### Mathematical Biosciences and Engineering

2022, Issue 4: 3337-3349. doi: 10.3934/mbe.2022154
Research article Special Issues

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

• 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.

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

### Related Papers:

• 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.

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沈阳化工大学材料科学与工程学院 沈阳 110142

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