation occurs when two branches of equilibrium solutions intersect. A periodic solution y^∗(t) is a solution that restates itself after a finite period T i.e., y^∗(t+T) =y^∗(t) for allt >0. The local asymptotic stability of the periodic solution is determined by the time integration opera- tor; which integrates the variational Equation (1.10) aroundy^∗(t)from timet=0 over the period. This operator is also called the monodromy operator and its eigenvalues (which are independent of the starting pointt=0) are called Floquet Multipliers. Additionally, if T ≥τ then the oprator is compact. The periodic solution is stable if all multipliers (except the trivial one) have modulus smaller than one and it is unstable if there exists a multipliers with modulus larger than one [40].

Example 1.3.1. To study the qualitative behaviour of the linear DDE, recall Equation (1.4)

dy

dt =µ₁y(t−τ) +µ₀y(t), t ≥0. (1.12)

The aim is to investigate the stability around the equilibrium solutiony=0. Consider the exponential solutiony(t) =Ce^λt, whereC is constant and the eigenvalueλ are solutions of the characteristic equation

λ−µ₀−µ₁e^{−λ τ}=0, (1.13)

which is a transcendental equation. Therefore,y=0 is asymptotically stable if all eigen- values of (1.13) have negative real parts. Letλ =ζ₀+iζ₁. Substituting in the character- istic Equation (1.13) then separating the real and the imaginary parts, one obtains

ζ0−µ0−µ1e^−ζ0τcosζ1τ =0, ζ1+µ1e^−ζ0τsinζ1τ=0. (1.14)

Notice that whenτ=0, the eigenvalue of the characteristic Equation (1.13) has a negative real number if λ =µ₀+µ₁<0. For fixed τ >0 the boundaries of the domains of the (µ₀,µ₁)−plane are formed by the lineµ₁=−µ₀and the parametric curveµ₀=ζ₁cotζ₁τ, µ₁=−ζ₁/sinζ₁τ whereζ₁∈R, for which ζ₀=0. Therefore, the stability condition is µ₀≤ −|µ₁|, that is independent ofτ>0; see Figure 1.1, which shows the stability region of (1.12) whenτ=1. For complex-valuedµ₀,µ₁, the solution are stable if|µ₁|<−ℜ(µ₀).

Restricting conditions onτsuch thatℜ(λ)changes from negative to positive. By the continuity, ifλ changes fromµ₀+µ₁to a certain value such thatℜ(λ) =ζ₀>0 asτ increases, there must be some threshold value ofτ, sayτ^∗, at whichℜλ(τ^∗) =ζ₀(τ^∗) =0;

In this case the characteristic Equation (1.13) must have a pair of purely imaginary roots

±iζ₁^∗,ζ₁^∗=ζ₁^∗(τ^∗). Therefore, having−µ₀−µ₁cosζ₁^∗τ=0, which implies

τ_k=cos⁻¹(−µ₀/µ₁) q

µ₁²−µ₀²

+ 2πk

q

µ₁²−µ₀²

,k∈Z. (1.15)

Noting that ζ₁^∗= q

µ₁²−µ₀² >0; Thus, when τ =τ^∗ =minτ_k, Equation (1.13) has a pair of purely imaginary roots. When 0<τ <τ^∗, all roots of (1.13) have negative real parts then the equilibriumy=0 is asymptotically stable. ifτ >τ^∗, theny=0 is unsta- ble. Assumeλ(τ) =ζ₀(τ) +iζ₁(τ)the root of Equation (1.13) satisfyingζ₀(τ_k) =0 and

ζ1(τ_k) =ζ₁^∗,k=0,1,2, . . .. In which the transversality condition as following

dℜ(λ) dτ

τ=τ^∗ =dζ₀ dτ

τ=τ^∗ = (ζ₁^∗)²>0, j=0,1,2, . . . .

A Hopf bifurcation occurs aty=0 with a period given byT = √^2π

µ₁²−µ₀², Figure 1.2 shows a stable solution for Equation (1.12) whenτ=0.8<τ^∗; Periodic solution where a Hopf bifurcation occurs asτ=τ^∗=1.209; the solution of (1.12) becomes unstable whenτ = 1.28>τ^∗. Now, one may discuss the stability of (1.12) when µ₀=0 i.e. pure DDE,

µ0

-5 -4 -3 -2 -1 0 1

µ 1

-5 -4 -3 -2 -1 0 1 2 3 4 5

(1/τ,-1/τ) µ1=µ

0

µ1=-µ

0

Stability Region

Figure 1.1: Stability region for scalar DDEy⁰(t) =µ₁y(t−τ) +µ₀y(t). Red line gives the real root crossingµ₁=−µ₀, while the blue line gives the imaginary root crossing; the dash line represents the equationµ1=µ0

Recall the Equation (1.8)

dy

dt =µ1y(t−τ), (1.16)

Therefore the characteristic Equation of (1.16) is

λ−µ₁e^{−λ τ} =0, (1.17)

Time t

0 5 10 15 20 25 30 35 40 45 50

y(t)

-3 -2 -1 0 1 2 3

τ=0.8 τ=1.209 τ=1.28

Figure 1.2: Numerical simulations for DDEy⁰(t) =µ1y(t−τ) +µ0y(t). Withµ1=−2 andµ0=

−1, where the initial functionφ(t) =1+t, whenτ =0.8 the steady statey=0 is stable; When τ =τ^∗=1.209 a Hopf bifurcation occurs; The solution of (1.12) becomes unstable when τ = 1.28>τ^∗

First, suppose thatλ is real, forµ₁>0 the equilibriumy=0 is unstable; Forµ₁<0, one can plotz=λ andz=µ₁e^{λ τ}, whenτ =1; there are three cases; single intersection when µ1=µ₁^∗=−e⁻¹ whenλ =−1, for µ1∈[µ₁^∗,0]there are two real negative eigenvalues, forµ₁<µ₁^∗ there are no real eigenvalues; See Figure 1.3. Assume thatλ =ω₀+iω₁ is complex, in the same manner for studying the stability for linear case one may have,

τj= π

2ω₁^∗+kπ

ω₁^∗,j∈Z, (1.18)

noting that ω₁^∗=−µ₁ >0; Thus, whenτ =τ⁰=minτj, Equation (1.17) has a pair of purely imaginary roots. When 0<τ<τ⁰, all roots of (1.17) have negative real parts then the equilibriumy=0 is asymptotically stable; ifτ>τ⁰, theny=0 is unstable. Assume λ(τ) =ω0(τ) +iω₁(τ)the root of Equation (1.17) satisfyingω0(τ_j) =0 andω1(τ_j) =ω₁^∗,

λ

-2 -1.5 -1 -0.5 0 0.5 1

z

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5

Figure 1.3: Number of real eigenvalues for the characteristic equation of DDEy⁰(t) =µ1y(t−τ).

Single intersection whenµ₁=µ₁^∗=−e⁻¹whenλ=−1, forµ₁∈[µ₁^∗,0]there are two real negative eigenvalues, forµ1<µ₁^∗there are no real eigenvalues

j=0,1,2, . . .. The transversality condition

dℜ(λ) dτ

τ=τ⁰= dω₀ dτ

τ=τ⁰= (ω₁^∗)²>0, j=0,1,2, . . . .

Whenτ= _2ω^π∗

1, a Hopf bifurcation occurs aty=0.

Noting from Figure 1.2 that the large time delay can induce instability and cause the solution to fluctuate when the time delay is larger than a critical value, the time delay can induce a stable limit cycle generated through the Hopf bifurcation and larger time delay can increase the amplitude of the oscillating orbits of the solution.

1.3.1 Lyapunov theory approach

Now, the author introduce the Lyapunov theory approach; by considering a more general type for DDE (1.9) with one delay. Thus, a functional delay differential system is

given by

˙

y(t) =f(t,y_t), t≥t₀,

y_t(θ) =y(t+θ), −τ≤θ ≤0, f(t,0)≡0, y_t0 =φ.

(1.19)

LetCn:[−τ,0]→Rⁿbe the set of continuous functions, whereτ>0 is fixed. Supposet∈ Randy:[t−τ,t]→Rⁿis continuous. Definey_t∈Cnbyy_t(θ) =y(t+θ)forθ ∈[−τ,0], where φ ∈Cn, such that one may consider the existence and uniqueness of solutions, without loss of generality, the solution y_t =0 is an equilibrium. In this approach, the idea is to consider a classical positive definite Lyapunov functionV(t,y(t)), such that its derivative with respect to time along the trajectories of System (1.19) is negative definite.

This concept is formalized in the following theorems.

Theorem 1.3.1. [72] Let u₁, u₂and u₃:R⁺→R⁺be nondecreasing functions such that u₁(θ) and u₂(θ) are strictly positive for all θ >0. Assume that the vector field f of (1.19) is bounded for bounded values of its arguments. If there exists a continuous and differentiable function V :R×Rⁿ→R⁺such that:

i) u₁(kφ(0)k)≤V(t,φ)≤u₂(kφk),

ii) V˙(t,φ)≤ −u₃(kφ(0)k)for all trajectories of (1.19) satisfying

V(t+θ,φ(t+θ))≤V(t,φ(t)), θ ∈[−τ,0], (1.20)

then the solutiony_t =0is uniformly stable for (1.19).

Additionally, ifu₃(θ)>0 and there exists a strictly increasing functionu₄:R+→ R+such thatu₂(θ)>θ as long withi)andii), verifying that

V(t+θ,y(t+θ))≤u₄(V(t,y(t))), θ ∈[−τ,0], (1.21)