# Cholera dynamics on community networks (13rit168)

Arriving in Banff, Alberta Sunday, August 4 and departing Sunday August 11, 2013

## Organizers

Zhisheng Shuai (University of Central Florida)

Joseph Tien (Ohio State University)

Pauline Van den Driessche (University of Victoria)

## Objectives

Background and previous work.

The study of population dynamics on community net- works has a long history in ecology and epidemiology, for example in the context of metapopulations [10, 12, 14, 17, 27]. Epidemiological studies on community networks have examined the ability of disease to invade [11, 15, 28], steady state distributions [2], phase relationships between oscillations in different locales [8, 11, 19, 27], and disease ‘fade-out’ and extinction [8, 9]. Developments in network theory [24, 37] and the explosion of data on empirical network structure in the last fifteen years has opened up new avenues of research in mathematical epidemiology. A central mathematical question is to understand how network structure and community (“patch”) characteristics combine to affect disease dynamics. Recently, we have started to make progress in answering this question for two specific aspects of cholera dynamics: invasibility, and global stability.

The ability of a disease to invade a susceptible population is determined by the basic reproduction number ${cal {R}}_0$, a fundamental quantity in mathematical epidemiology. The basic reproduction number for a community network can be defined as the dominant eigenvalue of a certain matrix, the "next generation" matrix [36]. Analytical expressions for ${cal {R}}_0$ for networks are generally difficult to derive. Recently we have preliminary results on computing ${cal {R}}_0$ for network cholera models as a function of the ratio $varepsilon$ of two time scales: the rate of pathogen decay in the environment, compared with the rate of movement between nodes of the network. In the limit of fast water movement, ${cal {R}}_0$ can be computed analytically as a weighted average of the ``patch risk" for the different patches, times an average pathogen lifetime in the environment. The weights in both averages correspond to the rooted spanning trees of the network. Away from this limit, ${cal {R}}_0$ can be expressed as a Laurent series in $varepsilon$. The next order correction to ${cal {R}}_0$ involves a generalization of the group inverse for the graph Laplacian matrix, reflects clustering of disease ``hot spots" in the network, but is not yet fully understood. These results are encouragingly simple, easy to interpret biologically, and touch upon several different areas in mathematics, including graph theory, Markov processes, and matrix theory. When ${cal {R}}_0>1$, the disease is able to invade. The resulting dynamics depend upon the disease incidence functions and distribution for infection-derived immunity. Under a broad set of incidence functions, and for exponentially distributed immune durations, the asymptotic behavior of the system is completely determined by ${cal {R}}_0$: for ${cal {R}}_0>1$, the system converges to a unique, globally stable endemic equilibrium, and for ${cal {R}}_0 leq 1$, the system converges to the disease-free equilibrium. To establish global stability of the endemic equilibrium, a global Lyapunov function for the whole community network model can be constructed as a weighted sum of Lyapunov functions of individual community models, with the weights in the sum also determined by the rooted spanning trees of the network.

Using the above results as a starting point, we propose this Banff Research Team to examine the following questions that relate to and extend our preliminary results: 1) Can we obtain a better understanding of the rooted spanning trees and generalized inverse of the graph Laplacian matrix for certain network motifs that are likely to arise in practice? 2) The results described in the previous section correspond to infected individuals being too sick to move. When this assumption is relaxed, two different networks arise, one corresponding to the movement of water, and the other to the movement of infected people. How do these two networks combine to affect ${cal {R}}_0$? 3) What is the relationship between the network structure and the final outbreak size? The classical result relating outbreak size and ${cal {R}}_0$ in single patch models (revisited in [20]) no longer holds for community network models. We expect the spanning trees of the network to once again play a role here, but the result in this setting has not yet been generally established. 4) Oscillations can occur in non-spatial cholera models in the absence of exogenous forcing, for example through ``cooperative" nonlinear incidence functions [7] or delayed loss of immunity [29]. Numerical simulations indicate that the dynamics can be quite complex, and the dependence on the immune distribution is not understood. As a first step towards understanding oscillations on networks, can we analyze the bifurcations underlying periodic orbits in the single patch setting for certain types of immune distributions, such as delta functions or gamma distributions? 5) What are the resulting network dynamics when the patches can support these types of oscillations in isolation? We might start, for example, by examining the phase relationship between oscillations at the different nodes, as a function of the network structure.

Team composition. The areas of expertise of our team members complement one an- other for addressing the research objectives described above. Prof. van den Driessche is an expert in matrix analysis and has extensive experience in mathematical epidemiology. Prof. Shuai brings expertise on Lyapunov functions, global stability analysis, and graph structure. Prof. Tien brings expertise on cholera dynamics, which has been the focus of his research for the past five years. Cholera in Haiti is of particular interest [35], for which we have case data coming from the Ministry of Health at the Department level, together with finer resolution data from Hoˆpital Albert Schweitzer in the Artibonite Department, as well as from internally displaced person camps in the Port-au-Prince region. We have a record of successfully collaborating together on cholera dynamics previously [29]. Indeed, the members of this proposed Banff Research Team first began collaborating following discussions at the BIRS Workshop on “Modelling and analysis of options for controlling persistent infectious diseases”, from February 27 to March 4, 2011.

The study of population dynamics on community net- works has a long history in ecology and epidemiology, for example in the context of metapopulations [10, 12, 14, 17, 27]. Epidemiological studies on community networks have examined the ability of disease to invade [11, 15, 28], steady state distributions [2], phase relationships between oscillations in different locales [8, 11, 19, 27], and disease ‘fade-out’ and extinction [8, 9]. Developments in network theory [24, 37] and the explosion of data on empirical network structure in the last fifteen years has opened up new avenues of research in mathematical epidemiology. A central mathematical question is to understand how network structure and community (“patch”) characteristics combine to affect disease dynamics. Recently, we have started to make progress in answering this question for two specific aspects of cholera dynamics: invasibility, and global stability.

The ability of a disease to invade a susceptible population is determined by the basic reproduction number ${cal {R}}_0$, a fundamental quantity in mathematical epidemiology. The basic reproduction number for a community network can be defined as the dominant eigenvalue of a certain matrix, the "next generation" matrix [36]. Analytical expressions for ${cal {R}}_0$ for networks are generally difficult to derive. Recently we have preliminary results on computing ${cal {R}}_0$ for network cholera models as a function of the ratio $varepsilon$ of two time scales: the rate of pathogen decay in the environment, compared with the rate of movement between nodes of the network. In the limit of fast water movement, ${cal {R}}_0$ can be computed analytically as a weighted average of the ``patch risk" for the different patches, times an average pathogen lifetime in the environment. The weights in both averages correspond to the rooted spanning trees of the network. Away from this limit, ${cal {R}}_0$ can be expressed as a Laurent series in $varepsilon$. The next order correction to ${cal {R}}_0$ involves a generalization of the group inverse for the graph Laplacian matrix, reflects clustering of disease ``hot spots" in the network, but is not yet fully understood. These results are encouragingly simple, easy to interpret biologically, and touch upon several different areas in mathematics, including graph theory, Markov processes, and matrix theory. When ${cal {R}}_0>1$, the disease is able to invade. The resulting dynamics depend upon the disease incidence functions and distribution for infection-derived immunity. Under a broad set of incidence functions, and for exponentially distributed immune durations, the asymptotic behavior of the system is completely determined by ${cal {R}}_0$: for ${cal {R}}_0>1$, the system converges to a unique, globally stable endemic equilibrium, and for ${cal {R}}_0 leq 1$, the system converges to the disease-free equilibrium. To establish global stability of the endemic equilibrium, a global Lyapunov function for the whole community network model can be constructed as a weighted sum of Lyapunov functions of individual community models, with the weights in the sum also determined by the rooted spanning trees of the network.

Using the above results as a starting point, we propose this Banff Research Team to examine the following questions that relate to and extend our preliminary results: 1) Can we obtain a better understanding of the rooted spanning trees and generalized inverse of the graph Laplacian matrix for certain network motifs that are likely to arise in practice? 2) The results described in the previous section correspond to infected individuals being too sick to move. When this assumption is relaxed, two different networks arise, one corresponding to the movement of water, and the other to the movement of infected people. How do these two networks combine to affect ${cal {R}}_0$? 3) What is the relationship between the network structure and the final outbreak size? The classical result relating outbreak size and ${cal {R}}_0$ in single patch models (revisited in [20]) no longer holds for community network models. We expect the spanning trees of the network to once again play a role here, but the result in this setting has not yet been generally established. 4) Oscillations can occur in non-spatial cholera models in the absence of exogenous forcing, for example through ``cooperative" nonlinear incidence functions [7] or delayed loss of immunity [29]. Numerical simulations indicate that the dynamics can be quite complex, and the dependence on the immune distribution is not understood. As a first step towards understanding oscillations on networks, can we analyze the bifurcations underlying periodic orbits in the single patch setting for certain types of immune distributions, such as delta functions or gamma distributions? 5) What are the resulting network dynamics when the patches can support these types of oscillations in isolation? We might start, for example, by examining the phase relationship between oscillations at the different nodes, as a function of the network structure.

Team composition. The areas of expertise of our team members complement one an- other for addressing the research objectives described above. Prof. van den Driessche is an expert in matrix analysis and has extensive experience in mathematical epidemiology. Prof. Shuai brings expertise on Lyapunov functions, global stability analysis, and graph structure. Prof. Tien brings expertise on cholera dynamics, which has been the focus of his research for the past five years. Cholera in Haiti is of particular interest [35], for which we have case data coming from the Ministry of Health at the Department level, together with finer resolution data from Hoˆpital Albert Schweitzer in the Artibonite Department, as well as from internally displaced person camps in the Port-au-Prince region. We have a record of successfully collaborating together on cholera dynamics previously [29]. Indeed, the members of this proposed Banff Research Team first began collaborating following discussions at the BIRS Workshop on “Modelling and analysis of options for controlling persistent infectious diseases”, from February 27 to March 4, 2011.