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On a two-strain epidemic model involving delay equations.

Mohammed Meziane1, Ali Moussaoui1, Vitaly Volpert2,3

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This study models two interacting virus strains, exploring scenarios with and without cross-immunity. Coexistence occurs when transmission rates are similar; otherwise, one strain dominates.

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disease durationdistributed recovery and death ratesepidemic modeltime delay

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Infectious Disease Dynamics

Background:

  • Understanding viral interactions is crucial for public health.
  • Cross-immunity significantly impacts disease dynamics.
  • Modeling disease spread requires accounting for recovery and death rates.

Purpose of the Study:

  • To develop and analyze an epidemiological model for two interacting virus strains.
  • To investigate the impact of cross-immunity on disease dynamics.
  • To explore scenarios with distributed versus point-wise delay models.

Main Methods:

  • Integro-differential equations to model susceptible, infectious, recovered, and dead compartments.
  • Simplification to ordinary differential equations (ODEs) for uniformly distributed rates.
  • Development of a point-wise delay model for delta-function approximated rates.
  • Analysis of solution positivity and determination of the basic reproduction number.

Main Results:

  • Established solution positivity for distributed delay models.
  • Determined the basic reproduction number and estimated the final epidemic size for the delay model.
  • Numerical simulations showed strain coexistence with similar transmission rates.
  • Demonstrated dominance of one strain when transmission rates differ significantly.

Conclusions:

  • The proposed model provides insights into the complex dynamics of competing viral strains.
  • Cross-immunity and transmission rate differences are key factors in determining strain coexistence or elimination.
  • The model's flexibility allows for analysis under various recovery and death rate distributions.