Fitting Parameters of ODE to Experimental Data SIR Model

Using observed data for the values S, I, and R, I am trying to solve for the parameters β and γ given by the simple SIR Model:

β and γ are represented by the parameter vector p and the values of S, I, and R are represented by vector y.

My ODE function to match the above equations is:

function dydt = odefun(t,y,p)
    dydt = zeros(3,1);
    p = zeros(2,1);
    dydt(1) = -p(1) * y(1) * y(2);
    dydt(2) = p(1) * y(1) * y(2) - p(2) * y(2);
    dydt(3) = p(2) * y(2);
end

My least-squares function is:

function err = odefit(exp_t,exp_y,p)
    [t,y] = ode45(@(t,y)odefun(t,y,p),exp_t,[1 1/5072000 0]); %initial conditions w/ 1 infected (i)
    err = sum((y-exp_y).^2);
end

Given observed values of y (exp_y) at times t (exp_t) [I've made these variables available at the end of the question if needed], I set:

exp_y = [s; i; r];
exp_t = 1:length(i);
p0 = [0.341 0.141]; %initial guess for beta and gamma, based on real world data

My estimate for the parameters is then:

p_estimate = fminsearch(@(p)odefit(exp_t,exp_y,p),p0);

I am getting a matrix dimensions must agree error in my odefit function.

My assumption is that I've made some kind of mistake in defining odefun, as I'm used to doing so with known constants instead of unknown constants. Any advice on how to proceed is appreciated, thanks!

----

My values for s, i, and r, if needed:

s = 0.999999802839117	0.999999408517350	0.999999605678233	0.999999408517350	0.999999605678233	0.999999605678233	0.999999408517350	0.999999408517350	0.999999211356467	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998028391167	0.999998028391167	0.999998028391167	0.999997634069401	0.999997634069401	0.999995859621451	0.999995859621451	0.999995859621451	0.999995465299685	0.999995465299685	0.999994873817035	0.999994479495268	0.999994479495268	0.999991522082019	0.999989353312303	0.999986395899054	0.999985015772871	0.999983044164038	0.999979889589905	0.999970031545741	0.999962145110410	0.999954850157729	0.999947358044164	0.999935528391167	0.999920149842271	0.999911080441640	0.999901025236593	0.999888801261830	0.999875000000000	0.999868690851735	0.999862184542587	0.999851537854890	0.999844045741325	0.999833004731861	0.999825512618297	0.999820189274448	0.999818809148265	0.999783517350158	0.999780954258675	0.999779179810726	0.999743296529968	0.999735607255521	0.999733635646688	0.999731861198738	0.999731072555205	0.999715102523659	0.999714905362776	0.999714511041010	0.999713328075710	0.999653194006309	0.999652405362776	0.999652405362776	0.999651419558360	0.999651419558360	0.999651025236593	0.999651025236593	0.999638209779180	0.999613170347003	0.999612578864353	0.999610804416404	0.999609227129338	0.999608241324921	0.999604692429022	0.999602326498423	0.999600946372240	0.999599960567823	0.999598974763407	0.999597791798107	0.999596805993691	0.999591876971609	0.999591876971609	0.999591482649842	0.999589116719243	0.999588722397476	0.999588525236593	0.999587736593060	0.999586553627760	0.999586356466877	0.999585764984227	0.999585567823344	0.999585173501577	0.999585173501577	0.999585173501577	0.999584582018927	0.999584384858044	0.999585962145110	0.999585567823344	0.999584187697161	0.999583990536278	0.999583990536278	0.999583990536278	0.999583793375394	0.999583793375394	0.999583399053628	0.999583596214511	0.999583596214511	0.999583004731861	0.999582807570978	0.999582610410095	0.999582413249211	0.999582216088328	0.999582216088328	0.999582018927445	0.999581821766562	0.999581624605678	0.999581624605678	0.999581624605678	0.999581624605678	0.999581230283912	0.999581230283912	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999580835962145	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580441640379	0.999580441640379	0.999580441640379	0.999580244479495	0.999580047318612	0.999579850157729	0.999579850157729	0.999579258675079	0.999579258675079	0.999579061514196	0.999579061514196	0.999579061514196	0.999579061514196	0.999579061514196	0.999578864353312	0.999578667192429	0.999578272870663	0.999578272870663	0.999577878548896	0.999577484227129	0.999577089905363	0.999577089905363	0.999577089905363	0.999577089905363	0.999577089905363
i = 1.97160883280757e-07	5.91482649842271e-07	3.94321766561514e-07	5.91482649842271e-07	3.94321766561514e-07	3.94321766561514e-07	5.91482649842271e-07	5.91482649842271e-07	7.88643533123028e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.97160883280757e-06	1.97160883280757e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.38012618296530e-06	1.38012618296530e-06	1.97160883280757e-06	2.36593059936909e-06	2.36593059936909e-06	5.32334384858044e-06	7.49211356466877e-06	1.04495268138801e-05	1.18296529968454e-05	1.38012618296530e-05	1.69558359621451e-05	2.68138801261830e-05	3.47003154574132e-05	4.19952681388013e-05	4.94873817034700e-05	6.13170347003155e-05	7.66955835962145e-05	8.57649842271293e-05	9.54258675078864e-05	0.000107649842271293	0.000121451104100946	0.000127365930599369	0.000133872239747634	0.000144518927444795	0.000152011041009464	0.000162263406940063	0.000169755520504732	0.000175078864353312	0.000176458990536278	0.000146687697160883	0.000149250788643533	0.000151025236593060	0.000119085173501577	0.000116127760252366	0.000117705047318612	0.000119479495268139	0.000120268138801262	0.000108635646687697	0.000108832807570978	0.000109227129337539	0.000110410094637224	5.34305993690852e-05	5.42192429022082e-05	5.42192429022082e-05	5.52050473186120e-05	5.52050473186120e-05	5.55993690851735e-05	5.55993690851735e-05	4.27839116719243e-05	1.93217665615142e-05	1.99132492113565e-05	1.85331230283912e-05	1.65615141955836e-05	1.55757097791798e-05	1.24211356466877e-05	1.04495268138801e-05	1.02523659305994e-05	1.12381703470032e-05	1.02523659305994e-05	9.85804416403786e-06	8.87223974763407e-06	3.94321766561514e-06	3.94321766561514e-06	3.54889589905363e-06	3.54889589905363e-06	3.54889589905363e-06	2.95741324921136e-06	3.35173501577287e-06	2.56309148264984e-06	2.36593059936909e-06	2.56309148264984e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.56309148264984e-06	2.76025236593060e-06	2.36593059936909e-06	2.36593059936909e-06	1.38012618296530e-06	1.18296529968454e-06	1.18296529968454e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.57728706624606e-06	9.85804416403786e-07	1.18296529968454e-06	9.85804416403786e-07	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.38012618296530e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	1.18296529968454e-06	1.18296529968454e-06	1.38012618296530e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.18296529968454e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06
r = 0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.77444794952681e-06	1.77444794952681e-06	1.77444794952681e-06	1.97160883280757e-06	1.97160883280757e-06	1.97160883280757e-06	1.97160883280757e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	3.48974763406940e-05	3.48974763406940e-05	3.48974763406940e-05	6.88091482649842e-05	7.41324921135647e-05	7.43296529968454e-05	7.43296529968454e-05	7.43296529968454e-05	8.81309148264984e-05	8.81309148264984e-05	8.81309148264984e-05	8.81309148264984e-05	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000159503154574132	0.000183753943217666	0.000183753943217666	0.000185331230283912	0.000187105678233438	0.000188091482649842	0.000191443217665615	0.000193611987381703	0.000194400630914827	0.000194400630914827	0.000195386435331230	0.000196175078864353	0.000197160883280757	0.000202089905362776	0.000202089905362776	0.000202484227129338	0.000203667192429022	0.000203864353312303	0.000204258675078864	0.000204455835962145	0.000205441640378549	0.000205638801261830	0.000205835962145110	0.000206033123028391	0.000206230283911672	0.000206230283911672	0.000206230283911672	0.000206427444794953	0.000206427444794953	0.000205835962145110	0.000206033123028391	0.000207216088328076	0.000207413249211356	0.000207413249211356	0.000207413249211356	0.000207610410094637	0.000207610410094637	0.000207807570977918	0.000207610410094637	0.000207413249211356	0.000208004731861199	0.000208004731861199	0.000208201892744480	0.000208201892744480	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208596214511041	0.000208596214511041	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000209187697160883	0.000209187697160883	0.000209187697160883	0.000209187697160883	0.000209384858044164	0.000209582018927445	0.000209582018927445	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209976340694006	0.000210173501577287	0.000210370662460568	0.000210370662460568	0.000210567823343849	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129

Answer 1

[Alan Stevens]

The following works, but you get a lousy fit because of the huge range (orders of magnitude) between s and i and r. The error differences in i and r make a negigible contribution to the fit.

s = [0.999999802839117	0.999999408517350	0.999999605678233	0.999999408517350	0.999999605678233	0.999999605678233	0.999999408517350	0.999999408517350	0.999999211356467	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998028391167	0.999998028391167	0.999998028391167	0.999997634069401	0.999997634069401	0.999995859621451	0.999995859621451	0.999995859621451	0.999995465299685	0.999995465299685	0.999994873817035	0.999994479495268	0.999994479495268	0.999991522082019	0.999989353312303	0.999986395899054	0.999985015772871	0.999983044164038	0.999979889589905	0.999970031545741	0.999962145110410	0.999954850157729	0.999947358044164	0.999935528391167	0.999920149842271	0.999911080441640	0.999901025236593	0.999888801261830	0.999875000000000	0.999868690851735	0.999862184542587	0.999851537854890	0.999844045741325	0.999833004731861	0.999825512618297	0.999820189274448	0.999818809148265	0.999783517350158	0.999780954258675	0.999779179810726	0.999743296529968	0.999735607255521	0.999733635646688	0.999731861198738	0.999731072555205	0.999715102523659	0.999714905362776	0.999714511041010	0.999713328075710	0.999653194006309	0.999652405362776	0.999652405362776	0.999651419558360	0.999651419558360	0.999651025236593	0.999651025236593	0.999638209779180	0.999613170347003	0.999612578864353	0.999610804416404	0.999609227129338	0.999608241324921	0.999604692429022	0.999602326498423	0.999600946372240	0.999599960567823	0.999598974763407	0.999597791798107	0.999596805993691	0.999591876971609	0.999591876971609	0.999591482649842	0.999589116719243	0.999588722397476	0.999588525236593	0.999587736593060	0.999586553627760	0.999586356466877	0.999585764984227	0.999585567823344	0.999585173501577	0.999585173501577	0.999585173501577	0.999584582018927	0.999584384858044	0.999585962145110	0.999585567823344	0.999584187697161	0.999583990536278	0.999583990536278	0.999583990536278	0.999583793375394	0.999583793375394	0.999583399053628	0.999583596214511	0.999583596214511	0.999583004731861	0.999582807570978	0.999582610410095	0.999582413249211	0.999582216088328	0.999582216088328	0.999582018927445	0.999581821766562	0.999581624605678	0.999581624605678	0.999581624605678	0.999581624605678	0.999581230283912	0.999581230283912	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999580835962145	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580441640379	0.999580441640379	0.999580441640379	0.999580244479495	0.999580047318612	0.999579850157729	0.999579850157729	0.999579258675079	0.999579258675079	0.999579061514196	0.999579061514196	0.999579061514196	0.999579061514196	0.999579061514196	0.999578864353312	0.999578667192429	0.999578272870663	0.999578272870663	0.999577878548896	0.999577484227129	0.999577089905363	0.999577089905363	0.999577089905363	0.999577089905363	0.999577089905363];
i = [1.97160883280757e-07	5.91482649842271e-07	3.94321766561514e-07	5.91482649842271e-07	3.94321766561514e-07	3.94321766561514e-07	5.91482649842271e-07	5.91482649842271e-07	7.88643533123028e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.97160883280757e-06	1.97160883280757e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.38012618296530e-06	1.38012618296530e-06	1.97160883280757e-06	2.36593059936909e-06	2.36593059936909e-06	5.32334384858044e-06	7.49211356466877e-06	1.04495268138801e-05	1.18296529968454e-05	1.38012618296530e-05	1.69558359621451e-05	2.68138801261830e-05	3.47003154574132e-05	4.19952681388013e-05	4.94873817034700e-05	6.13170347003155e-05	7.66955835962145e-05	8.57649842271293e-05	9.54258675078864e-05	0.000107649842271293	0.000121451104100946	0.000127365930599369	0.000133872239747634	0.000144518927444795	0.000152011041009464	0.000162263406940063	0.000169755520504732	0.000175078864353312	0.000176458990536278	0.000146687697160883	0.000149250788643533	0.000151025236593060	0.000119085173501577	0.000116127760252366	0.000117705047318612	0.000119479495268139	0.000120268138801262	0.000108635646687697	0.000108832807570978	0.000109227129337539	0.000110410094637224	5.34305993690852e-05	5.42192429022082e-05	5.42192429022082e-05	5.52050473186120e-05	5.52050473186120e-05	5.55993690851735e-05	5.55993690851735e-05	4.27839116719243e-05	1.93217665615142e-05	1.99132492113565e-05	1.85331230283912e-05	1.65615141955836e-05	1.55757097791798e-05	1.24211356466877e-05	1.04495268138801e-05	1.02523659305994e-05	1.12381703470032e-05	1.02523659305994e-05	9.85804416403786e-06	8.87223974763407e-06	3.94321766561514e-06	3.94321766561514e-06	3.54889589905363e-06	3.54889589905363e-06	3.54889589905363e-06	2.95741324921136e-06	3.35173501577287e-06	2.56309148264984e-06	2.36593059936909e-06	2.56309148264984e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.56309148264984e-06	2.76025236593060e-06	2.36593059936909e-06	2.36593059936909e-06	1.38012618296530e-06	1.18296529968454e-06	1.18296529968454e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.57728706624606e-06	9.85804416403786e-07	1.18296529968454e-06	9.85804416403786e-07	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.38012618296530e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	1.18296529968454e-06	1.18296529968454e-06	1.38012618296530e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.18296529968454e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06];
r = [0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.77444794952681e-06	1.77444794952681e-06	1.77444794952681e-06	1.97160883280757e-06	1.97160883280757e-06	1.97160883280757e-06	1.97160883280757e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	3.48974763406940e-05	3.48974763406940e-05	3.48974763406940e-05	6.88091482649842e-05	7.41324921135647e-05	7.43296529968454e-05	7.43296529968454e-05	7.43296529968454e-05	8.81309148264984e-05	8.81309148264984e-05	8.81309148264984e-05	8.81309148264984e-05	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000159503154574132	0.000183753943217666	0.000183753943217666	0.000185331230283912	0.000187105678233438	0.000188091482649842	0.000191443217665615	0.000193611987381703	0.000194400630914827	0.000194400630914827	0.000195386435331230	0.000196175078864353	0.000197160883280757	0.000202089905362776	0.000202089905362776	0.000202484227129338	0.000203667192429022	0.000203864353312303	0.000204258675078864	0.000204455835962145	0.000205441640378549	0.000205638801261830	0.000205835962145110	0.000206033123028391	0.000206230283911672	0.000206230283911672	0.000206230283911672	0.000206427444794953	0.000206427444794953	0.000205835962145110	0.000206033123028391	0.000207216088328076	0.000207413249211356	0.000207413249211356	0.000207413249211356	0.000207610410094637	0.000207610410094637	0.000207807570977918	0.000207610410094637	0.000207413249211356	0.000208004731861199	0.000208004731861199	0.000208201892744480	0.000208201892744480	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208596214511041	0.000208596214511041	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000209187697160883	0.000209187697160883	0.000209187697160883	0.000209187697160883	0.000209384858044164	0.000209582018927445	0.000209582018927445	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209976340694006	0.000210173501577287	0.000210370662460568	0.000210370662460568	0.000210567823343849	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129];

exp_y = [s(:); i(:); r(:)];
exp_t = 1:length(i);
p0 = [0.341 0.141]; %initial guess for beta and gamma, based on real world data

p_estimate = fminsearch(@(p)odefit(exp_t,exp_y,p),p0);

[~,Y] = ode45(@(t,y)odefun(t,y,p_estimate),exp_t,[1 1/5072000 0]);

S = Y(:,1); I = Y(:,2); R = Y(:,3);
plot(exp_t,i,'o',exp_t,r,'*',exp_t,I,exp_t,R)
legend('i','r','I','R')
figure
plot(exp_t,s,'o',exp_t,S)
legend('S')


function err = odefit(exp_t,exp_y,p)
    [~,y] = ode45(@(t,y)odefun(t,y,p),exp_t,[1 1/5072000 0]); %initial conditions w/ 1 infected (i)
    err = sum((y(:) - exp_y).^2);   % This gives a lousy fit!
end

function dydt = odefun(~,y,p)
         
         s = y(1);
         i = y(2);
        %r = y(3);
       
    dydt = [-p(1) * s * i;
             p(1) * s * i - p(2) * i;
             p(2) * i];
    
end

[Alan Stevens]

On reflection, I think the poor fit means the SIR model doesn't represent the phenomenon very well! In particular, it assumes recovery begins immediately, though the data clearly sshows a delay. The following code attempts to estimate the delay, though the fit is still not wonderful!

s = [0.999999802839117	0.999999408517350	0.999999605678233	0.999999408517350	0.999999605678233	0.999999605678233	0.999999408517350	0.999999408517350	0.999999211356467	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999999014195584	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998817034700	0.999998028391167	0.999998028391167	0.999998028391167	0.999997634069401	0.999997634069401	0.999995859621451	0.999995859621451	0.999995859621451	0.999995465299685	0.999995465299685	0.999994873817035	0.999994479495268	0.999994479495268	0.999991522082019	0.999989353312303	0.999986395899054	0.999985015772871	0.999983044164038	0.999979889589905	0.999970031545741	0.999962145110410	0.999954850157729	0.999947358044164	0.999935528391167	0.999920149842271	0.999911080441640	0.999901025236593	0.999888801261830	0.999875000000000	0.999868690851735	0.999862184542587	0.999851537854890	0.999844045741325	0.999833004731861	0.999825512618297	0.999820189274448	0.999818809148265	0.999783517350158	0.999780954258675	0.999779179810726	0.999743296529968	0.999735607255521	0.999733635646688	0.999731861198738	0.999731072555205	0.999715102523659	0.999714905362776	0.999714511041010	0.999713328075710	0.999653194006309	0.999652405362776	0.999652405362776	0.999651419558360	0.999651419558360	0.999651025236593	0.999651025236593	0.999638209779180	0.999613170347003	0.999612578864353	0.999610804416404	0.999609227129338	0.999608241324921	0.999604692429022	0.999602326498423	0.999600946372240	0.999599960567823	0.999598974763407	0.999597791798107	0.999596805993691	0.999591876971609	0.999591876971609	0.999591482649842	0.999589116719243	0.999588722397476	0.999588525236593	0.999587736593060	0.999586553627760	0.999586356466877	0.999585764984227	0.999585567823344	0.999585173501577	0.999585173501577	0.999585173501577	0.999584582018927	0.999584384858044	0.999585962145110	0.999585567823344	0.999584187697161	0.999583990536278	0.999583990536278	0.999583990536278	0.999583793375394	0.999583793375394	0.999583399053628	0.999583596214511	0.999583596214511	0.999583004731861	0.999582807570978	0.999582610410095	0.999582413249211	0.999582216088328	0.999582216088328	0.999582018927445	0.999581821766562	0.999581624605678	0.999581624605678	0.999581624605678	0.999581624605678	0.999581230283912	0.999581230283912	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999581033123028	0.999580835962145	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580638801262	0.999580441640379	0.999580441640379	0.999580441640379	0.999580244479495	0.999580047318612	0.999579850157729	0.999579850157729	0.999579258675079	0.999579258675079	0.999579061514196	0.999579061514196	0.999579061514196	0.999579061514196	0.999579061514196	0.999578864353312	0.999578667192429	0.999578272870663	0.999578272870663	0.999577878548896	0.999577484227129	0.999577089905363	0.999577089905363	0.999577089905363	0.999577089905363	0.999577089905363];
i = [1.97160883280757e-07	5.91482649842271e-07	3.94321766561514e-07	5.91482649842271e-07	3.94321766561514e-07	3.94321766561514e-07	5.91482649842271e-07	5.91482649842271e-07	7.88643533123028e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	7.88643533123028e-07	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.97160883280757e-06	1.97160883280757e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.38012618296530e-06	1.38012618296530e-06	1.97160883280757e-06	2.36593059936909e-06	2.36593059936909e-06	5.32334384858044e-06	7.49211356466877e-06	1.04495268138801e-05	1.18296529968454e-05	1.38012618296530e-05	1.69558359621451e-05	2.68138801261830e-05	3.47003154574132e-05	4.19952681388013e-05	4.94873817034700e-05	6.13170347003155e-05	7.66955835962145e-05	8.57649842271293e-05	9.54258675078864e-05	0.000107649842271293	0.000121451104100946	0.000127365930599369	0.000133872239747634	0.000144518927444795	0.000152011041009464	0.000162263406940063	0.000169755520504732	0.000175078864353312	0.000176458990536278	0.000146687697160883	0.000149250788643533	0.000151025236593060	0.000119085173501577	0.000116127760252366	0.000117705047318612	0.000119479495268139	0.000120268138801262	0.000108635646687697	0.000108832807570978	0.000109227129337539	0.000110410094637224	5.34305993690852e-05	5.42192429022082e-05	5.42192429022082e-05	5.52050473186120e-05	5.52050473186120e-05	5.55993690851735e-05	5.55993690851735e-05	4.27839116719243e-05	1.93217665615142e-05	1.99132492113565e-05	1.85331230283912e-05	1.65615141955836e-05	1.55757097791798e-05	1.24211356466877e-05	1.04495268138801e-05	1.02523659305994e-05	1.12381703470032e-05	1.02523659305994e-05	9.85804416403786e-06	8.87223974763407e-06	3.94321766561514e-06	3.94321766561514e-06	3.54889589905363e-06	3.54889589905363e-06	3.54889589905363e-06	2.95741324921136e-06	3.35173501577287e-06	2.56309148264984e-06	2.36593059936909e-06	2.56309148264984e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.56309148264984e-06	2.76025236593060e-06	2.36593059936909e-06	2.36593059936909e-06	1.38012618296530e-06	1.18296529968454e-06	1.18296529968454e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.57728706624606e-06	9.85804416403786e-07	1.18296529968454e-06	9.85804416403786e-07	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.38012618296530e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	1.18296529968454e-06	1.18296529968454e-06	1.38012618296530e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	1.18296529968454e-06	1.18296529968454e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.18296529968454e-06	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	9.85804416403786e-07	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06	1.38012618296530e-06];
r = [0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.97160883280757e-07	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.57728706624606e-06	1.77444794952681e-06	1.77444794952681e-06	1.77444794952681e-06	1.97160883280757e-06	1.97160883280757e-06	1.97160883280757e-06	1.97160883280757e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	2.36593059936909e-06	3.48974763406940e-05	3.48974763406940e-05	3.48974763406940e-05	6.88091482649842e-05	7.41324921135647e-05	7.43296529968454e-05	7.43296529968454e-05	7.43296529968454e-05	8.81309148264984e-05	8.81309148264984e-05	8.81309148264984e-05	8.81309148264984e-05	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000146687697160883	0.000159503154574132	0.000183753943217666	0.000183753943217666	0.000185331230283912	0.000187105678233438	0.000188091482649842	0.000191443217665615	0.000193611987381703	0.000194400630914827	0.000194400630914827	0.000195386435331230	0.000196175078864353	0.000197160883280757	0.000202089905362776	0.000202089905362776	0.000202484227129338	0.000203667192429022	0.000203864353312303	0.000204258675078864	0.000204455835962145	0.000205441640378549	0.000205638801261830	0.000205835962145110	0.000206033123028391	0.000206230283911672	0.000206230283911672	0.000206230283911672	0.000206427444794953	0.000206427444794953	0.000205835962145110	0.000206033123028391	0.000207216088328076	0.000207413249211356	0.000207413249211356	0.000207413249211356	0.000207610410094637	0.000207610410094637	0.000207807570977918	0.000207610410094637	0.000207413249211356	0.000208004731861199	0.000208004731861199	0.000208201892744480	0.000208201892744480	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208399053627760	0.000208596214511041	0.000208596214511041	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208793375394322	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000208990536277603	0.000209187697160883	0.000209187697160883	0.000209187697160883	0.000209187697160883	0.000209384858044164	0.000209582018927445	0.000209582018927445	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209779179810726	0.000209976340694006	0.000210173501577287	0.000210370662460568	0.000210370662460568	0.000210567823343849	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129	0.000210764984227129];

exp_y = [s(:); i(:); r(:)];
exp_t = 1:length(i);
p0 = [0.2 0.2 60]; %initial guess for beta, gamma and the delay

p_estimate = fminsearch(@(p)odefit(exp_t,exp_y,p),p0);

[~,Y] = ode45(@(t,y)odefun(t,y,p_estimate),exp_t,[1 1/5072000 0]);

S = Y(:,1); I = Y(:,2); R = Y(:,3);
plot(exp_t,i,'o',exp_t,r,'*',exp_t,I,exp_t,R)
legend('i','r','I','R')
figure
plot(exp_t,s,'o',exp_t,S)
legend('S')


function err = odefit(exp_t,exp_y,p)
    [~,y] = ode45(@(t,y)odefun(t,y,p),exp_t,[1 1/5072000 0]); %initial conditions w/ 1 infected (i)
    err = sum((y(:) - exp_y).^2);   
end

function dydt = odefun(t,y,p)
         
        s = y(1);
        i = y(2);
       % r = y(3);
       beta = p(1); gamma = p(2); deltat = p(3);
       if t<deltat
          gamma = 0;
       end
       
       
    dydt = [-beta * s * i;
             beta * s * i - gamma * i;
             gamma * i];
    
end

[Shawn Cooper]

Thank you for the help and the observations. They were both really useful!

[Faihatuz Zuhairoh]

I changed the above step with the SVIRS model but it became an error.

Error section:

p_estimate = fminsearch (@(p) odefit (exp_t, exp_y, p), p0);

Please the solution

[Alan Stevens]

?! You need to upload your code and specify more precisely exactly what you've done.

[Santanu Bhattacharya]

Hi Alan, I have a similar issue regarding parameter estimation of Covid 19 SIRV model. Can you help me please...

[Alan Stevens]

You will need to upload the code you've used so far and explain your problem.