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Epidemic cycling in a multi-strain SIRS epidemic network model.

Zhang XS - Theor Biol Med Model (2016)

Bottom Line: The standard pair approximation is applied to describe the changing numbers of individuals in different infection states and contact pairs.We show that spatial correlation due to contact network and interactions between strains through both ecological interference and immune response interact to generate epidemic cycling.Our results suggest that co-circulation of multiple strains within a contact network provides an explanation for epidemic cycling.

View Article: PubMed Central - PubMed

Affiliation: Department of Statistics, Modelling and Economics, Centre for Infectious Disease Surveillance and Control, Public Health England, 61 Colindale Avenue, London, NW9 5EQ, UK. xu-sheng.zhang@phe.gov.uk.

ABSTRACT

Background: One common observation in infectious diseases caused by multi-strain pathogens is that both the incidence of all infections and the relative fraction of infection with each strain oscillate with time (i.e., so-called Epidemic cycling). Many different mechanisms have been proposed for the pervasive nature of epidemic cycling. Nevertheless, the two facts that people contact each other through a network rather than following a simple mass-action law and most infectious diseases involve multiple strains have not been considered together for their influence on the epidemic cycling.

Methods: To demonstrate how the structural contacts among people influences the dynamical patterns of multi-strain pathogens, we investigate a two strain epidemic model in a network where every individual randomly contacts with a fixed number of other individuals. The standard pair approximation is applied to describe the changing numbers of individuals in different infection states and contact pairs.

Results: We show that spatial correlation due to contact network and interactions between strains through both ecological interference and immune response interact to generate epidemic cycling. Compared to one strain epidemic model, the two strain model presented here can generate epidemic cycling within a much wider parameter range that covers many infectious diseases.

Conclusion: Our results suggest that co-circulation of multiple strains within a contact network provides an explanation for epidemic cycling.

No MeSH data available.


Related in: MedlinePlus

Phase diagram in the (λ,σ) plane for pair approximation model of two strain SIRS model. Other parameters: κ = 4, μ = 0.0005, and ψ = 0.0. The boundary of the one strain model of [16] is included for comparison. Region I is the constant endemics phase where the number of new infections is balanced by the number of recoveries and region II the oscillatory epidemics phase. The critical infection rate that separates disease-free phase and region I is λc ≈ 0.5. Data for the four childhood infectious diseases are from [16]. As the infectious period is used as the unit of time, the birth and death rate of μ = 0.0005 is equivalent to a life span of about 50 years in the model system for infectious diseases which have infectious periods of 1-2 weeks (such as Measles, chickenpox, rubella). It is worth mentioning that our predicted threshold waning rate of immunity in one strain SIRS model with μ = 0.0005 (i.e., the dashed line) are only slightly smaller than that presented in [16] who do not consider the birth rate
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Fig1: Phase diagram in the (λ,σ) plane for pair approximation model of two strain SIRS model. Other parameters: κ = 4, μ = 0.0005, and ψ = 0.0. The boundary of the one strain model of [16] is included for comparison. Region I is the constant endemics phase where the number of new infections is balanced by the number of recoveries and region II the oscillatory epidemics phase. The critical infection rate that separates disease-free phase and region I is λc ≈ 0.5. Data for the four childhood infectious diseases are from [16]. As the infectious period is used as the unit of time, the birth and death rate of μ = 0.0005 is equivalent to a life span of about 50 years in the model system for infectious diseases which have infectious periods of 1-2 weeks (such as Measles, chickenpox, rubella). It is worth mentioning that our predicted threshold waning rate of immunity in one strain SIRS model with μ = 0.0005 (i.e., the dashed line) are only slightly smaller than that presented in [16] who do not consider the birth rate

Mentions: For simplicity, the time scale is set so that γ =1 (i.e. the average infectious duration is taken as the time unit). The numerical calculations show that the final dynamic patterns of epidemic time series are independent of the initial conditions. The phase diagram of the two strain SIRS model is shown in Fig. 1, which is divided into three parts as that for one strain model (see Fig. 1 of [16]). When infection rate λ is less than a critical value λc ≈ (σ + 1)/[(κ − 2) + σ(κ − 1)] from [16], disease cannot survive (disease-free phase). For a given infection rate λ that is larger than λc, only a steady endemic with constant incidence is possible if immunity waning rate σ is larger than a critical value σc (region I: constant incidence phase); otherwise, the sustained oscillatory epidemic emerges (region II: oscillatory incidence phase). Comparison of our model with another two strain model that assumes homogeneous mixing [50] suggests that the spatial correlation due to network structure induces the sustained epidemic cycling as in the one strain model [16]. From Fig. 1, it is obvious that the critical value σc for the two strain model is much higher than that for the one strain model. (Note that resonant amplification of stochastic fluctuations due to a finite population size can only slightly increase the values of σc in the one strain deterministic model [16]). This indicates that under the circumstance of the same epidemic characteristics, the two strain model allows for oscillatory epidemics in infectious diseases that have much shorter immunity periods, and thus expands model parameter range for oscillatory epidemics.Fig. 1


Epidemic cycling in a multi-strain SIRS epidemic network model.

Zhang XS - Theor Biol Med Model (2016)

Phase diagram in the (λ,σ) plane for pair approximation model of two strain SIRS model. Other parameters: κ = 4, μ = 0.0005, and ψ = 0.0. The boundary of the one strain model of [16] is included for comparison. Region I is the constant endemics phase where the number of new infections is balanced by the number of recoveries and region II the oscillatory epidemics phase. The critical infection rate that separates disease-free phase and region I is λc ≈ 0.5. Data for the four childhood infectious diseases are from [16]. As the infectious period is used as the unit of time, the birth and death rate of μ = 0.0005 is equivalent to a life span of about 50 years in the model system for infectious diseases which have infectious periods of 1-2 weeks (such as Measles, chickenpox, rubella). It is worth mentioning that our predicted threshold waning rate of immunity in one strain SIRS model with μ = 0.0005 (i.e., the dashed line) are only slightly smaller than that presented in [16] who do not consider the birth rate
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC4836137&req=5

Fig1: Phase diagram in the (λ,σ) plane for pair approximation model of two strain SIRS model. Other parameters: κ = 4, μ = 0.0005, and ψ = 0.0. The boundary of the one strain model of [16] is included for comparison. Region I is the constant endemics phase where the number of new infections is balanced by the number of recoveries and region II the oscillatory epidemics phase. The critical infection rate that separates disease-free phase and region I is λc ≈ 0.5. Data for the four childhood infectious diseases are from [16]. As the infectious period is used as the unit of time, the birth and death rate of μ = 0.0005 is equivalent to a life span of about 50 years in the model system for infectious diseases which have infectious periods of 1-2 weeks (such as Measles, chickenpox, rubella). It is worth mentioning that our predicted threshold waning rate of immunity in one strain SIRS model with μ = 0.0005 (i.e., the dashed line) are only slightly smaller than that presented in [16] who do not consider the birth rate
Mentions: For simplicity, the time scale is set so that γ =1 (i.e. the average infectious duration is taken as the time unit). The numerical calculations show that the final dynamic patterns of epidemic time series are independent of the initial conditions. The phase diagram of the two strain SIRS model is shown in Fig. 1, which is divided into three parts as that for one strain model (see Fig. 1 of [16]). When infection rate λ is less than a critical value λc ≈ (σ + 1)/[(κ − 2) + σ(κ − 1)] from [16], disease cannot survive (disease-free phase). For a given infection rate λ that is larger than λc, only a steady endemic with constant incidence is possible if immunity waning rate σ is larger than a critical value σc (region I: constant incidence phase); otherwise, the sustained oscillatory epidemic emerges (region II: oscillatory incidence phase). Comparison of our model with another two strain model that assumes homogeneous mixing [50] suggests that the spatial correlation due to network structure induces the sustained epidemic cycling as in the one strain model [16]. From Fig. 1, it is obvious that the critical value σc for the two strain model is much higher than that for the one strain model. (Note that resonant amplification of stochastic fluctuations due to a finite population size can only slightly increase the values of σc in the one strain deterministic model [16]). This indicates that under the circumstance of the same epidemic characteristics, the two strain model allows for oscillatory epidemics in infectious diseases that have much shorter immunity periods, and thus expands model parameter range for oscillatory epidemics.Fig. 1

Bottom Line: The standard pair approximation is applied to describe the changing numbers of individuals in different infection states and contact pairs.We show that spatial correlation due to contact network and interactions between strains through both ecological interference and immune response interact to generate epidemic cycling.Our results suggest that co-circulation of multiple strains within a contact network provides an explanation for epidemic cycling.

View Article: PubMed Central - PubMed

Affiliation: Department of Statistics, Modelling and Economics, Centre for Infectious Disease Surveillance and Control, Public Health England, 61 Colindale Avenue, London, NW9 5EQ, UK. xu-sheng.zhang@phe.gov.uk.

ABSTRACT

Background: One common observation in infectious diseases caused by multi-strain pathogens is that both the incidence of all infections and the relative fraction of infection with each strain oscillate with time (i.e., so-called Epidemic cycling). Many different mechanisms have been proposed for the pervasive nature of epidemic cycling. Nevertheless, the two facts that people contact each other through a network rather than following a simple mass-action law and most infectious diseases involve multiple strains have not been considered together for their influence on the epidemic cycling.

Methods: To demonstrate how the structural contacts among people influences the dynamical patterns of multi-strain pathogens, we investigate a two strain epidemic model in a network where every individual randomly contacts with a fixed number of other individuals. The standard pair approximation is applied to describe the changing numbers of individuals in different infection states and contact pairs.

Results: We show that spatial correlation due to contact network and interactions between strains through both ecological interference and immune response interact to generate epidemic cycling. Compared to one strain epidemic model, the two strain model presented here can generate epidemic cycling within a much wider parameter range that covers many infectious diseases.

Conclusion: Our results suggest that co-circulation of multiple strains within a contact network provides an explanation for epidemic cycling.

No MeSH data available.


Related in: MedlinePlus