# Simulation of COVID-19 Spread in Nepal

A longstanding global pursuit to accurately predict the spread of infections disease across human population is currently undergoing a pivotal step. The recent outbreak of COVID-19 in Wuhan has triggered researchers and scientists alike to leapfrog into developing scientific tools to predict risks posed by the spread of this disease at an individual as well as community level. Simulations has historically been seen to be one of the most powerful tools to model disaster spread among people and between cities. In the next few months, we will see a high volume of activity within this domain of scientific exploration, where teams of researchers across borders will be racing to build a practical disease-spread simulation toolkit.

We are one such group of researchers cooperatively working towards building such a toolkit. Given the unprecedented lack of facilities poor countries face in terms of responding to a global crisis like this, we have focused in designing our tool in the context of one such country – Nepal. Our goal is to help the government of Nepal make informed policies which will potentially trigger intelligent responses at different levels against the spread of Coronavirus across the nation.

What started off as a fun project one night to code up a disaster spread model of [L. Buzna  et. al. Physical Review E 75, 056107 (2007)] ended up garnering interests from tens of people from my social network. One thing led to another, and now we are a team of national and international scientists and researchers with expertise in machine learning, big data analysis, chemical physics, quantum simulations, public health research and natural language processing working towards a common goal to fight the Coronavirus crisis.

## Mathematical Treatment

The disaster spread model is based on the research of aforementioned paper, and the central algorithm has been put into the simulation framework by myself in python. It consists of a graph with $N$ nodes $\{x_1 ... x_N\}$ and $R$ links which connect $x_i$  to $x_j$. The overall health of nodes (e.g. cities or villages) evolves in time. Initially, at time $t=0$, we assume all nodes are at state $x_i(0)=0$ . This represents the normal functioning of the nodes. This functioning is disturbed due to the interaction with neighboring nodes, resulting in $x_i(t) >0$, which signifies a diseased node. When the infection level of the node exceeds a certain threshold parameter, we consider it to fail. This threshold parameter can be tuned by changing $\sigma_i$. The time dynamics of the spread of disease can be modeled by the following equation,

$\frac{d x_i}{d t}=-\frac{x_i}{\tau_i} +\mathrm{Erf}\Big(\frac{1}{\sigma_i}\sum_{ij}\frac{M_{ji} x_j(t-t_{ji})}{f(O_j)}\mathrm{e}^{-\beta t_{ji}}\Big)$

The above equation is similar to the model of the paper but is not the same.  The first term in it represents the ability of a node to heal itself once it is diseased. However, the second term has been modified by introducing the Gauss error function, and represents the net affect towards a node. One can alternatively directly follow the model of the paper, and approximate the second part of the equation with a sigmoidal function  $\Theta(y)=\frac{1-\mathrm{e}^{-\alpha y}}{1+\mathrm{e}^{-\alpha (y-\theta_i)}}$, where $\theta_i$ is the threshold parameter and $\alpha$ is the gain parameter. Each node is assumed to self-heal in time, and does so exponentially with rate $1/\tau_i$. $M_{ji}$ is the flow volume of a link connecting $x_i$ to $x_j$, while $f(O_j) = (a O_j)/(1+b O_j)$, where $O_j$ is the out-degree of node $x_j$. Finally, we have parameter $t_{ji}$ which reports on the time-delay in disease-spread.

As an example, we consider the situation in which the outbreak of the Coronavirus first occurred in Kathmandu, the capital city of Nepal.

Fig 1. We assume that the only infected city at time zero is Kathmandu, which has been marked in brown. At this point, the disease level in this city has exceeded the threshold level, and is able to spread to other cities. Healthy nodes are indicated by purple blobs, and the dotted lines signify airways.

The solutions to the coupled differential equations yield the state of nodes at different times. The time dynamics of the health of the nodes is not trivial, and involves concerted influence from the neighboring nodes – the health of each of the nodes is changing in time depending on the health of its neighboring counterparts. There is a competition between the first term and the second term in the equation. When $\tau_i >0$, the first term brings node $x_i$ towards the healthier state. On the contrary,  as the disturbances from the neighboring nodes add up, $x_i$ increases, and becomes infected. Fig 2 shows a snapshot of the state of the health of the cities in Nepal at some later time during the evolution.

Fig 2.  At this point, the disease has spread to other cities in Nepal. The nodes in intermediate color signify cases of infection that are at the intermediate level (neither fully failed nor fully recovered).

## Future Work and Collaboration

It is said that a beautiful physical model is almost parameter-free in its formulation. This means that a solid physical expression yields a sensible output $f(x)$ for every sensible input $x$ without introducing or tuning a single parameter. Unfortunately,  when it comes to simulating human interactions in societies, parameter-free model is a luxury, and not everyone can afford it. Given the sheer complexity of  our problem, one relies on a number of approximations and large volume of data to find the best fitting parameters. We plan to achieve this through a collaborative effort maintained with people, institutions and companies from many walks of lives in Nepal and abroad. To this end, the major players and contributers in this humanitarian endeavor are the following,

NAAMII – This non-profit organization (led by Dr. Bishesh Khanal and Dr. Suresh Manandhar) has been vital in creating a nascent ecosystem of research and scientific collaboration in Nepal in the field of informatics, applied mathematics and artificial intelligence.

Risav Karna – A computer scientist by training, his research expertise and interests span across disciplines, from physics to software architecture.

Dr. Kiran Raj Pandey – A physician and researcher, he has taken the lead with regards to integrating the most prevalent disease spread model, SEIR, into the existing framework.

Nepali Telecom Service Provider – The largest telecom service providers in Nepal will take the lead in regards to providing large volume of human mobility data, which will be useful in estimating parameters such as $M_{ji}$

With this collaboration, the technical goal of our team is to scale the simulation by introducing more nodes and links in our system, and determine the best fitting parameters of our model using the available data. This will make our framework ever-more realistic, and allow for systematic quantification of the impact of human mobility in disease spreading. In addition, we want to hybridize our approach by combining other models such as SEIR into our toolkit. In the future, it might therefore be necessary to make use of high-performance computer resources that are maintained in servers at international companies such as IBM, which has recently called for a proposal to perform COVID19-related researches.

In the next few months, we aim to work tireless as a team to fight the Coronavirus crisis in Nepal through our simulation tool. We will design and incorporate intervention policies as well as mitigating strategies which shall be achieved through intelligent deployment of resources to the vulnerable cities. We have already had valuable responses from the government officials of Nepal and the United Nations about the possibility of further collaboration. This is very encouraging, and will only boost the morale of our team members. To learn more about our team, please click here.

## Call for Action

The following is the executive summary prepared by Risav Karna requesting the citizens of Nepal and government authorities to act responsibly by taking a collaborative approach to fight the Coronavirus crisis.

### Contact:

manish.thapa@phys.chem.ethz.ch