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MBW:The Impact of information transmission on epidemic outbreaks

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Review of article[1] with expansion/analysis of model by Inom Mirzaev and Ash Same

Executive Summary

In the original paper[1], a mathematical model for disease transmission is introduced. Unlike most older models, it takes into account the changes in individual responsiveness (taking measures to avoid infection, or seeking treatment early if infected) as information about the disease is made known to the community, either through various media (with level of dissemination increasing with disease prevalence) or passed individual to individual. It also considers the decay in responsiveness as individuals become less cautious over periods of time without any infection or without worsening symptoms.

The model is analyzed (considering disease free and endemic steady states, and their stabilities), and the implications of spreading information about disease prevention/treatment are discussed with respect to the disease outbreak threshold, the dynamics of the disease, and the long term behavior. The effects of changes in individual to individual information transmission vs population-wide dissemination levels are compared.

We extend the model. The formula for individual to individual information distribution is altered. The disease free steady states are found, and their stabilities explored to see how the dynamics are affected. The existence conditions for the endemic steady state are found using numerical methods. The implications of different methods of spreading information are compared with the results from the original paper. In both cases we see how different levels of information dissemination can help to reduce the prevalence of, or even eradicate, the disease.

Context/Biological Phenomenon Under Consideration

Many models of disease transmission do not consider the behavioral changes of individuals as information about the disease is distributed among the community. This can occur through targeted campaigns, the media, or passed from individual to individual. As an individual learns about the dangers of a disease, and the ways in which they can avoid becoming infected (or can be tested/treated for it), there is a good chance that they will take measures that will decrease their chance of becoming infected (or decrease their recovery time).

In the paper being studied, a model was constructed that takes this behavior into account, specifically in the context of sexually transmitted infections (STIs). It was assumed that people are generally aware of the risks associated with STIs, and ways in which they can avoid infection. But information from both the media and other individuals will increase the willingness of people to act, as seen in certain cases in the past. Susceptible individuals can be willing to respond to information and take measures to reduce their chances of infection, or not. Infected individuals can be willing to respond to information and seek treatment early (decreasing the recovery time), or not. Campaigns are aimed at increasing this responsiveness.

Mathematical Model With Variable/Parameter Definitions

In this paper they began with the simple SIRS model. The susceptible and infected states were each divided into two classes, based on the responsiveness of the individuals to information, and whether or not they are taking measures to prevent or treat infection. The heterogeneity in the number of contacts for different individuals was not considered.

The model uses five different states

S_{{nr}}: Susceptible and non-responsive. S_{{r}}: Susceptible and responsive. I_{{nr}}: Infectious and non-responsive. I_{{r}}: Infectious and responsive. T: Being treated.

Transitions

Becoming sick (mass-action)

S_{{nr}}\rightarrow I_{{nr}} based on contact between S_{{nr}} individuals and all infected individuals, \beta _{{nr}}(I_{{nr}}+I_{r}){\frac  {S_{{nr}}}{N}}.

S_{r}\rightarrow I_{r} based on contact between S_{r} individuals and all infected individuals, \beta _{r}(I_{{nr}}+I_{r}){\frac  {S_{r}}{N}}.

\beta _{r}<\beta _{{nr}} since responsive individuals will have less contact and/or be more protected on contact.

Becoming treated (mass-action)

I_{{nr}}\rightarrow T based on the number of I_{{nr}} individuals and rate of seeking treatment, \gamma _{{nr}}I_{{nr}}.

I_{r}\rightarrow T based on the number of I_{r} individuals and rate of seeking treatment, \gamma _{r}I_{r}.

\gamma _{r}>\gamma _{{nr}} since responsive individuals will seek treatment sooner.

Becoming well (mass-action)

T\rightarrow S_{{nr}} based on the number of T individuals, rate of healing and proportion that remain non-responsive following their experience with the disease, prT.

T\rightarrow S_{r} based on the number of T individuals, rate of healing and proportion that become responsive after learning the risks and ways to prevent sickness, (1-p)rT.

For certain STIs some individuals may have no symptoms, resulting in a high p value.

Becoming responsive via information passed from individual to individual (mass-action)

S_{{nr}}\rightarrow S_{r} based on contact between S_{{nr}} individuals and all responsive individuals or individuals in treatment (the individuals that would explain the risks and measures that can be taken), \alpha _{s}{\frac  {S_{{nr}}(S_{r}+I_{r}+T)}{N}}.

I_{{nr}}\rightarrow I_{r} based on contact between I_{{nr}} individuals and all responsive individuals or individuals in treatment, \alpha _{i}{\frac  {I_{{nr}}(S_{r}+I_{r}+T)}{N}}.

Becoming responsive via information disseminated throughout the population

S_{{nr}}\rightarrow S_{r} based on the number of S_{{nr}} individuals and the amount of information sent out. This amount is assumed to increase as the disease becomes more prevalent (based on the number of infected people) and the importance of stopping it increases. It is also assumed to saturate at a maximum value, since there is only so much information that can be distributed, and information that covers the same topic over and over will lose its effectiveness. So, they use \delta _{s}S_{{nr}}{\frac  {(I_{{nr}}+I_{r})^{n}}{K+(I_{{nr}}+I_{r})^{n}}}, with n\geq 1 and K>0.

I_{{nr}}\rightarrow I_{r} based on the number of I_{{nr}} individuals and the amount of information sent out, \delta _{i}I_{{nr}}{\frac  {(I_{{nr}}+I_{r})^{n}}{K+(I_{{nr}}+I_{r})^{n}}}.

Becoming non-responsive as the value of information decays over time (mass-action)

Over time the value of information will decay. Individuals become less willing to be responsive. Susceptible individuals that don’t become infected will become less cautious over time. The same is true of infected individuals whose symptoms don’t get worse.

S_{r}\rightarrow S_{{nr}} based on the number of S_{r} individuals and rate of decay, d_{s}S_{r}.

I_{r}\rightarrow I_{{nr}} based on the number of I_{r} individuals and rate of decay, d_{i}I_{r}.


The system

Figure 1.jpg

Figure 2.jpg

Figure 3.jpg

(S_{{nr}}+S_{r}+I_{{nr}}+I_{r}+T)(t)=N for all t\geq 0, population size N.

Analysis

In the paper, the disease related parameter values are selected based on Chlamydia data. They study the dynamical behavior of the system. Two disease free steady states (one trivial and one non-trivial) are found analytically, along with their stabilities for different parameter values. These are also expressed in terms of basic reproduction number R_{0}. An endemic steady state, along with its existence, uniqueness and stability, is determined numerically. Finally, the global behavior and parameter sensitivities are analyzed.

Disease free steady states and stabilities

The model is non-dimensionalized by N to get a new set of equations. The disease free steady states are found by setting i_{{nr}}, i_{r}, \tau , ds_{{nr}}/dt, and ds_{r}/dt to 0. They are (s_{{nr}},s_{r},i_{{nr}},i_{r},\tau )=(1,0,0,0,0) and, if \alpha _{s}>d_{s},({\frac  {d_{s}}{\alpha _{s}}},{\frac  {\alpha _{s}-d_{s}}{\alpha _{s}}},0,0,0).

The \tau variable is removed (\tau =1-s_{{nr}}-s_{r}-i_{{nr}}-i_{r}) and Jacobians are found for each steady state. Eigenvalues show that (1, 0, 0, 0) is stable iff \alpha _{s}<d_{s} and \beta _{{nr}}<\gamma _{{nr}}. ({\frac  {d_{s}}{\alpha _{s}}},{\frac  {\alpha _{s}-d_{s}}{\alpha _{s}}},0,0) is stable iff \alpha _{s}>d_{s} and detM>0, where

Figure 4.jpg

Endemic steady state

Through numerical analysis (and analytical analysis of a simplified system) it is shown that there is a stable unique endemic equilibrium when both disease free steady states are unstable. As either become stable the endemic equilibrium merges with one or the other. Since both disease free steady states can't be stable at the same time, there is always one, and only one, stable steady state.

Global behaviour

The steady states are shown to be globally stable whenever they are locally stable. This is proven analytically for a simplified system.

The following image, found numerically, shows the three possible cases for the stable global attractor (trivial DFSS, non-trivial DFSS, or endemic) for different values of infection rate and information transmission rate.

Figure 5.jpg

Parameter sensitivity

Individual to individual information transmission

The values \alpha _{s} and \alpha _{i} are involved in all of the inequalities that determine which is the stable steady state. So the rate of information transmission from individual to individual can actually affect which is the stable attractor. Changing these rates can potentially change an endemic steady state to a disease free one, preventing spread of the disease. Increased rates can also decrease the level of infection at the endemic equilibrium.

Population wide information transmission

The values \delta _{s}, \delta _{i} and K (as well as p) do not feature in the inequalities that determine the stable steady state. So the level of information given publicly cannot alter which is the stable attractor. In the case that it is endemic, the best that increased public information can do is decrease the level of infection, not eradicate it completely. The effect of changing these parameters on the level of infection and behavior of the system is examined numerically. Increased population-wide transmission can still make the infection levels very low. Changing the p value has less of an effect.

Modified (New) Model

Our modified model differs from the original one in the choice of function used for the spread of information from individual to individual, coming from those that are aware and responsive (i.e. S_{r},I_{r} and T). We define it as follows

f_{s}(S_{{nr}};S_{r},I_{r},T)=f_{i}(I_{{nr}};S_{r},I_{r},T)=f(X;S_{r},I_{r},T)=X\left({\frac  {S_{r}+I_{r}+T}{N}}\right)^{{\theta }}

One can think of it as \theta responsive individuals having to have contact with one non-responsive individual in order to convert them from non-responsive to responsive. Obviously, for \theta >1 this reduces the rate of contact-based information transmission, and for \theta <1 it increases the transmission rate. We will investigate this new model by finding the disease-free steady states (DFSSs) and conducting stability analysis for \theta =2.

Number of DFSSs for the New Model

DFSSs can be obtained by setting i_{{nr}},i_{r} and \tau to zero and determining the values of s_{{nr}} and s_{r} such that the first two equations are at equilibrium (i.e., ds_{{nr}}/dt=ds_{r}/dt=0). Then we get

s_{0}\left({\frac  {d_{s}}{\alpha _{s}}}-s_{0}^{{\theta -1}}(1-s_{0})\right)=0,

where each of its roots define a DFSS (s_{{nr}},s_{r},i_{{nr}},i_{r},\tau )=(1-s_{0},s_{0},0,0,0). For each \theta there is a trivial DFSS (s_{0}=0) and non-trivial DFSSs found from the roots of F(s_{0},\theta )={\frac  {d_{s}}{\alpha _{s}}}-s_{0}^{{\theta -1}}(1-s_{0})=0. The picture below shows the behavior of this function for various \theta values.

A l.
Figure 1 Three different behaviors of function F(s_{0},\theta ) for d_{s}=0.25\alpha _{s} (a) 0<\theta <1 (b) \theta =1 (c) \theta >1

As we can see in Figure 1, the model has 1 non-trivial DFSS for 0<\theta \leq 1 and 2 non-trivial DFSSs for \theta >1. There must be a bifurcation point at \theta =1.

Stability Analysis of DFSSs for \theta =2

In the case \alpha _{s}<4d_{s} there is only the trivial disease-free steady state, (s_{{nr}},s_{r},i_{{nr}},i_{r},\tau )=(1,0,0,0,0). Provided that \alpha _{s}>4d_{s}, there are also two non-trivial DFSSs given by DFSS_{{non-triv}}=(s_{{nr}},s_{r},i_{{nr}},i_{r},\tau )=(1-s_{0},s_{0},0,0,0), where s_{0} can take 2 values. One possible value is s_{0}={\frac  {1}{2}}(1+{\sqrt  {1-4d_{s}/\alpha _{s}}})>{\frac  {1}{2}}, and the other s_{0}={\frac  {1}{2}}(1-{\sqrt  {1-4d_{s}/\alpha _{s}}})<{\frac  {1}{2}}.
The linear stability analysis can be carried out more easily when the equations are considered as a four variable system with \tau =1-s_{{nr}}-s_{r}-i_{{nr}}-i_{r}. Then the DFSSs can be written in the form (1-s_{0},s_{0},0,0), where s_{0}=0 for the trivial DFSS and s_{0}={\frac  {1}{2}}(1\pm {\sqrt  {1-4d_{s}/\alpha _{s}}}) for the non-trivial DFSS. Then the 4\times 4 Jacobian for each DFSS takes the form Failed to parse (lexing error): J=\left (\begin{array}{cc} L&N\\ 0&M\\ \end{array}•\right )

with Failed to parse (lexing error): L=\left (\begin{array}{cc} \alpha_ss_0(2-3s_0)-pr&d_s-pr\\ -\alpha_ss_0(2-3s_0)+pr-r&-d_s+pr-r\\ \end{array}•\right )
and Failed to parse (lexing error): M=\left (\begin{array}{cc} \beta_{nr}(1-s_0)-\alpha_is_0^2-\gamma_{nr}&\beta_{nr}(1-s_0)+d_i\\ \beta_rs_0+\alpha_is_0^2&\beta_rs_0-\gamma_r-d_i\\ \end{array}•\right )

.

The block diagonal form of the matrix J yields that its eigenvalues are the eigenvalues of L and M. The eigenvalues of L are denoted by \lambda _{1},\lambda _{2} and those of M are denoted by \lambda _{3},\lambda _{4}.

For the trivial DFSS, when s_{0}=0, the eigenvalues are

\lambda _{1}=-r,\quad \lambda _{2}=-d_{s},\quad \lambda _{3}=\beta _{{nr}}-\gamma _{{nr}},\quad \lambda _{4}=-\gamma _{r}-d_{i}.

Therefore, the trivial DFSS is stable if and only if \beta _{{nr}}<\gamma _{{nr}} and \alpha _{s}<4d_{s} .

For the non-trivial DFSSs, when s_{0}={\frac  {1}{2}}(1\pm {\sqrt  {1-4d_{s}/\alpha _{s}}}), the eigenvalues are

\lambda _{1}=-r,\quad \lambda _{2}=-\alpha _{s}s_{0}(3s_{0}-2)-d_{s},\quad \lambda _{{3,4}}={\frac  {1}{2}}\left(Tr(M)\pm {\sqrt  {(Tr(M))^{2}-4det(M)}}\right).
For the stability of each non-trivial DFSS we should have
\lambda _{2}<0\Leftrightarrow s_{0}(1-s_{0})={\frac  {d_{s}}{\alpha _{s}}}>s_{0}(2-3s_{0})\Leftrightarrow s_{0}>{\frac  {1}{2}}.
Thus the second non-trivial DFSS is an unstable equilibrium.

For the stability of the first non-trivial DFSS we also need Re(\lambda _{{3,4}})<0\Leftrightarrow det(M)>0,\quad Tr(M)<0.

Since Tr(M)=-\gamma _{{nr}}-d_{i}-\gamma _{r}+\beta _{{nr}}(1-s_{0})+\beta _{r}s_{0}-a_{i}s_{0}^{2} and det(M)=\gamma _{{nr}}(d_{i}+\gamma _{r})+a_{i}s_{0}^{2}\gamma _{r}-\beta _{{nr}}(d_{i}+a_{i}s_{0}^{2}+\gamma _{r})(1-s_{0})-\beta _{r}(d_{i}+\gamma _{{nr}}+a_{i}s_{0}^{2})s_{0}, one can prove that det(M)>0 implies that Tr(M)<0. Therefore, the first non-trivial DFSS is stable if and only if 4d_{s}<\alpha _{s} and det(M)>0.

We can summarize above results in single proposition.
Proposition There are two different cases for the disease-free steady states of the system of differential equations: a trivial DFSS (1,0, 0, 0, 0) if \alpha _{s}<4d_{s}, or in the case \alpha _{s}>4d_{s}, two extra non-trivial DFSSs (1-s_{0},\ s_{0},\ 0,\ 0,\ 0) with s_{0}={\frac  {1}{2}}(1\pm {\sqrt  {1-4d_{s}/\alpha _{s}}}).

  1. The trivial DFSS is locally stable if and only if \beta _{{nr}}<\gamma _{{nr}} and \alpha _{s}<4d_{s}.
  2. The first non-trivial DFSS is locally stable if and only if 4d_{s}<\alpha _{s} and \gamma _{{nr}}(d_{i}+\gamma _{r})+a_{i}s_{0}^{2}\gamma _{r}-\beta _{{nr}}(d_{i}+a_{i}s_{0}^{2}+\gamma _{r})(1-s_{0})-\beta _{r}(d_{i}+\gamma _{{nr}}+a_{i}s_{0}^{2})s_{0}>0.
  3. The second non-trivial DFSS is unstable for all parameter values.
  4. Whenever the above conditions are not satisfied there is an endemic steady state. This is the same as in the original model.

Comparison and Interpretation

Steady states and stabilities

The original model had either one (\alpha _{s}<d_{s}) or two disease-free steady states. For \alpha _{s}<d_{s} there is only the trivial DFSS, where everybody is non-responsive. As \alpha _{s} increases past d_{s} a non-trivial disease-free steady state forms (which is stable if detM>0). This steady state originally has everybody non-responsive, but as \alpha _{s} increases further the proportion of responsive individuals increases more and more, toward the case where everybody is responsive.

The new model has one (\alpha _{s}<4d_{s}) or three DFSSs. The \alpha _{s} value has to be higher now before there are any (stable) non-trivial DFSSs with responsive individuals. The second non-trivial DFSS (s_{0}<0.5) is unstable for all values of the parameters, so we can disregard this one. The first non-trivial DFSS is stable under conditions which are similar to those required for the non-trivial DFSS in the original model. This steady state originally (\alpha _{s}=4d_{s}) has 1/2 of the individuals responsive, but as \alpha _{s} increases further the proportion of responsive individuals increases, toward the case where everybody is responsive. However the number of responsive individuals (1/2 for \alpha _{s}=4d_{s}) is always less than in the original model (3/4 for \alpha _{s}=4d_{s}).

These results are as expected. The rate of information transmission between individuals has been decreased, so the proportion of responsive individuals is always lower for a given \alpha _{s}. But as \alpha _{s} increases it's still possible to move toward a completely responsive population.

Also, as in the original model, whenever the stability conditions for the DFSSs are not satisfied we have an endemic steady state.

The numerical results in Figure 2 confirm the above results.

A l.
Figure 2 Three different model outcomes, (a) trivial DFSS (b) first non-trivial DFSS (c) endemic steady state, illustrated in terms of s_{{nr}} (red line), s_{r} (green line), i_{{nr}}+i_{r} (dashed line). Parameter values are \gamma _{{nr}}=1/26,\gamma _{r}=1/13,r=1,d_{s}=d_{i}=1/12,\delta _{s}=d_{s},\delta _{i}=d_{i},p=0.5,k=0.01 a) \beta _{{nr}}=0.7\gamma _{{nr}},\beta _{r}=0.5\beta _{{nr}},\alpha _{s}=2d_{s},\alpha _{i}=5d_{i}, b) \beta _{{nr}}=0.7\gamma _{{nr}},\beta _{r}=0.5\beta _{{nr}},\alpha _{s}=5d_{s},\alpha _{i}=5d_{i}, c) \beta _{{nr}}=3\gamma _{{nr}},\beta _{r}=0.5\beta _{{nr}},\alpha _{s}=2d_{s},\alpha _{i}=5d_{i}

Parameter sensitivity

The values \alpha _{s} and \alpha _{i} are still involved in the inequalities that determine the stable steady state. The values \delta _{s}, \delta _{i} and K (as well as p) are still not involved. So the rate of information transmission from individual to individual still affects which is the stable attractor, while the level of information given publicly doesn't. Both can still affect the level of infection in endemic steady states.

Even though individual to individual communication has less of an effect, it can still control the stable steady state, with the ability to eradicate an endemic. Public information distribution still cannot control this. This could be expected, since there was little change to the form of the system.

Discussion

In both the original and new models, the value of R={\frac  {d_{s}}{\alpha _{s}}} plays key role in the stability of the disease-free steady states. For the original model R=1 is the point where the stability of the DFSS changes from the non-trivial to the trivial DFSS. For the new model R=1/4 is the point where the stability of the DFSS changes from the non-trivial to the trivial DFSS. So, for both models, the value of R={\frac  {d_{s}}{\alpha _{s}}} is serving as a bifurcation parameter. Further investigations can be carried out on the existence of other bifurcation points.

Since \alpha _{s} (and \alpha _{i}) plays an important role in the existence and stabilities of steady states, we can see the importance of information dissemination via direct contact between individuals. This form of information transfer should be considered just as highly as population-wide dissemination when making choices about disease control. The importance of this is particularly relevant but not limited to cases like SARS in China, where initially no information was made available by the governing bodies[2]. Or AIDS in the United Kingdom, where informal information campaigns had a much greater effect than the wider population information campaigns that came later.

Further analysis could help us to determine just how the different forms of information transmission and the rate of decay of information over time affect disease dynamics.

Supplementary Information

For the Mathematica file associated with this project, which contains the code for the above plots, click here. For the wiki page summarizing the main ideas in the original model, click here.

References

  1. 1.0 1.1 Kiss, I. Z., Cassell, J., Recker, M., & Simon, P. L. (2010). The impact of information transmission on epidemic outbreaks. Mathematical biosciences, 225(1), 1–10. doi:10.1016/j.mbs.2009.11.009
  2. Sebastian Funk, Erez Gilad, Chris Watkins, and Vincent A. A. Jansen The spread of awareness and its impact on epidemic outbreaks PNAS 2009; published ahead of print March 30, 2009, doi:10.1073/pnas.0810762106