Global stability mathematical analysis for virus transmission model with latent age structure

Shanjing Ren; Lingling Li; Shanjing Ren; Lingling Li

doi:10.3934/mbe.2022154

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 4: 3337-3349. doi: 10.3934/mbe.2022154

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Global stability mathematical analysis for virus transmission model with latent age structure

Shanjing Ren ^{1
,
,},
Lingling Li ²

1.
School of Mathematics and Big Data, Guizhou Education University, Guiyang 550018, China
2.
School of Science, Xi'an Polytechnic University, Xi'an 710048, China

Academic Editor: Simon James Fong

Received: 29 November 2021 Revised: 10 January 2022 Accepted: 13 January 2022 Published: 24 January 2022

Background and objective
Mathematical model is a very important method for the control and prevention of disease transmissing. Based on the communication characteristics of diseases, it is necesssery to add fast and slow process into the model of infectious diseases, which more effectively shows the transmission mechanism of infectious diseases.

Methods
This paper proposes an age structure epidemic model with fast and slow progression. We analyze the model's dynamic properties by using the stability theory of differential equation under the assumption of constant population size.

Results
The very important threshold $ R_{0} $ was calculated. If $ R_{0} < 1 $, the disease-free equilibrium is globally asymptotically stable, whereas if $ R_{0} > 1 $, the Lyapunov function is used to show that endemic equilibrium is globally stable. Through more in-depth analysis for basic reproduction number, we obtain the greater the rate of slow progression of an infectious disease, the fewer the threshold results. In addition, we also provided some numerical simulations to prove our result.

Conclusions
Vaccines do not provide lifelong immunity, but can reduce the mortality of those infected. By vaccinating, the rate of patients entering slow progression increases and the threshold is correspondingly reduced. Therefore, vaccination can effectively control the transmission of Coronavirus. The theoretical incidence predicted by mathematical model can provide evidence for prevention and controlling the spread of the epidemic.
- age structure,
- Lyapunov function,
- infection equilibrium,
- global stability,
- prevention and control
Citation: Shanjing Ren, Lingling Li. Global stability mathematical analysis for virus transmission model with latent age structure[J]. Mathematical Biosciences and Engineering, 2022, 19(4): 3337-3349. doi: 10.3934/mbe.2022154

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Abstract

Background and objective

Mathematical model is a very important method for the control and prevention of disease transmissing. Based on the communication characteristics of diseases, it is necesssery to add fast and slow process into the model of infectious diseases, which more effectively shows the transmission mechanism of infectious diseases.

Methods

This paper proposes an age structure epidemic model with fast and slow progression. We analyze the model's dynamic properties by using the stability theory of differential equation under the assumption of constant population size.

Results

The very important threshold $ R_{0} $ was calculated. If $ R_{0} < 1 $, the disease-free equilibrium is globally asymptotically stable, whereas if $ R_{0} > 1 $, the Lyapunov function is used to show that endemic equilibrium is globally stable. Through more in-depth analysis for basic reproduction number, we obtain the greater the rate of slow progression of an infectious disease, the fewer the threshold results. In addition, we also provided some numerical simulations to prove our result.

Conclusions

Vaccines do not provide lifelong immunity, but can reduce the mortality of those infected. By vaccinating, the rate of patients entering slow progression increases and the threshold is correspondingly reduced. Therefore, vaccination can effectively control the transmission of Coronavirus. The theoretical incidence predicted by mathematical model can provide evidence for prevention and controlling the spread of the epidemic.

References

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