Original articles
Dynamical analysis for the impact of asymptomatic infective and infection delay on disease transmission

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Abstract

The influence of asymptomatic patients on disease transmission has attracted more and more attention, but the mechanism of some factors affecting disease transmission needs to be studied urgently. Considering the self-healing rate of asymptomatic patients, the cure rate of symptomatic patients, the transformation rate from asymptomatic to symptomatic and the infection delay, a type of infectious disease dynamics model SIsIaS with asymptomatic infection and infection delay is established in this paper. It is found that both the infection delay and the difference size between the cure rate and the self-healing rate not only affect the minimum value of the total number of patients in the persistent state of the disease, but also lead to disease extinction to be controlled by the proportion of symptomatic patients in patients. Moreover, the infection delay can lead to local Hopf bifurcation of periodic solutions. By using the normal form and center manifold theory the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are discussed. At last, sensitivity analysis shows that the infection delay can change the correlation of the proportion of symptomatic patients in patients and the transformation rate to the total number of patients.

Introduction

In recent years, various infectious diseases have appeared in the world. These infectious diseases not only affect the development of social economy, but also bring serious impact on the survival and development of human beings. Take Hepatitis B virus, Influenza virus A/H1N1, Dengue virus, Mumps virus and SARS-CoV-2 as an example, these infectious diseases have a common feature, and there are asymptomatic patients in the development process [7], [10], [15], [24], [41]. However, there are still some asymptomatic patients with infectious diseases that have not attracted public attention. Since asymptomatic patients have deviation in their own body cognition, they may spread the disease in the process of contact with susceptibles [38]. At the same time, asymptomatic patients are difficult to be found in the population and have a certain invisibility. Hence, the influence of asymptomatic patients cannot be ignored.

As can be seen from the survey of asymptomatic patients with novel coronavirus, asymptomatic patients are divided into two types: one type of population does not show clinical symptoms from beginning to the end, and the other type of population has no symptoms for the time being, but clinical symptoms will appear later [23]. In other words, the first type of people will not become symptomatic patients, and the second type of people will develop into symptomatic patients. Since these two types of asymptomatic patients are infectious, it is of great practical significance to study their impact on disease transmission.

Many studies have shown that although asymptomatic patients have no symptoms, they have similar viral load as symptomatic patients, and they are also at risk of transmission and belong to patients [40]. The first type of asymptomatic patients of many infectious diseases show no symptoms from the beginning to the end, so they will not receive treatment. They can only rely on their own immune system to fight the disease [9], [13], [29]. Therefore, this paper considers the self-healing rate of asymptomatic patients. The second type of asymptomatic patients will transform into symptomatic patients after a period of time, and will receive treatment at this time [28], [30], so this paper also considers the cure rate of symptomatic patients. For the self-healing rate and the cure rate, since external treatment cannot guarantee better than autoimmunity or whether it will affect autoimmunity. Therefore, this paper considers the situation where the size relationship of the self-healing rate and the cure rate is unknown.

The development process of many diseases shows that the development and change of diseases are not only dependent on the current state, but also related to the past state. This situation is called the phenomenon of delay [33]. Therefore, in the construction of many infectious disease models, time delays are usually introduced to more accurately describe the regularity of disease development [33], [35]. In general, the delay term often affects the stability of the system, and the system will also have bifurcation phenomenon and periodic solutions [39]. In [20], Ma et al. studied the permanence and global asymptotic stability of the SIR epidemic model with time delay. In [37], Xu constructed a predator–prey model with time delay and obtained the Hopf bifurcation. Moreover, permanence and extinction of disease has important implications for studying disease prevalence [17], [27]. In this paper, we intend to introduce a exponential infection delay, and establish a model with delay, and analyze the bifurcation situation and periodic stability at the equilibrium.

Regarding infectious diseases with asymptomatic patients, some scholars have established different mathematical models to reveal the epidemic law of infectious diseases. In [21], Martin Grunnill found that there is asymptomatic infection in dengue fever infection, and a SAIR model was constructed. The results of the study show that if the infectivity level of asymptomatic infection is higher than that of symptomatic infection, dengue fever will develop into an endemic disease. In [18], Liu et al. took mumps disease as an example and considered asymptomatic infection. When asymptomatic individuals are less infectious, the spread of mumps can be effectively suppressed. Otherwise, the more the number of asymptomatic individuals is, the more serious the epidemic is, and the stronger the infectivity of asymptomatic individuals is. In [1], Idris Ahmed et al. constructed an ODEs model with asymptomatic patients to describe the outbreak of COVID-19. The model considers the process of symptom diagnosis and does not consider the mortality of asymptomatic patients. In [3], Berlinda Batista et al. suggest that if the presence of asymptomatic patients is not considered, quarantine on the ship will lead to serious disease transmission. In the above literature, literature [1], [18], [21] consider asymptomatic patients, but does not consider the transformation of asymptomatic patients to symptomatic patients. The literature [3], [19] consider asymptomatic patients and the transformation of asymptomatic patients to symptomatic patients, but does not consider the existence of infection delay, nor does it consider the difference between the cure rate and the self-healing rate.

In summary, the impact of asymptomatic infection on disease transmission has been noted by many researchers, but many models of the transition from asymptomatic to symptomatic have not been involved. At the same time, the difference between the self-healing rate of asymptomatic patients and the cure rate of symptomatic patients is unclear, and none of the models paid attention to the impact of this factor on the dynamics of disease transmission. Hence, considering the differences between symptomatic and asymptomatic patients, the infection delay and the transformation from asymptomatic patients to symptomatic patients, we construct a three-dimensional dynamic model of infectious disease with exponential delay.

The structure of this article is as follows. In Section 2, the infectious disease dynamics model with asymptomatic infection and the infection delay is presented. In Section 3, the basic reproduction number and the existence of equilibria are obtained. In Section 4, based on the size relationship between the cure rate and the self-healing rate, the sufficient conditions for the permanence and extinction of the disease are obtained. In Section 5, the local stability and global stability of disease-free equilibrium are analyzed. In Section 6, the local stability of endemic equilibrium and the properties of Hopf bifurcation are discussed. In Section 7, the sensitivity of each parameter to the number of patients at the endemic equilibrium and the basic reproduction number are analyzed. In Section 8, numerical simulations are performed. Finally, we give the conclusion of this paper.

Section snippets

Model formulation

Firstly, according to whether they have symptoms or not, patients are divided into symptomatic patients (Is) and the asymptomatic patients (Ia). Susceptibles (S) become infected from enough contact with symptomatic or asymptomatic patients. Since there is a group of asymptomatic patients who can be transformed into symptomatic patients, we consider the transformation rate from asymptomatic patients to the symptomatic patients. Since asymptomatic patients do not know their illness and do not go

Existence of equilibria

Firstly, it is easy to get dNdt=ΛμN. Therefore, the invariant region of system (2.1) [2], [34] is D={(S(t),Is(t),Ia(t))R3:0S,Is,Ia,S+Is+IaΛμ}. Then, we can get the disease-free equilibrium of the model (2.1) given by E0=(Λμ,0,0).

Following the definition and computation procedure in Diekmann et al. [5] and Van Den Driessche and Watmough [8], we can rewrite system (2.1) as follows: dY(t)dt=(Yt)V(Y(t)),t0where Yt(θ)=Y(t+θ),θ[τ,0],:(([τ,0],R3))R3, V:(([τ,0],R3))R3, Y(t)=(S(t),Is(t),I

Permanence and extinction of disease

In this section, we discuss the permanence and extinction of the disease under different conditions, since the size relationship between the cure rate of symptomatic patients and the self-healing rate of asymptomatic patients is unknown.

Stability of disease-free equilibrium E0

In this section, we discuss the local stability and the global stability of the disease-free equilibrium of system (2.1) by analyzing the corresponding characteristic equation and constructing a reasonable Lyapunov function, respectively.

Stability of endemic equilibrium E1

In this section, we will study the local asymptotic stability at the endemic equilibrium E1 and the existence of Hopf bifurcation, and further discuss the direction of the Hopf bifurcation and stability of bifurcated periodic solutions arising through this Hopf bifurcation.

Sensitivity analysis

The sensitivity indices can help us get the relative change of state variables when the parameters change. These indices can be positive or negative. The absolute value of the indices indicates the strength of the relationship and the positive and negative properties of the indices indicate the positive and negative correlation. Next, we will use the PRCC method to study the influence of parameters on the solution of the original model, as well as the influence of parameters on the basic

Numerical simulation

In this section, some numerical simulations are performed to verify the existence of equilibria, the local stability and global stability of each equilibrium and the Hopf bifurcation of the endemic equilibrium. The results are presented in Table 5. In the following we will use dde23 in the mathematical software matlab to solve delay differential equations [31], [32].

When the parameters of the model satisfy Number 1 in Table 5, and the initial value is S(0)=40,Is(0)=15,Ia(0)=10, we have R0=0.5398

Discussion

In order to study the effects of asymptomatic infection and the infection delay on the model, we construct a three-dimensional dynamic model of infectious disease with exponential delay, considering the transformation from asymptomatic patients to symptomatic patients. We obtain the disease transmission threshold R0 with asymptomatic infection and the infection delay, and obtain some sufficient conditions for the permanence and extinction of the disease under different relationships (Theorem 4.1

Acknowledgments

This research is supported by the National Natural Science Foundation of China (11401002), the Natural Science Foundation of Anhui Province, PR China (2008085MA02), the Natural Science Foundation for Colleges and Universities in Anhui Province, PR China (KJ2018A0029), the Teaching Research Project of Anhui University, PR China (ZLTS2016065), quality engineering project of colleges and universities in Anhui Province, PR China (2020jyxm0103) and the Science Foundation of Anhui Province

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